# Decay rates of convergence for Fokker-Planck equations with confining drift

Alessio Porretta Dipartimento di Matematica, Università di Roma Tor Vergata.
Via della Ricerca Scientifica 1, 00133 Roma, Italy. Email:
<EMAIL_ADDRESS>The author is member of GNAMPA research group of
Indam.

###### Abstract

We consider Fokker-Planck equations in the whole Euclidean space, driven by
Levy processes, under the action of confining drifts, as in the classical
Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or
sub-exponential decay rates, as time goes to infinity, of zero average
solutions, under some diffusivity condition on the Levy process, which
includes the fractional Laplace operator as a model example. Our approach
relies on the long time oscillation estimates of the adjoint problem and
applies to (the possible superposition of) both local and nonlocal diffusions,
as well as to strongly or weakly confining drifts. Our results extend, with a
unifying perspective, many previous works based on different analytic or
probabilistic methods, with several interesting connections. On one hand, we
make a link between the (nonlinear) PDE methods used for the long time
behavior of Hamilton-Jacobi equations and the decay estimates of Fokker-Planck
equations; on another hand, we give a purely analytical approach towards some
oscillation decay estimates which were obtained so far only with probabilistic
coupling methods.

##  1 Introduction

This article is concerned with Fokker-Planck equations in ${\mathbb{R}}^{d}$
and with the issue of time-decay rates of zero average solutions, which
results as effect of the presence of a confining drift. To fix the ideas, the
model problem that we will discuss is the following

$\begin{cases}m_{t}+\left(-\Delta\right)^{\frac{\sigma}{2}}m-{\rm
div}(b(t,x)m)=0&\text{ in }(0,T)\times{\mathbb{R}}^{d},\\\
m(0,x)=m_{0}(x),\qquad\,&\text{ in }{\mathbb{R}}^{d}\end{cases}$ (1.1)

where $\left(-\Delta\right)^{\frac{\sigma}{2}}m$ is the usual Laplace operator
for $\sigma=2$ or the fractional Laplacian for $\sigma\in(0,2)$.

If $b(t,x)=b(x)$, the decay of zero average solutions is equivalent, in
particular, to the problem of long time convergence of unit mass solutions
towards stationary invariant measures. This is a fundamental question in the
study of Fokker-Planck equations, which attracted a huge interest all over the
years. As is well-known, the confining property of the drift plays a crucial
role in this convergence, being related to the ergodicity property of the
underlying stochastic process. The typical result which is proved in this
context, for $m_{0}$ with zero average, is that

$\|m(t)\|_{X}\leq\varpi(t)\|m_{0}\|_{X}\,,\qquad\forall t>0\,,\quad\hbox{with
$\varpi(t)\mathop{\to}\limits^{t\to\infty}0$,}$ (1.2)

where $\varpi(t)$ decays exponentially if the confinement force is
sufficiently strong ($b\cdot x\gtrsim|x|^{2}$ at infinity), or it possibly
decays sub-exponentially if $b\cdot x\gtrsim|x|^{\gamma}$ with
$\gamma\in(0,2)$. The space $X$ is typically a $L^{p}$ weighted space, with
$p=1$ being the most natural choice, in a way that (1.2) implies decay of the
total mass.

It is impossible here to mention all relevant contributions in the literature,
where an estimate such as (1.2) has been proved. Let us just point out two
main approaches that were developed so far for this kind of analysis.

Starting from the case of Laplace operator and drift $b=\nabla V(x)$, a first
approach relies on functional analytic methods, exploiting spectral gap
results in $L^{2}$ (weighted) spaces, logarithmic Sobolev (and Poincaré)
inequalities and, more generally, entropy-energy methods. We refer the reader
to many influential papers such as [2], [26], [33], [38], [39]. Further
refinements of spectral gap methods, relying on ideas in semigroup theory,
were developed more recently, see e.g. [25], [30]. Even if the above results
are mostly related to local operators, methods based on entropy decay or
spectral gaps were also used for nonlocal diffusions, see [11], [24], [40].

A second approach, which is different in spirit, stands on ideas and
probabilistic methods developed in the study of recurrence and decay
properties of Markov processes. This is often referred to as the Meyn &
Tweedie approach (see e.g. [21], [36]) and this is intrinsically related to
Harris’ ergodic theorem on Markov chains, as presented in the enlightening
version of Hairer and Mattingly [27]. The main tools in this approach are the
existence of a Lyapunov function at infinity and the (local) strict positivity
of the operator, see e.g. [18], [12]. Extensions to nonlocal diffusions were
recently developed in [31]. We also address the reader to the influential
paper [3] for a comparison between the different approaches mentioned above.
Finally, it is to mention that similar estimates can be also deduced with a
probabilistic approach from the study of decay rates of the transition
probabilities associated to the process, see e.g. [19], [20], [35]. We will
comment more about that, in what follows.

The purpose of this article is to introduce a new method for proving decay
estimates such as (1.2), for solutions of (1.1). This is an entirely PDE
method, relatively simple, which applies to a variety of different regimes,
ranging from local to nonlocal diffusions and including both geometrical and
subgeometrical rates of convergence.

The motivation of our work comes from the study of the long time behavior of
mean-field game systems [22]; those are systems of PDEs coupling Fokker-Planck
and Hamilton-Jacobi equations, which describe Nash equilibria in large
populations of interacting agents (see [32], [28], [13]). In that context, the
Fokker-Planck equation is driven by the optimal strategies chosen by the
agents, which results in the fact that the drift term in (1.1) depends on the
solution of the coupled Hamilton-Jacobi equation. With this in mind, the goal
is to obtain decay estimates for Fokker-Planck equations under hopefully cheap
conditions on the drift term $b(t,x)$; in particular, our results apply to
time-dependent fields $b$ and avoid any direct assumption on ${\rm div}(b)$.

Before explaining the novelty of our approach, let us give a sample of the
kind of results that we prove, when specialized to the model problem (1.1). We
start with the case of purely local diffusion, where the results are slightly
more general.

###### Theorem 1.1

Let us set $\langle x\rangle:=\sqrt{1+|x|^{2}}$. For $k>0$, assume that
$m_{0}\in L^{1}({\mathbb{R}}^{d},\langle x\rangle^{k}\,dx)$ be such that
${\int_{{\mathbb{R}}^{d}}}m_{0}\,dx=0$. Let $m$ be a weak solution to the
Cauchy problem

$\begin{cases}m_{t}-\Delta m-{\rm div}(b(t,x)m)=0&\text{ in
}(0,T)\times{\mathbb{R}}^{d},\\\ m(0,x)=m_{0}(x),\qquad\,&\text{ in
}{\mathbb{R}}^{d}\end{cases}$ (1.3)

where we assume that $b(t,x)$ is a locally bounded function satisfying

$\exists\,\,\alpha\,,\,R>0\,:\,\quad b(t,x)\cdot
x\geq\alpha\,|x|^{\gamma}\qquad\forall x\in{\mathbb{R}}^{d}:\,|x|\geq
R\,,\,\forall t>0\,,$ (1.4)

and

$\exists\,\,c_{0}>0\,:\,\quad(b(t,x)-b(t,y))\cdot(x-y)\geq-
c_{0}|x-y|\qquad\forall x,y\in{\mathbb{R}}^{d}\,,\,\forall t>0\,.$ (1.5)

Then we have:

  * (a)

If $\gamma\geq 2$ in (1.4), then

$\|m(t)\|_{L^{1}(\langle x\rangle^{k})}\leq K\,e^{-\omega
t}\,\|m_{0}\|_{L^{1}({\langle x\rangle^{k}})}$

for some constant $K,\omega$ only depending on $\alpha,c_{0},k,d$.

  * (b)

If $\gamma\in(0,2)$ in (1.4), then for any $\bar{k}>k$ such that $m_{0}\in
L^{1}({\mathbb{R}}^{d},\langle x\rangle^{\bar{k}}\,dx)$ we have

$\|m(t)\|_{L^{1}(\langle x\rangle^{k})}\leq
K\,(1+t)^{-q}\,\|m_{0}\|_{L^{1}({\langle
x\rangle^{\bar{k}}})}\qquad\hbox{where $q=\frac{\bar{k}-k}{2-\gamma}$.}$

for some $K$ depending on $\alpha,\gamma,c_{0},k,\bar{k},d$.

Theorem 1.1 recovers, with a unifying approach, the results which are known in
the literature, including the case of slow, or degenerate, confining drift,
which was studied previously both with the entropy method by Toscani and
Villani [39] and with more sophisticated tools of semigroup theory by Kavian,
Mischler and Dao [30]. We stress however that we avoid any $C^{1}$ regularity
of the drift $b$ and any requirement on ${\rm div}(b)$ or $Db$; the one-sided
condition (1.5) plays somehow a similar role, although it appears to be a
milder requirement which, for instance, allows for any (space-time) bounded
perturbation of dissipative vector fields. We also mention that, at the
expense of a small restriction on the parameter $k$ in the weight function,
the previous result holds even replacing (1.5) with more general conditions,
e.g. assuming $(b(t,x)-b(t,y))\cdot(x-y)\geq-c_{0}(|x-y|\vee|x-y|^{2})$, see
Remark 3.7.

Whilst Theorem 1.1 only gives a sample of the results that we prove, the main
outcome of our approach appears in the extension of this kind of estimates to
general Levy processes, including possibly nonlocal diffusions. This is the
model result that we prove for the specific example of the fractional
Laplacian, to be compared with [31, Thm 1.4].

###### Theorem 1.2

Let $\sigma\in(0,2)$ and $k\in(0,\sigma)$. Assume that $m_{0}\in
L^{1}({\mathbb{R}}^{d},\langle x\rangle^{k}\,dx)$ be such that
${\int_{{\mathbb{R}}^{d}}}m_{0}\,dx=0$. Let $m$ be a weak solution to the
Cauchy problem (1.1), where we assume that $b$ is a locally Lipschitz vector
field satisfying (1.4) and either of the following conditions: (i)
$\sigma\in(1,2)$ and (1.5) holds true. (ii) $\sigma\in(0,1]$ and
$(b(t,x)-b(t,y))\cdot(x-y)\geq-c_{0}|x-y|(|x-y|\wedge 1)^{1-\sigma+\delta}$,
for some $\delta>0$, $c_{0}>0$. Then we have:

  * (a)

If $\gamma\geq 2$ in (1.4), then

$\|m(t)\|_{L^{1}(\langle x\rangle^{k})}\leq K\,e^{-\omega
t}\,\|m_{0}\|_{L^{1}({\langle x\rangle^{k}})}$

for some constant $K,\omega$ only depending on
$\alpha,\sigma,\delta,c_{0},k,d$.

  * (b)

If $\gamma\in(((2-\sigma)\vee 1),2)$ in (1.4), then for any $k<\bar{k}<\sigma$
such that $m_{0}\in L^{1}({\mathbb{R}}^{d},\langle x\rangle^{\bar{k}}\,dx)$ we
have

$\|m(t)\|_{L^{1}(\langle x\rangle^{k})}\leq
K\,(1+t)^{-q}\,\|m_{0}\|_{L^{1}({\langle
x\rangle^{\bar{k}}})}\qquad\hbox{where $q=\frac{\bar{k}-k}{2-\gamma}$.}$

for some $K$ depending on $\alpha,\gamma,\sigma,\delta,c_{0},k,\bar{k},d$.

Let us stress that the local Lipschitz character of the drift, which is
assumed in the above statement, was only used to make sure that $m(t)\in
L^{1}$. More general statements, which extend Theorem 1.2, will be given in
Section 4 for Levy operators: we refer the reader to Theorem 4.2, in Section
4. In that case, the decay is formulated in the total variation norm for
measure-valued solutions. Correspondingly, we will adopt a suitable
formulation of solution which can be applied to a far more general setting; as
an example, one can consider the case of fractional Laplacian with additional
second order degenerate diffusions.

Compared with the existing literature, our strategy is certainly close to (and
even inspired from) the probabilistic approach mentioned above, although we
develop a purely PDE method which has an interest in its own. There are two
main cornerstones in the approach that we suggest:

(i) the decay estimates for the Fokker-Planck equation (1.1) are entirely
deduced by duality from oscillation decay estimates obtained on the solutions
of the adjoint problem

$\begin{cases}u_{t}+\left(-\Delta\right)^{\frac{\sigma}{2}}u+b(t,x)\cdot
Du=0&\text{ in }(0,T)\times{\mathbb{R}}^{d},\\\
u(0,x)=u_{0}(x),\qquad\,&\text{ in }{\mathbb{R}}^{d}\,.\end{cases}$ (1.6)

(ii) for solutions of (1.6), we prove decay estimates which are dual to
estimates of the type (1.2). Namely, we show that

$\begin{split}&\qquad[u(t)]_{\langle
x\rangle^{k}}\leq\varpi(t)\,[u_{0}]_{\langle x\rangle^{k}}\qquad\hbox{with
$\varpi(t)\mathop{\to}\limits^{t\to\infty}0$,}\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr&\hbox{where $[u]_{\langle
x\rangle^{k}}=\sup\limits_{x,y\in{\mathbb{R}}^{2d}}\,\,\frac{|u(x)-u(y)|}{\langle
x\rangle^{k}+\langle y\rangle^{k}}$.}\end{split}$ (1.7)

Estimates such as (1.7) are an evidence of the ergodicity of the underlying
process. In the PDE perspective, global (in time) bounds for the oscillation
of solutions are usually exploited in the study of long time behavior of
Hamilton-Jacobi equations and the convergence to the ergodic problem (see e.g.
[10], [6], [9], [14]). Those results typically stand on either classical
gradient estimates (through the celebrated Bernstein method) or doubling
variables techniques developed in the theory of viscosity solutions,
specifically by Ishii and Lions [29] when coming at oscillation estimates.
This kind of methods was later extended to nonlocal operators in [5]. However,
quantitative estimates like (1.7) are new in this framework, compared to those
earlier works. Here, our key idea is to obtain (1.7) by proving that

$u(t,x)-u(t,y)\leq\varpi(t)\,[u_{0}]_{\langle x\rangle^{k}}\big{[}\langle
x\rangle^{k}+\langle y\rangle^{k}+\psi(|x-y|)\big{]}$

for some bounded function $\psi(r)$ which is locally Hölder near $r=0$. Notice
that this estimate combines the effects of the Lyapunov function at infinity
(used to control long-range interactions) with the local ellipticity used to
control short-range interactions, when $|x-y|$ is small.

It seems remarkable that the decay of Fokker-Planck equations is deduced from
this single estimate on the dual problem (1.6). This is the main difference of
our method compared to the Meyn-Tweedy, or Harris-type approach, where the
local ellipticity is exploited directly in Harnack-type properties of the
Fokker-Planck equation (or strong positivity, say irreducibility conditions,
on the associated semigroup). Otherwise, estimates as (1.7) can be found in
the probabilistic literature in the form of contraction estimates for
transition probabilities in Wasserstein’s metrics, see e.g. [19], [20], [35],
[41]. In that context, the proofs rely on probabilistic tools based on
coupling methods ([15], [34]). From this perspective, our proof can be seen as
a PDE method which is alternative to the probabilistic approach.

Let us mention that the idea of an analytic version of the coupling method was
already suggested in [37] for local diffusions. On that occasion, we explained
that the doubling variables approach developed in the theory of viscosity
solutions was parallel, and conceptually equivalent, to the coupling method
used in probability. In the present article, we come back on this viewpoint
and introduce coupling ideas to get a crucial improvement in the doubling
variable method which is used for Levy operators. This allows us to give
entirely PDE proofs of some contraction estimates that were only proved with
probabilistic methods so far, with the advantage of relaxing the assumptions
and making them ready to several possible extensions.

In the present paper, in order not to increase the level of technicalities
which could be induced by generalizations, we confine ourselves to the study
of Levy processes, which contain both local diffusion and jump processes,
giving evidence of the generality of the method.

We also show how the suggested strategy can be adapted to different Lyapunov
functions in order to obtain exponential or sub-exponential rates of
convergence. We refer the reader to Theorem 5.5 for a general version of our
results, which is not limited to power-type Lyapunov functions. In particular,
we show how different decay rates are proved for the weighted norm
$\|m(t)\|_{{\mathcal{M}}_{\varphi}}$ (the total variation of $m(t)\,\varphi$)
according to whether the Lyapunov function $\varphi$ satisfies stronger, or
weaker, conditions of super-solution. This way, Theorem 5.5 will provide with
a general recipe which may please the readers inclined towards statements in a
more abstract form. As an example of application of similar generalizations,
we will obtain exponential convergence of solutions to (1.3) even for slowly
confining drifts, up to using stronger Lyapunov weights rather than powers: we
refer to Corollary 5.6, which extends recent results obtained in [30].

To conclude, let us briefly mention the organization of the paper. In Section
2, we introduce the notation and the assumptions that will be used, as well as
the notions of solutions for the evolution problems to be considered. Section
3 is devoted to the oscillation decay estimates for problems like (1.6); this
is where we use doubling variables methods (with coupling ideas embedded) to
derive estimates as (1.7), with power-type weight functions. In Section 4, we
deduce by duality the decay of Fokker-Planck equations, which generalize
Theorem 1.1 and 1.2. Section 5 is devoted to a more general version of the
results, where we consider Lyapunov functions of general type, and we give
exponential or sub-exponential decay rates accordingly. We also mention the
application of our results to the convergence of unit mass solutions towards
stationary measures. Finally, in the Appendix, Section 6, we give the proof of
the existence and uniqueness of solutions for the Fokker-Planck equation,
under the (non standard) setting of assumptions which is used in the paper.

##  2 Notation, assumptions and preliminary tools

We denote by ${\mathbb{R}}^{d}$, $d\geq 1$, the $d$-dimensional Euclidean
space, with Lebesgue measure $dx$. $I_{d}$ denotes the identity matrix in
${\mathbb{R}}^{d}$. Given two real numbers $a,b$, we use the standard notation
$a\vee b=\max(a,b)$ and $a\wedge b=\min(a,b)$. For a positive continuous
function $\varphi(x)$, we denote by $L^{1}(\varphi(x))$ the usual Lebesgue
space $L^{1}({\mathbb{R}}^{d},\mu)$ defined in terms of the measure
$\mu=\varphi(x)dx$, and by $L^{\infty}(\varphi(x))$ the space of measurable
functions $u:{\mathbb{R}}^{d}\to{\mathbb{R}}$ such that $u\,\varphi$ is
essentially bounded (i.e. $u\varphi\in L^{\infty}({\mathbb{R}}^{d})$).

We will use systematically the weight function

$\langle x\rangle:=\sqrt{1+|x|^{2}}\,$

replacing the role of $|x|$. Notice that $D\langle x\rangle=\frac{x}{\langle
x\rangle}$.

The space ${\mathcal{M}}({\mathbb{R}}^{d})$ denotes the space of finite
(signed) Borel measures $m$ endowed with the total variation norm
$\|m\|_{TV}=|m|({\mathbb{R}}^{d})$; here $|m|=m^{+}-m^{-}$, where $m^{\pm}$
are positive measures giving the Hahn decomposition of $m$. The set
${\mathcal{P}}({\mathbb{R}}^{d})$ denotes the space of probability measures,
where we consider the Kantorovich-Rubinstein distance

$d_{1}(m,\tilde{m}):=\sup\\{{\int_{{\mathbb{R}}^{d}}}\varphi\,d(m-\tilde{m})\,,\,\,\varphi\,\hbox{Lipschitz,}\,\,\|\varphi\|_{\infty},\|D\varphi\|_{\infty}\leq
1\\}\,.$

Finally, we denote by ${\mathcal{M}}_{k}({\mathbb{R}}^{d})$ the subset of
measures $m$ with finite $k$ moments, i.e. $\langle x\rangle^{k}\in
L^{1}({\mathbb{R}}^{d},d|m|)$. For $m\in{\mathcal{M}}_{k}$, we use the
notation

$\|m\|_{{\mathcal{M}}_{k}}:={\int_{{\mathbb{R}}^{d}}}\langle
x\rangle^{k}\,d|m|$

to denote the total variation of $\langle x\rangle^{k}\,m$. We use a similar
notation with possibly different weights, setting
${\mathcal{M}}_{\varphi}({\mathbb{R}}^{d})$ the subset of measures such that
$\varphi\in L^{1}({\mathbb{R}}^{d},d|m|)$ for some positive function
$\varphi$, with
$\|m\|_{{\mathcal{M}}_{\varphi}}={\int_{{\mathbb{R}}^{d}}}\varphi\,d|m|$.

For given $T>0$, we set $Q_{T}:=(0,T)\times{\mathbb{R}}^{d}$. We denote by
$C^{1,2}(Q_{T})$ the functions defined on $Q_{T}$ which are $C^{1}$ in $t$ and
$C^{2}$ in $x$. The notation $C_{b}(U)$ refers to continuous bounded functions
on a set $U$, and $C_{c}(U)$ to continuous compactly supported functions. We
denote by $C([0,T];{\mathcal{M}}({\mathbb{R}}^{d})^{*})$ the functions $m$
from $[0,T]$ into ${\mathcal{M}}({\mathbb{R}}^{d})$ which are continuous with
respect to the weak∗ topology of measures. When working with probability
measures, we denote $C([0,T];{\mathcal{P}}({\mathbb{R}}^{d}))$ the functions
$m$ which are continuous from $[0,T]$ into ${\mathcal{P}}({\mathbb{R}}^{d})$,
endowed with the $d_{1}$ distance.

Associated to a continuous function $\varphi$ such that $\varphi(x)\geq 1$ for
all $x\in{\mathbb{R}}^{d}$, we introduce suitable weighted seminorms, defined
as

$[u]_{\varphi}=\sup_{x,y\in{\mathbb{R}}^{2d}}\,\,\frac{|u(x)-u(y)|}{\varphi(x)+\varphi(y)}\,.$

In particular, we will set

$[u]_{\langle
x\rangle^{k}}:=\sup_{x,y\in{\mathbb{R}}^{2d}}\,\,\frac{|u(x)-u(y)|}{\langle
x\rangle^{k}+\langle y\rangle^{k}}\,.$

With a slight abuse of notation, when considering the case $k=0$, we denote

$[u]_{0}:=\sup_{x,y\in{\mathbb{R}}^{2d}}\,\,\left(|u(x)-u(y)|\right)\,$

the seminorm measuring the oscillation of a bounded function $u$. Given a
continuous matrix $\Sigma\,:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d\times\ell}$,
$\Sigma^{*}$ denotes its transposed matrix; then we define the second order
operator

$\begin{split}{\mathcal{L}}_{0}(u):&=-\lambda_{0}\Delta u-{\rm
tr}(\Sigma(x)\Sigma^{*}(x)D^{2}u)\\\
&=-\lambda_{0}\sum_{i=1}^{d}u_{x_{i}x_{i}}-\sum_{i,j=1}^{d}\sum_{k=1}^{\ell}\Sigma_{ik}(x)\Sigma_{jk}(x)u_{x_{i}x_{j}}\
\end{split}$ (2.1)

where $\lambda_{0}\geq 0$. Without going in search of the most generality, we
will assume $\Sigma$ to be bounded and Lipschitz continuous:

$\exists\,\,\sigma_{0},\sigma_{1}>0\,:\quad\|\Sigma(x)\|\leq\sigma_{0}\,,\qquad\|\Sigma(x)-\Sigma(y)\|\leq\sigma_{1}|x-y|\quad\forall
x,y\in{\mathbb{R}}^{d}\,.$ (2.2)

Now we define a pure jump Levy operator as

$\mathcal{I}(x,[u]):=\int_{{\mathbb{R}}^{d}}\\{u(x+z)-u(x)-(Du(x)\cdot
z)\mathds{1}_{|z|\leq 1}\\}\nu(dz)$ (2.3)

in which the measure $\nu$ is absolutely continuous and satisfies

$\exists\lambda,\Lambda\geq
0\,,\sigma\in(0,2)\,:\,\quad\,\frac{\lambda}{|z|^{d+\sigma}}\leq\frac{d\nu}{dz}\leq\,\frac{\Lambda}{|z|^{d+\sigma}}\qquad\forall
z\in{\mathbb{R}}^{d}\,.$ (2.4)

Typical examples of operators which satisfy (2.3)-(2.4) are given by the
fractional Laplacian $(-\Delta)^{\frac{\sigma}{2}}$ ($\sigma\in(0,2)$) or more
generally by the generators of $\alpha$-stable jump processes. We define the
operator ${\mathcal{L}}$ as

${\mathcal{L}}[u]:={\mathcal{L}}_{0}(u)-\mathcal{I}(x,[u])\,.$ (2.5)

It is well known (see e.g. [1]) that ${\mathcal{L}}[u]+b\cdot Du$ is the
infinitesimal generator of a general Levy process where the drift-diffusion
part is generated by ${\mathcal{L}}_{0}+b\cdot D$ and the jump part is
represented by ${\mathcal{I}}$. This is associated with the stochastic SDE

$\begin{cases}dX_{t}=-b(X_{t})dt+\sqrt{2\lambda_{0}}dB_{t}+\sqrt{2}\Sigma(X_{t})dB_{t}+dP_{t}&\\\
X_{0}=x\in{\mathbb{R}}^{d}&\end{cases}$ (2.6)

where $B_{t}$ is a standard Brownian motion and $P_{t}$ is a compound Poisson
process with measure $\nu$.

We point out that more general conditions upon the diffusion matrix
$\Sigma(x)$ and the Levy measure $\nu(z)$ would be possible, but we keep
things simpler here for the reader’s convenience. Under the assumptions (2.2),
(2.4), and for a continuous vector field $b(t,x)$, we consider the evolution
problem

$\begin{cases}\partial_{t}u+{\mathcal{L}}[u]+b(t,x)\cdot Du=0&\text{ in
}(0,T)\times{\mathbb{R}}^{d},\\\ u(0,x)=u_{0}(x),\qquad\,&\text{ in
}{\mathbb{R}}^{d}\end{cases}$ (2.7)

It is well known that, if $b$ is sufficiently regular, then the unique
solution of (2.7) is characterized as $u(t,x)={\mathbb{E}}(X_{t})$, where
$X_{t}$ is the solution of (2.6). The associated Fokker-Planck equation is
defined through the adjoint problem of (2.7). To this purpose, we set

$\mathcal{I}^{*}(x,[m]):=\int_{{\mathbb{R}}^{d}}\\{m(x+z)-m(x)-(Dm(x)\cdot
z)\mathds{1}_{|z|\leq 1}\\}\nu^{*}(dz)$ (2.8)

where we define $\nu^{*}(B):=\nu(-B)$, for every Borel set
$B\subseteq{\mathbb{R}}^{d}$. Then the operator ${\mathcal{L}}^{*}$ is defined
as

$\begin{split}{\mathcal{L}}^{*}[m]:&={\mathcal{L}}_{0}^{*}(m)-\mathcal{I}^{*}(x,[m])\\\
&=-\lambda_{0}\Delta
m-\sum_{i,j=1}^{d}\sum_{k=1}^{\ell}\left(\Sigma_{ik}\Sigma_{jk}m\right)_{x_{i}x_{j}}-\int_{{\mathbb{R}}^{d}}\\{m(x+z)-m(x)-(Dm(x)\cdot
z)\mathds{1}_{|z|\leq 1}\\}\nu^{*}(dz)\,.\end{split}$ (2.9)

This allows us to introduce the Fokker-Planck equation

$\begin{cases}\partial_{t}m+{\mathcal{L}}^{*}[m]-{\rm div}(b(t,x)\,m)=0&\text{
in }(0,T)\times{\mathbb{R}}^{d},\\\ m(0)=m_{0},\qquad\,&\text{ in
}{\mathbb{R}}^{d}\end{cases}$ (2.10)

where $m_{0}\in{\mathcal{M}}({\mathbb{R}}^{d})$. Suitable formulations of
problems (2.7) and (2.10) are given below. In order to ease notations, in the
following we set, for $\varphi\in C^{2}({\mathbb{R}}^{d})$,

${\mathcal{L}}^{b}[\varphi]:={\mathcal{L}}[\varphi]+b(t,x)\cdot
D\varphi\qquad\,;\qquad\qquad{\mathcal{L}}_{0}^{b}[\varphi]:={\mathcal{L}}_{0}[\varphi]+b(t,x)\cdot
D\varphi\,.$ (2.11)

The key-assumption, throughout the whole paper, is the following confining
condition on the drift:

$\exists\,\,\alpha\,,\,R>0\,:\,\quad b(t,x)\cdot
x\geq\alpha\,|x|^{\gamma}\qquad\forall x\in{\mathbb{R}}^{d}:\,|x|\geq
R\,,\,\forall t>0\,,$ (2.12)

where we assume that $\gamma>0$. In the literature, there are possibly
different approaches to deal with three different regimes: $\gamma\geq 2$,
which is the coercive case, $\gamma\in[1,2)$, which is often referred to as a
slowly confining case, and $\gamma\in(0,1)$, which may be considered as a
degenerate confining case. Our goal is to give a unifying approach which
provides with optimal decay estimates in the whole range of $\gamma$.

We also assume a one-sided control for the accretivity of $b$. Two slightly
different conditions will be assumed depending on the ellipticity of the
nonlocal operator, say whether $\sigma$ in (2.4) is bigger or smaller than
$1$. Namely, we suppose that $b$ satisfies either

$\exists\,\,c_{0}>0\,:\,\quad(b(t,x)-b(t,y))\cdot(x-y)\geq-
c_{0}|x-y|\qquad\forall x,y\in{\mathbb{R}}^{d}\,,\,\forall t>0\,,$ (2.13)

or the strongest condition (when $\sigma\leq 1$):

$\sigma\in(0,1]\,\,\hbox{and $\exists\,\delta\in(0,1)$:
\quad$(b(t,x)-b(t,y))\cdot(x-y)\geq-c_{0}|x-y|(|x-y|\wedge
1)^{1-\sigma+\delta}$,}$ (2.14)

for some $c_{0}>0$. Let us mention that further generalizations of (2.13),
(2.14) are also possible; we refer to Remark 3.7.

###  2.1 Viscosity solutions

It is convenient to use the framework of viscosity solutions for problem
(2.7). In our setting, where the operator is defined through (2.5), we could
use slightly different formulations of the notion of viscosity solution, which
all turn out to be equivalent; we refer the reader to [8] for a discussion of
this issue, and we pick just one among those possible versions. Moreover,
since our focus is on regularity estimates, we will simplify some requirement
by only considering continuous solutions. For the interested reader, upper
semi-continuous (USC) subsolutions, and lower semi-continuous (LSC)
supersolutions, are defined accordingly in [8].

###### Definition 2.1

A continuous function $u\in C^{0}([0,T]\times{\mathbb{R}}^{d})$ is a viscosity
solution of (2.7) if $u\in L^{\infty}((0,T);\langle x\rangle^{-m})$ for some
$m\in(0,\sigma)$, if $u(0,x)=u_{0}(x)$ in ${\mathbb{R}}^{d}$ and if $u$
satisfies the following two requirements:

  * (i)

for every $\varphi\in C^{1,2}(Q_{T})$ such that $\varphi\in
L^{\infty}((0,T);\langle x\rangle^{-k})$ for some $k\in(0,\sigma)$, if
$(t_{0},x_{0})\in Q_{T}$ is a global maximum point of $u-\varphi$, then

$\partial_{t}\varphi(t_{0},x_{0})+{\mathcal{L}}[\varphi](t_{0},x_{0})+b(t_{0},x_{0})\cdot
D\varphi(t_{0},x_{0})\leq 0\,.$

  * (ii)

for every $\varphi\in C^{1,2}(Q_{T})$ such that $\varphi\in
L^{\infty}((0,T);\langle x\rangle^{-k})$ for some $k\in(0,\sigma)$, if
$(t_{0},x_{0})\in Q_{T}$ is a global minimum point of $u-\varphi$, then

$\partial_{t}\varphi(t_{0},x_{0})+{\mathcal{L}}[\varphi](t_{0},x_{0})+b(t_{0},x_{0})\cdot
D\varphi(t_{0},x_{0})\geq 0\,.$

###### Remark 2.2

We stress that the requirement made in Definition 2.1 concerning the growth at
infinity, for both the solution and the test function, is only due to the
nonlocal term ${\mathcal{I}}$. Due to assumption (2.4), some limitation in the
growth at infinity is required in order that ${\mathcal{I}}(x,[u])$,
${\mathcal{I}}(x,[\varphi])$ be well-defined. Alternative definitions,
formulated with a more local character, are suggested in [8], which turn out
to be equivalent.

Another alternative definition of solution can be given using sub-jets and
super-jets of the function $u$, as introduced in [17], denoted by
${J}^{2,-}_{Q_{T}}u$, ${J}^{2,+}_{Q_{T}}u$ respectively. The notation
${\overline{J}}^{2,\pm}_{Q_{T}}u$ denotes the closure of elements in the
sub/super-jets.

The following lemma is the extension to nonlocal operators of the classical
Ishii-Jensen’s lemma in viscosity solutions’s theory [17, Thm 8.3]. The
following nonlocal version follows from [8, Corollary 2]; we use here a
slightly more readable version, which is obtained asymptotically, see e.g. [8,
Proposition 2].

###### Theorem 2.3

Let $u$ be an USC sub-solution of (2.7), let $v$ be a LSC super-solution of
(2.7), and let $(\hat{t},\hat{x},\hat{y})\in(0,T)\times({\mathbb{R}}^{d})^{2}$
be a local maximum point of the map
$(t,x,y)\mapsto[u(t,x)-v(t,y)-\Phi(t,x,y)]$, for some function $\Phi\in
C^{1,2}((0,T)\times({\mathbb{R}}^{d})^{2})$.

Then, for every $n$ sufficiently large, there exist real numbers
$a,b\in{\mathbb{R}}$ and matrices $X,Y\in{\mathcal{S}}_{N}$ such that

$\displaystyle a-b=\partial_{t}\Phi(\hat{t},\hat{x},\hat{y})$
$\displaystyle\,,\,\,(a,D_{x}\Phi(\hat{t},\hat{x},\hat{y}),X)\in{\overline{J}}^{2,+}_{Q_{T}}u(\hat{t},\hat{x}),\,\,(b,-D_{y}\Phi(\hat{t},\hat{x},\hat{y}),Y)\in{\overline{J}}^{2,-}_{Q_{T}}v(\hat{t},\hat{y})$
$\displaystyle-(n+c_{d}\|A\|)I\leq\begin{pmatrix}X&0\\\ 0&-Y\end{pmatrix}\leq
A+\frac{1}{n}A^{2}$

where $A=D^{2}_{x,y}\Phi(\hat{t},\hat{x},\hat{y})$. Moreover, we have

$\displaystyle a-{\rm tr}(Q(\hat{x})X)+b(\hat{t},\hat{x})\cdot
D_{x}\Phi(\hat{t},\hat{x},\hat{y})-{\mathcal{I}}(\hat{x},u(\hat{t}),D_{x}\Phi(\hat{t},\hat{x},\hat{y}))\leq
0$ $\displaystyle b-{\rm tr}(Q(\hat{y})Y)-b(\hat{t},\hat{y})\cdot
D_{y}\Phi(\hat{t},\hat{x},\hat{y})-{\mathcal{I}}(\hat{y},v(\hat{t}),-D_{y}\Phi(\hat{t},\hat{x},\hat{y}))\geq
0$

where we have denoted, for $x,p\in{\mathbb{R}}^{d}$:

$\begin{split}&Q(x)=\lambda_{0}I_{d}+\Sigma\Sigma^{*}(x)\\\
&{\mathcal{I}}(x,u,p)={\int_{{\mathbb{R}}^{d}}}[u(x+z)-u(x)-(p\cdot
z)\mathds{1}_{{|z|\leq 1}}]d\nu(z)\,.\end{split}$ (2.15)

Let us stress that the previous theorem also implies the consistency of the
viscosity formulation; if $u\in C^{1,2}(Q_{T})$ is a viscosity solution, then
it is also a classical solution. Otherwise, if $u_{0}$ is a continuous
function with polynomial growth, problem (2.7) admits a unique viscosity
solution; we refer to Proposition 6.1 for a statement of this kind.

###  2.2 Duality solutions of Fokker-Planck equations

Solutions of the Fokker-Planck equation (2.10) will be defined here in duality
with the notion of viscosity solution for ${\mathcal{L}}^{b}$.

###### Definition 2.4

Let $m_{0}\in{\mathcal{M}}({\mathbb{R}}^{d})$. A function $m\in
C^{0}([0,T);{\mathcal{M}}({\mathbb{R}}^{d})^{*})$ is a solution to (2.10) if

$\displaystyle{\int_{{\mathbb{R}}^{d}}}\xi\,dm(t)+\int_{0}^{t}{\int_{{\mathbb{R}}^{d}}}f\,dm(\tau)\,d\tau={\int_{{\mathbb{R}}^{d}}}\varphi(0,x)\,dm_{0}$
$\displaystyle\qquad\qquad\forall\,t\in(0,T)\,,\,\xi\in
C_{b}({\mathbb{R}}^{d}),\,f\in C_{b}(Q_{t})\,,\varphi\in
C_{b}([0,t]\times{\mathbb{R}}^{d})\quad\mbox{such that}$
$\displaystyle\qquad\qquad\qquad\hbox{$\varphi$ is a viscosity solution of
}\quad\mbox{\bf(B)}\,\,\begin{cases}-\partial_{t}\varphi+{\mathcal{L}}^{b}[\varphi]=f&\text{
in }(0,t)\times{\mathbb{R}}^{d},\\\ \varphi(t,x)=\xi(x),\qquad\,&\text{ in
}{\mathbb{R}}^{d}\end{cases}$

Let us point out that, since classical solutions are viscosity solutions, then
a solution $m$ according to Definition 2.4 is also a weak solution in the
usual distributional sense, which means

$\int_{0}^{T}{\int_{{\mathbb{R}}^{d}}}[-\partial_{t}\varphi+{\mathcal{L}}^{b}[\varphi]]\,dm(t)\,dt={\int_{{\mathbb{R}}^{d}}}\varphi(0,x)\,dm_{0}\qquad\forall\varphi\in
C^{1,2}_{c}([0,T)\times{\mathbb{R}}^{d})\,.$

The converse statement, namely that weak solutions enjoy the duality property
of Definition 2.4, is more delicate under the general setting in which we
consider the drift term $b$ and the nonlocal diffusion ${\mathcal{I}}$.
However, this is certainly true for second order operators (i.e. when
$\lambda_{0}>0$ in (2.1)) or for locally Lipschitz vector fields $b$. Indeed,
weak solutions satisfy Definition 2.4 whenever they are unique. This is a
consequence of the following existence and uniqueness result, which fully
justifies the previous notion of solution in the present setting. We recall
that ${\mathcal{M}}_{k}({\mathbb{R}}^{d})$ denotes the subset of measures in
${\mathbb{R}}^{d}$ having finite $k$-th moment.

###### Theorem 2.5

Let $m_{0}\in{\mathcal{M}}_{k}({\mathbb{R}}^{d})$, for some $0<k<\sigma$. Let
${\mathcal{L}}$ be defined by (2.5) where $\Sigma$ satisfies (2.2), the Levy
measure in ${\mathcal{I}}$ satisfies (2.4), and $\lambda_{0}+\lambda>0$.
Assume that $b\in C^{0}(Q_{T})$ is such that $b(t,x)\cdot x$ is bounded below
and either of the following conditions hold: (i) $\lambda_{0}>0$ or
$\sigma\in(1,2)$, and (2.13) holds true whenever $|x-y|\leq 1$. (ii)
$\lambda_{0}=0$, $\sigma\in(0,1]$ and (2.14) holds true whenever $|x-y|\leq
1$.

Then there exists a unique solution $m$ of (2.10) in the sense of Definition
2.4, and $m(t)\in{\mathcal{M}}_{k}$ for all $t>0$. Moreover, $m$ is obtained
as the limit of solutions $m_{\varepsilon}$ corresponding to the operator
$({\mathcal{L}}^{b})^{*}-\varepsilon\Delta$.

Finally, the application $m_{0}\mapsto m(t)$ preserves positivity and mass.

The proof of Theorem 2.5 will be given in the Appendix, for the sake of
completeness. It stems from the simple idea that, in duality by what happens
for problem (B), the notion of solution given in Definition 2.4 is stable by
vanishing viscosity limits. Thus, even if at first sight the formulation of
Definition 2.4 sounds a bit unusual, the reader can see that this is actually
a quite natural setting in which problem (2.10) can be formulated (and proved
to be well-posed) outside a standard set of assumptions. In fact, the
uniqueness result, and the characterization of the solution as the limit of
Fokker-Planck equations with vanishing viscosity, show the consistency of this
formulation with all known results in terms of weak or classical solutions. In
particular, one can readily deduce that $m\in
C([0,T];L^{1}({\mathbb{R}}^{d}))$ when $\lambda_{0}>0$ (second order
problems), or for the case of fractional Laplacian (or more generally,
symmetric kernels) under some extra condition on ${\rm div}(b)$ (see e.g. [31,
Thm 1.2]).

Finally, we point out that the Definition 2.4 has been given for signed
measures, because we will deal with zero average measures. Due to the Hahn
decomposition, and the linearity of the formulation, every solution $m$ can be
split as $m=m_{1}-m_{2}$, where $m_{1},m_{2}$ are the solutions corresponding
to $m_{0}^{+}$, $m_{0}^{-}$ respectively.

##  3 Oscillation decay estimates

In this Section we give the oscillation estimates for problem (2.7), where we
assume that $u_{0}$ is a continuous function in ${\mathbb{R}}^{d}$, and
${\mathcal{L}},b$ satisfy the assumptions (2.2), (2.4), (2.12) and (2.13) (or
(2.14)).

A crucial role here is played by the explicit Lyapunov functions $\langle
x\rangle^{\beta}:=(\sqrt{1+|x|^{2}})^{\beta}$. The properties of those super-
solutions are listed below.

###### Lemma 3.1

Assume (2.12). Let us denote $\varphi(x)=\langle x\rangle^{\beta}$, with
$\beta>0$. Then we have:

  * (i)

for every $\varepsilon>0$ there exists $K_{\varepsilon}>0$ such that

$\mathcal{L}_{0}[\varphi]+b(t,x)\cdot
D\varphi\geq(\alpha-\varepsilon)\beta\,\frac{\varphi(x)}{\langle
x\rangle^{2-\gamma}}-K_{\varepsilon}\qquad\forall
x\in{\mathbb{R}}^{d}\,,t>0\,.$ (3.1)

  * (ii)

Let the Levy measure $\nu$ satisfy (2.4). If $0<\beta<\sigma$, then for every
$\varepsilon>0$ there exists $K_{\varepsilon}>0$ such that

$\mathcal{L}[\varphi]+b(t,x)\cdot
D\varphi\geq(\alpha-\varepsilon)\beta\,\frac{\varphi(x)}{\langle
x\rangle^{2-\gamma}}-K_{\varepsilon}\qquad\forall x\in{\mathbb{R}}^{d}\,,t>0,$
(3.2)

provided $\beta\leq 1$ or $\gamma>1$ in (2.12).

Proof. We compute $D\varphi=\beta\langle x\rangle^{\beta-1}\frac{x}{\langle
x\rangle}$, $D^{2}\varphi=\beta\langle
x\rangle^{\beta-2}\left(I_{d}+(\beta-1)\frac{x}{\langle
x\rangle}\otimes\frac{x}{\langle x\rangle}\right)$ and so, if
$Q(x)=\lambda_{0}I_{d}+\Sigma\Sigma^{*}(x)$, we have from (2.2)

$\left|{\rm
tr}(Q(x)D^{2}\varphi)\right|\leq\beta\,c_{\beta}\,(\sigma_{0}^{2}+\lambda_{0})\langle
x\rangle^{\beta-2}\,$

for some $c_{\beta}$ only depending on $\beta$ and $d$. Hence

${\mathcal{L}}_{0}^{b}[\varphi]\geq\beta\langle
x\rangle^{\beta-2}\left(b(t,x)\cdot
x-c_{\beta}(\sigma_{0}^{2}+\lambda_{0})\right)\,.$

Using (2.12), we get (3.1).

As for the Levy operator ${\mathcal{I}}$, the difference is that we need to
handle the nonlocal term for large values of $|z|$. Indeed, we have

$\displaystyle{\mathcal{I}}(x,[\varphi])$
$\displaystyle=\int_{|z|>1}\\{\varphi(x+z)-\varphi(x)\\}\nu(dz)+\int_{|z|\leq
1}\\{\varphi(x+z)-\varphi(x)-(D\varphi(x)\cdot z)\\}\nu(dz)$
$\displaystyle\leq c_{\beta}\,\int_{|z|>1}\max(\langle
x\rangle^{\beta-1}\,|z|\,,\,|z|^{\beta})\nu(dz)+c_{\beta}\langle
x\rangle^{\beta-2}\int_{|z|\leq 1}|z|^{2}\,\nu(dz)$

and, due to (2.4), the first term is integrable provided $\beta<\sigma$. If
$\sigma\leq 1$, this also implies that $\beta\leq 1$, so we deduce that
${\mathcal{I}}(x,[\varphi])$ is bounded above. Hence we get

${\mathcal{L}}^{b}[\varphi]\geq\beta\langle x\rangle^{\beta-2}\,b(t,x)\cdot
x-C_{\beta}$

for some different constant $C_{\beta}$, possibly depending on $\lambda_{0}$
and $\sigma_{0}$ as well. Using (2.12), we conclude with (3.2). The same
remains true if $\sigma>1$ and $\beta\leq 1$.

If $\sigma>1$ and $\beta>1$, we estimate, for some $c_{\beta}>0$,

${\mathcal{I}}(x,[\varphi])\leq c_{\beta}(\langle x\rangle^{\beta-1}+1)$

and we get

${\mathcal{L}}[\varphi]\geq\beta\langle x\rangle^{\beta-2}\,b(t,x)\cdot
x-C_{\beta}(1+\langle x\rangle^{\beta-1})\,.$ (3.3)

Using (2.12), the conclusion follows provided $\gamma>1$.

###### Remark 3.2

Let us observe, from the previous Lemma, the coercive case given by
$\gamma\geq 2$. In this range, we have that
$\frac{{\mathcal{L}}[\varphi]}{\varphi}\to+\infty$ as $|x|\to\infty$. This
holds for every $\beta>0$ in the local case ${\mathcal{L}}={\mathcal{L}}_{0}$,
otherwise it holds for every $\beta<\sigma$ if ${\mathcal{I}}$ satisfies
(2.4).

By contrast, if $\gamma<2$, then ${\mathcal{L}}[\varphi]\to+\infty$ as
$|x|\to\infty$ only for $\beta>2-\gamma$. This introduces a natural threshold
for nonlocal operators satisfying (2.4): since $\beta<\sigma$ is required for
the behavior of $\nu$ at infinity, this will lead to require $\gamma>2-\sigma$
for the confinement force to play a significant role.

Notice that $\gamma+\beta-2>0$ already implies $\gamma>1$ whenever $\beta\leq
1$. This always occurs if $\sigma\leq 1$, since $\beta<\sigma$. Otherwise,
when $\sigma>1$, it is possible to consider $\beta>1$; nevertheless the
restriction $\gamma>1$ is still needed to obtain (3.2), because one needs the
drift term to dominate the diffusion coming from the jumps involving large
values of $x$ (see (3.3)). An interpretation is that the jump part contains
terms which are comparable to drift components; so they would be controlled by
a confining force only if this blows-up at infinity (say, $\gamma>1$).

To sum up this remark, we observe that requiring $\langle x\rangle^{\beta}$ to
be a Lyapunov function (i.e. ${\mathcal{L}}[\langle
x\rangle^{\beta}]\to+\infty$ as $|x|\to\infty$) leads to the restriction that
$\gamma>1$ and $\gamma>2-\sigma$.

However, we stress that the restriction $\beta<\sigma$ only comes from the
behavior at infinity of the Levy measure $\nu(z)$, and the reader may easily
think at generalizing this result to different settings for $\nu$.

###  3.1 Doubling variables methods

As we mentioned in the Introduction, two ingredients yield the oscillation
estimates of this Section. While Lemma 3.1 exploits the existence of Lyapunov
functions using the confinement property of the drift, the second ingredient
comes out by handling the local oscillation of solutions through the maximum
principle. Here we exploit in full strength the Ishii-Lions doubling variables
method, revisited through the ideas of coupling along the lines suggested in
[37]. For the sake of clarity, we state separately the contribution of local
and nonlocal diffusion.

###### Lemma 3.3

Let $u$ be a viscosity solution of (2.7) and suppose that $(x,y)$ is a local
maximum point of the function

$u(t,x)-u(t,y)-\zeta(t,x,y)\,,\qquad\zeta(t,\cdot,\cdot)=[\varphi(x)+\varphi(y)]+\psi(|x-y|)$

where $\varphi$ is a $C^{2}$ function and $\psi:[0,\infty)\to{\mathbb{R}}_{+}$
is a positive, increasing function which is $C^{2}$ in a neighborhood of
$|x-y|$.

For $n>0$, let $X_{n},Y_{n}\in{\mathcal{S}}^{N}$ be the matrices given by
Theorem 2.3 where $A=D^{2}_{(x,y)}\zeta$. Then, setting
$Q(x):=\lambda_{0}I_{d}+\Sigma(x)\Sigma^{*}(x)$, we have

$\begin{split}\limsup_{n\to\infty}\,\,{\rm tr}(Q(x)X_{n}-Q(y)Y_{n})&\leq
4\lambda_{0}\psi^{\prime\prime}(|x-y|)+\frac{\psi^{\prime}(|x-y|)}{|x-y|}\|\Sigma(x)-\Sigma(y)\|^{2}\\\
&\qquad\qquad+{\rm tr}(Q(x)D^{2}\varphi(x))+{\rm
tr}(Q(y)D^{2}\varphi(y))\,.\end{split}$ (3.4)

Proof. This is a consequence of the method introduced in [29]. In the above
precised form, inequality (3.4) stems from [37, Prop. 2.4] up to a specific
computation involving the very definition of $D^{2}\zeta$ (see [37, formula
(3.29)]).

The next step is an extension of the previous lemma to nonlocal operators, as
defined in (2.3). Generalizations of the Ishii-Lions method to nonocal
operators have been developed in [5], [7], but we need here an improvement
which is crucial for the purpose of our results. This improvement relies on
the idea of coupling (by reflection) for jump operators, and we believe this
is interesting in its own. Compared to the probabilistic approach, here the
coupling is naturally inserted in the analytical doubling variables method, as
it was in [37] for local operators. Let us mention that some related ideas
also appeared later in [23], in a rather informal way, .

In the following, we omit the time dependence of the function $u$, since it is
irrelevant here. Moreover, we assume that all functions involved are
integrable at infinity with respect to the Levy measure $\nu$ (in our context,
this is satisfied by all functions which have power growth of order less than
$\sigma$).

###### Lemma 3.4

Let the operator $\mathcal{I}$ be defined by (2.3), where $\nu$ satisfies
(2.4). Suppose that $(x,y)$ is a maximum point of the function

$u(x)-u(y)-\zeta(x,y)\,,\qquad\zeta:=[\varphi(x)+\varphi(y)]+\psi(|x-y|)$

where $\varphi$ is a $C^{2}$ function and $\psi:[0,\infty)\to{\mathbb{R}}_{+}$
is a positive, increasing function which is $C^{2}$ in a neighborhood of
$|x-y|$.

Then, for every $\delta\leq\left(\frac{|x-y|}{2}\wedge 1\right)$, we have
(recall the notations in (2.15)):

$\begin{split}&\mathcal{I}(x,u(x),D_{x}\zeta(x,y))-\mathcal{I}(y,u(y),-D_{y}\zeta(x,y))\leq\mathcal{I}(x,\varphi(x),D\varphi(x))+\mathcal{I}(y,\varphi(y),D\varphi(y))\\\
&\quad\qquad\qquad+4\lambda\int_{0}^{1}(1-s)\int_{B_{\delta}}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds,\end{split}$ (3.5)

where $B_{\delta}:=\\{z\in{\mathbb{R}}^{d}\,:\,|z|<\delta\\}$.

Proof. Let us set

$H(x,y):=u(x)-u(y)-\zeta(x,y)=u(x)-u(y)-[\varphi(x)+\varphi(y)]-\psi(|x-y|)\,.$

Since $H(x,y)\geq H(x+z,y+z)$ for every $z$, we have

$\begin{split}&[u(x+z)-u(x)]-[u(y+z)-u(y)]\\\
&\qquad\qquad\leq\left\\{\varphi(x+z)-\varphi(x)+\varphi(y+z)-\varphi(y)]\right\\}\qquad\forall
z\in{\mathbb{R}}^{d}\,.\end{split}$ (3.6)

In addition, we have

$D_{x}\zeta+D_{y}\zeta=D\varphi(x)+D\varphi(y)\,.$

Therefore, for any $A\subseteq{\mathbb{R}}^{d}$ and any Levy measure
$\tilde{\nu}$, we have

$\begin{split}&\int_{A}\\{u(x+z)-u(x)-\langle
D_{x}\zeta,z\rangle\mathds{1}_{|z|\leq
1}\\}\tilde{\nu}(dz)-\int_{A}\\{u(y+z)-u(y)+\langle
D_{y}\zeta,z\rangle\mathds{1}_{|z|\leq 1}\\}\tilde{\nu}(dz)\\\
&\quad\leq\int_{A}\\{\varphi(x+z)-\varphi(x)-\langle
D\varphi(x),z\rangle\mathds{1}_{|z|\leq
1}\\}\tilde{\nu}(dz)+\int_{A}\\{\varphi(y+z)-\varphi(y)-\langle
D\varphi(y),z\rangle\mathds{1}_{|z|\leq 1}\\}\tilde{\nu}(dz)\,.\end{split}$
(3.7)

Notice in particular that (3.7) with $A={\mathbb{R}}^{d}$ implies (3.5) when
$\lambda=0$. But now we wish to exploit the ellipticity, and to this purpose
we split the nonlocal terms for $|z|\geq r/2$ and $|z|<r/2$, where $r=|x-y|$.
Moreover, in order to handle the small  jumps we decompose the measure $\nu$
as

$\nu=\lambda\frac{dz}{|z|^{d+\sigma}}+\mu\,.$

By assumption (2.4), $\mu$ is nonnegative, and is itself a Levy measure. We
define the ball

$B:=\left\\{z\in{\mathbb{R}}^{d}\,:\,|z|<\left(1\wedge\frac{r}{2}\right)\right\\}$

and we decompose the nonlocal integrals in three parts, using that

$\nu=\nu\mathds{1}_{B^{c}}+\mu\mathds{1}_{B}+\lambda\frac{dz}{|z|^{d+\sigma}}\mathds{1}_{B\,.}$

For the first two measures, we only use (3.7), applied for both
$\tilde{\nu}=\nu\mathds{1}_{B^{c}}$ and $\tilde{\nu}=\mu\mathds{1}_{B}$. Hence

$\begin{split}&\int_{{\mathbb{R}}^{d}}\\{u(x+z)-u(x)-(D_{x}\zeta\cdot
z)\mathds{1}_{|z|\leq
1}\\}\nu(dz)-\int_{{\mathbb{R}}^{d}}\\{u(y+z)-u(y)+(D_{y}\zeta\cdot
z)\mathds{1}_{|z|\leq 1}\\}\nu(dz)\\\
&\leq\int_{{\mathbb{R}}^{d}}\left\\{\varphi(x+z)-\varphi(x)-(D\varphi(x)\cdot
z)\mathds{1}_{|z|\leq
1}\right\\}\nu(dz)+\int_{{\mathbb{R}}^{d}}\left\\{\varphi(y+z)-\varphi(y)-(D\varphi(y)\cdot
z)\mathds{1}_{|z|\leq 1}\right\\}\nu(dz)\\\
&\quad+\lambda\int_{B}\\{u(x+z)-u(x)-(D_{x}\zeta\cdot
z)\\}\frac{dz}{|z|^{d+\sigma}}-\lambda\int_{B}\\{u(y+z)-u(y)+(D_{y}\zeta\cdot
z)\\}\frac{dz}{|z|^{d+\sigma}}\\\
&\quad-\lambda\int_{B}\\{\varphi(x+z)-\varphi(x)-(D\varphi(x)\cdot
z)\\}\frac{dz}{|z|^{d+\sigma}}-\lambda\int_{B}\\{\varphi(y+z)-\varphi(y)-(D\varphi(y)\cdot
z)\\}\frac{dz}{|z|^{d+\sigma}}\,.\end{split}$ (3.8)

We will use a different strategy for the measure
$\lambda\frac{dz}{|z|^{d+\sigma}}\mathds{1}_{B}$, which provides with the
desired ellipticity. Since $H$ achieves its maximum at $(x,y)$, then
$H(x,y)\geq H(x+z,y+Az)$ where $A$ is the $d\times d$ matrix defined as

$A:=I_{d}-2(\widehat{x-y}\otimes\widehat{x-y})\,.$

We notice that, for a given $z\in{\mathbb{R}}^{d}$ (corresponding to a jump in
the $x$-process), $Az$ yields the reflection of $z$ through the hyperplane
$\\{z\cdot\widehat{x-y}=0\\}$. This encodes the idea to use a reflected
process in the $y$-variable to optimize the estimate, as in the probabilistic
coupling methods.

Thus we have, by the maximality condition,

$\begin{split}&[u(x+z)-u(x)]-[u(y+Az)-u(y)]\\\
&\qquad\qquad\leq[\psi(|x-y+z-Az|)-\psi(|x-y|)]\\\
&\qquad+\left\\{\varphi(x+z)-\varphi(x)+\varphi(y+Az)-\varphi(y)]\right\\}\qquad\forall
z\in{\mathbb{R}}^{d}\,.\end{split}$ (3.9)

Integrating (3.9) on $B$ against the measure $\frac{dz}{|z|^{d+\sigma}}$ we
get

$\displaystyle\int_{B}\left\\{u(x+z)-u(x)-D_{x}\zeta\cdot
z\right\\}\frac{dz}{|z|^{d+\sigma}}-\int_{B}\\{u(y+Az)-u(y)+D_{y}\zeta\cdot
Az\\}\frac{dz}{|z|^{d+\sigma}}$
$\displaystyle\leq\int_{B}\left\\{\varphi(x+z)-\varphi(x)-D\varphi(x)\cdot
z\right\\}\frac{dz}{|z|^{d+\sigma}}+\int_{B}\left\\{\varphi(y+Az)-\varphi(y)-D\varphi(y)\cdot
Az\right\\}\frac{dz}{|z|^{d+\sigma}}$
$\displaystyle\qquad+\int_{B}\left\\{\psi(|x-y+z-Az|)-\psi(|x-y|)-\psi^{\prime}(|x-y|)\widehat{x-y}\cdot(z-Az)\right\\}\frac{dz}{|z|^{d+\sigma}}\,.$

Now we use that $A$ is an orthogonal matrix and the measure
$\frac{dz}{|z|^{d+\sigma}}$ is invariant by rotation. Therefore, the previous
inequality reads as follows:

$\begin{split}&\int_{B}\left\\{u(x+z)-u(x)-D_{x}\zeta\cdot
z\right\\}\frac{dz}{|z|^{d+\sigma}}-\int_{B}\\{u(y+z)-u(y)+D_{y}\zeta\cdot
z\\}\frac{dz}{|z|^{d+\sigma}}\\\
&\leq\int_{B}\left\\{\varphi(x+z)-\varphi(x)-D\varphi(x)\cdot
z\right\\}\frac{dz}{|z|^{d+\sigma}}+\int_{B}\left\\{\varphi(y+z)-\varphi(y)-D\varphi(y)\cdot
z\right\\}\frac{dz}{|z|^{d+\sigma}}\\\
&\qquad+\int_{B}\left\\{\psi(|x-y+z-Az|)-\psi(|x-y|)-\psi^{\prime}(|x-y|)\widehat{x-y}\cdot(z-Az)\right\\}\frac{dz}{|z|^{d+\sigma}}\,.\end{split}$
(3.10)

Putting together (3.8) and (3.10), we deduce the following inequality

$\begin{split}&\mathcal{I}(x,u(x),D_{x}\zeta(x,y))-\mathcal{I}(y,u(y),-D_{y}\zeta(x,y))\leq\mathcal{I}(x,\varphi(x),D\varphi(x))+\mathcal{I}(y,\varphi(y),D\varphi(y))\\\
&\quad\qquad\qquad+\lambda\int_{B}\left\\{\psi(|x-y+z-Az|)-\psi(|x-y|)-\psi^{\prime}(|x-y|)\widehat{x-y}\cdot(z-Az)\right\\}\frac{dz}{|z|^{d+\sigma}}\,.\end{split}$
(3.11)

Recalling the definition of $A$, we compute

$\displaystyle\int_{B}\left\\{\psi(|x-y+z-Az|)-\psi(|x-y|)-\psi^{\prime}(|x-y|)\widehat{x-y}\cdot(z-Az)\right\\}\frac{dz}{|z|^{d+\sigma}}$
$\displaystyle\qquad=\int_{B}\left\\{\psi(|r+2(\widehat{x-y}\cdot
z)|)-\psi(r)-2\psi^{\prime}(r)(\widehat{x-y}\cdot
z)\right\\}\frac{dz}{|z|^{d+\sigma}}\,$

where, we recall, $r=|x-y|$. Now we estimate the right-hand side. To this
purpose, notice that for $|z|<\frac{r}{2}$ we have $r+2\widehat{x-y}\cdot
z\geq 0$. Hence, Taylor’s expansion implies

$\displaystyle\int_{B}\left\\{\psi(|r+2(\widehat{x-y}\cdot
z)|)-\psi(r)-2\psi^{\prime}(r)(\widehat{x-y}\cdot
z)\right\\}\frac{dz}{|z|^{d+\sigma}}$ $\displaystyle\leq
4\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,.$

Finally, coming back to (3.11), we deduce (3.5).

###  3.2 Exponential rate

We consider here the case that the drift is strongly confining, in the sense
that $\gamma\geq 2$ in (2.12). This provides with an exponential decay
estimate for a weighted oscillation of the solution of (2.7). For the sake of
clarity, we distinguish two different statements for the case of local
operators and the case of general Levy processes with jumps.

###### Theorem 3.5

Let ${\mathcal{L}}_{0}$ be defined by (2.1), where $\lambda_{0}>0$ and
$\Sigma$ satisfies (2.2). Assume that $b$ satisfies (2.12) with $\gamma\geq
2$, and (2.13). Let $u$ be a (viscosity) solution of

$\begin{cases}\partial_{t}u+{\mathcal{L}}_{0}[u]+b(t,x)\cdot Du=0&\text{ in
}(0,T)\times{\mathbb{R}}^{d},\\\ u(0,x)=u_{0}(x),\qquad\,&\text{ in
}{\mathbb{R}}^{d}\end{cases}$ (3.12)

such that $|u(t,x)|\leq C(1+|x|)^{m}$ for some $m>0$ and $C>0$ (uniformly in
time).

For any $k>0$ such that $u_{0}\in L^{\infty}(\langle x\rangle^{-k})$, there
exist positive constants $K,\omega>0$ (depending on
$\alpha,\gamma,\lambda_{0},\sigma_{0},\sigma_{1},c_{0},d,k$) such that $u$
satisfies

$[u(t)]_{\langle x\rangle^{k}}\leq K\,e^{-\omega t}\,[u_{0}]_{\langle
x\rangle^{k}}.$ (3.13)

In case of nonlocal operators with $\sigma$ fractional growth, the statement
is similar except for a restriction in the weight which is used (and the
growth of solutions).

###### Theorem 3.6

Let ${\mathcal{L}}$ be defined by (2.5) where $\Sigma$ satisfies (2.2), the
Levy measure in ${\mathcal{I}}$ satisfies (2.4), and $\lambda_{0}+\lambda>0$.
Assume that $b$ satisfies (2.12) with $\gamma\geq 2$ and either of the
following conditions hold:

(i) $\lambda_{0}>0$ or $\sigma\in(1,2)$, and (2.13) holds true.

(ii) $\lambda_{0}=0$ and $\sigma\in(0,1]$, and (2.14) holds true.

Let $u$ be a (viscosity) solution of (2.7) such that $|u(t,x)|\leq
C(1+|x|)^{m}$ for some $m\in(0,\sigma)$ and $C>0$ (uniformly in time).

For any $k\in(0,\sigma)$ such that $u_{0}\in L^{\infty}(\langle
x\rangle^{-k})$, there exist positive constants $K,\omega>0$ (depending on
$\alpha,\gamma,\sigma,(\lambda_{0}+\lambda)^{-1},\sigma_{0},\sigma_{1},\Lambda,c_{0},d,k$)
such that $u$ satisfies

$[u(t)]_{\langle x\rangle^{k}}\leq K\,e^{-\omega t}\,[u_{0}]_{\langle
x\rangle^{k}}.$ (3.14)

Proof of Theorem 3.5 and Theorem 3.6. Let us set $\varphi(x)=\langle
x\rangle^{k}$, $\Phi(x)=\langle x\rangle^{p}$, where $0<k<\sigma$ and
$m<p<\sigma$ (no restriction on $k,p$ is needed where there is no jump part,
i.e. for Theorem 3.5). In particular, we have that $u(t,x)=o(\Phi(x))$ as
$|x|\to\infty$, uniformly in time. As in [14], we also use the explicit
auxiliary function

$\psi(r)=C_{1}(1-e^{-C_{2}r^{\theta}})$ (3.15)

where $\theta\in(0,1)$ and $C_{1},C_{2}$ will be chosen later. Notice that
$\psi$ is a bounded, concave function, which is smooth in $(0,\infty)$ and
satisfies $\psi\simeq C_{1}C_{2}\,r^{\theta}$ for $r\to 0$. Moreover,
$\psi\leq C_{1}$ for every $r$.

We claim that, for suitably chosen large constants $K,C_{1}$ and $C_{2}$, and
for a convenient choice of $\omega$, we have

$u(t,x)-u(t,y)\leq e^{-\omega
t}\left\\{K\,(\varphi(x)+\varphi(y))+\psi(|x-y|)+\varepsilon[\Phi(x)+\Phi(y)]\right\\}+\frac{\varepsilon}{T-t}$
(3.16)

for every $\varepsilon>0$ sufficiently small.

As usual, we observe that if (3.16) were not true, then the function

$u(t,x)-u(t,y)-e^{-\omega
t}\underbrace{\left\\{K[\varphi(x)+\varphi(y)]+\psi(|x-y|)+\varepsilon[\Phi(x)+\Phi(y)]\right\\}}_{\zeta(x,y)}-\frac{\varepsilon}{T-t}$

has a positive maximum in $[0,T]\times{\mathbb{R}}^{2d}$ (because
$u(t,x)=o(\Phi(x))$ as $|x|\to\infty$), and this positive maximum is attained
at some point $(t,x,y)$ such that, obviously, $t<T$ and $x\neq y$. In
addition, if $K\geq[u_{0}]_{\langle x\rangle^{k}}$, then the positive maximum
cannot either be attained at $t=0$. As a consequence, the point $(t,x,y)$ is a
local maximum and we can use the viscosity inequalities satisfied by $u(t,x)$
and $u(t,y)$, thanks to Theorem 2.3. In particular, we deduce the existence of
matrices $X_{n},Y_{n}\in{\mathcal{S}}^{N}$ such that

$\begin{pmatrix}X_{n}&0\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&-Y_{n}\end{pmatrix}\leq e^{-\omega
t}D^{2}_{(x,y)}\zeta+\frac{1}{n}\left(e^{-\omega
t}D^{2}_{(x,y)}\zeta\right)^{2}$

and we have

$\displaystyle\frac{\varepsilon}{(T-t)^{2}}-\omega e^{-\omega
t}\left(K[\varphi(x)+\varphi(y)]+\psi(|x-y|)+\varepsilon[\Phi(x)+\Phi(y)]\right)$
$\displaystyle\quad+e^{-\omega
t}\left((b(t,x)-b(t,y))\cdot\widehat{x-y}\right)\psi^{\prime}(|x-y|)$
$\displaystyle\quad+e^{-\omega t}\left\\{b(t,x)\cdot(KD\varphi(x)+\varepsilon
D\Phi(x))+b(t,y)\cdot(KD\varphi(y)+\varepsilon D\Phi(y))\right\\}$
$\displaystyle\qquad\leq e^{-\omega t}{\rm
tr}(Q(x)X_{n})-Q(y)Y_{n})+{\mathcal{I}}(x,u(t),e^{-\omega
t}D_{x}\zeta)-{\mathcal{I}}(y,u(t),-e^{-\omega t}D_{y}\zeta)$

where $\widehat{x-y}=\frac{x-y}{|x-y|}$ and
$Q(x)=\lambda_{0}I_{d}+\Sigma\Sigma^{*}(x)$.

The right-hand side is estimated through Lemma 3.3 for the local diffusion and
Lemma 3.4 for the nonlocal part. Therefore, letting $n\to\infty$, we obtain

$\displaystyle\frac{\varepsilon}{(T-t)^{2}}-\omega e^{-\omega
t}\left(K[\varphi(x)+\varphi(y)]+\psi(|x-y|)+\varepsilon[\Phi(x)+\Phi(y)]\right)$
$\displaystyle\quad+e^{-\omega
t}\left((b(t,x)-b(t,y))\cdot\widehat{x-y}\right)\psi^{\prime}(|x-y|)$
$\displaystyle\quad+e^{-\omega t}\left\\{b(t,x)\cdot(KD\varphi(x)+\varepsilon
D\Phi(x))+b(t,y)\cdot(KD\varphi(y)+\varepsilon D\Phi(y))\right\\}$
$\displaystyle\qquad\leq e^{-\omega
t}\left\\{4\lambda_{0}\psi^{\prime\prime}(|x-y|)+\frac{\psi^{\prime}(|x-y|)}{|x-y|}\|\Sigma(x)-\Sigma(y)\|^{2}\right\\}$
$\displaystyle\quad+e^{-\omega t}\left\\{{\rm
tr}(Q(x)[KD^{2}\varphi+\varepsilon D^{2}\Phi](x))+{\rm
tr}(Q(y)[KD^{2}\varphi+\varepsilon D^{2}\Phi](y))\right\\}$
$\displaystyle\quad+e^{-\omega
t}K\left\\{\,\mathcal{I}(x,\varphi(x),D\varphi(x))+\mathcal{I}(y,\varphi(y),D\varphi(y))\right\\}+e^{-\omega
t}\varepsilon\left\\{\,\mathcal{I}(x,\Phi(x),D\Phi(x))+\mathcal{I}(y,\Phi(y),D\Phi(y))\right\\}$
$\displaystyle\quad\qquad\qquad+e^{-\omega
t}\,4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(|x-y|+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,,$

where

$B:=\left\\{z\in{\mathbb{R}}^{d}\,:\,|z|<\left(1\wedge\frac{|x-y|}{4}\right)\right\\}\,.$

Henceforth we set $r=|x-y|$, we use (2.2) and we rearrange terms,
reconstructing the operator ${\mathcal{L}}^{b}$ on the auxiliary functions
$\varphi,\Phi$. Then we get

$\begin{split}&K\left\\{\mathcal{L}^{b}[\varphi](x)-\omega\varphi(x)+\mathcal{L}^{b}[\varphi](y)-\omega\varphi(y)\right\\}+\varepsilon\left(\mathcal{L}^{b}[\Phi](x)-\omega\Phi(x)+\mathcal{L}^{b}[\Phi](y)-\omega\Phi(y)\right)\\\
&\quad+\left((b(t,x)-b(t,y))\cdot\widehat{x-y}\right)\psi^{\prime}(r)<4\lambda_{0}\psi^{\prime\prime}(r)+(2\sigma_{0}\wedge\sigma_{1}r)\sigma_{1}\,\psi^{\prime}(r)+\omega\,\psi(r)\\\
&\qquad\qquad\qquad+4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,.\end{split}$ (3.17)

Now we show that the previous inequality leads to a contradiction for suitable
choices of $K,C_{1},C_{2},\omega$.

We first analyze (3.17) for the case that $r=|x-y|\geq 1$. Using (2.13) or
(2.14) (they reduce to the same for $|x-y|\geq 1$), and dropping the terms
with $\psi^{\prime\prime}$ (which is negative), we obtain

$\begin{split}&K\left\\{\mathcal{L}^{b}[\varphi](x)-\omega\varphi(x)+\mathcal{L}^{b}[\varphi](y)-\omega\varphi(y)\right\\}+\varepsilon\left(\mathcal{L}^{b}[\Phi](x)-\omega\Phi(x)+\mathcal{L}^{b}[\Phi](y)-\omega\Phi(y)\right)\\\
&\qquad\qquad<(c_{0}+2\sigma_{0}\sigma_{1})\psi^{\prime}(r)+\omega\,\psi(r)\,.\end{split}$
(3.18)

Let us suppose that $r=|x-y|\geq r_{1}$, for a sufficiently large $r_{1}>1$.
This implies that $\max(|x|,|y|)\geq\frac{r_{1}}{2}$. Due to Lemma 3.1,
choosing $\omega$ sufficiently small (depending only on $\alpha,k$) we have

$\displaystyle\mathcal{L}^{b}[\varphi](x)-\omega\varphi(x)\geq\frac{\alpha
k}{2}\varphi(x)\qquad\hbox{for large $|x|$}$
$\displaystyle\mathcal{L}^{b}[\varphi](x)-\omega\varphi(x)\geq-
k_{0}\qquad\hbox{for every $x$,}$

for some $k_{0}>0$, and so is for the function $\Phi$. Notice that $k_{0}$
also depends on $\alpha,k$ (and possibly on $\sigma,d,\sigma_{0}$, but this
will not be relevant). Therefore, we can choose $r_{1}$ sufficiently large (so
far, only depending on $\alpha,k$) such that

$\mathcal{L}[\varphi]^{b}(x)-\omega\varphi(x)+\mathcal{L}^{b}[\varphi](y)-\omega\varphi(y)\geq
1\qquad\forall x,y:|x-y|\geq r_{1}\,.$

If we also estimate $\psi^{\prime}(r)\leq\psi^{\prime}(r_{1})$ for $r\geq
r_{1}$ (by concavity) and $\psi(r)\leq C_{1}$, we deduce from (3.18)

$K-2\varepsilon k_{0}-(c_{0}+2\sigma_{0}\sigma_{1})\psi^{\prime}(r_{1})-\omega
C_{1}<0\qquad\forall r\geq r_{1}.$ (3.19)

We now fix

$K:=\omega
C_{1}+(c_{0}+2\sigma_{0}\sigma_{1})\psi^{\prime}(r_{1})+2\varepsilon
k_{0}+[u_{0}]_{\langle x\rangle^{k}}$ (3.20)

and we see that inequality (3.19) cannot hold. Therefore, we are left with the
possibility that $r<r_{1}$. In this case we go back to (3.17) and we need to
distinguish two cases according to the diffusivity of the nonlocal term
${\mathcal{I}}$. (A) Case $\lambda_{0}>0$ or (2.4) with $\sigma\in(1,2)$. Here
we use condition (2.13), and we deduce from (3.17) (using also
$\mathcal{L}^{b}[\varphi]-\omega\varphi\geq-k_{0}$,
$\mathcal{L}^{b}[\Phi]-\omega\Phi\geq-k_{0}$)

$\begin{split}&-(2Kk_{0}+2\varepsilon
k_{0})<4\lambda_{0}\psi^{\prime\prime}(r)+(c_{0}+2\sigma_{0}\sigma_{1})\psi^{\prime}(r)+\omega\,\psi(r)\\\
&\qquad+4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,.\end{split}$ (3.21)

We observe that, using once more the concavity of $\psi$, we have
$\psi^{\prime}(r_{1})\leq\psi^{\prime}(r)$ for $r<r_{1}$, so the choice of $K$
in (3.20) implies

$K\leq\omega C_{1}+2\varepsilon k_{0}+[u_{0}]_{\langle
x\rangle^{k}}+(c_{0}+2\sigma_{0}\sigma_{1})\psi^{\prime}(r)\qquad\forall
r<r_{1}\,.$

Therefore (3.21) yields

$\begin{split}&-(2k_{0}(\omega C_{1}+[u_{0}]_{\langle
x\rangle^{k}})+2\varepsilon
k_{0}(1+2k_{0}))<4\lambda_{0}\psi^{\prime\prime}(r)+(1+2k_{0})(c_{0}+2\sigma_{0}\sigma_{1})\psi^{\prime}(r)+\omega\,\psi(r)\\\
&\qquad+4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,.\end{split}$ (3.22)

We point out that, if $\lambda_{0}>0$, we could conclude from here using only
the local term $\psi^{\prime\prime}(r)$ (this would be similar as in [37]). So
here the main novelty is in the use of nonlocal diffusion, neglecting the term
with $\lambda_{0}$. Hence, using also $\psi\leq C_{1}$, we reduce (3.22) to
the following:

$\begin{split}&-((1+2k_{0})\omega C_{1}+2k_{0}[u_{0}]_{\langle
x\rangle^{k}}+2\varepsilon
k_{0}(1+2k_{0}))<(1+2k_{0})(c_{0}+2\sigma_{0}\sigma_{1})\psi^{\prime}(r)\\\
&\qquad+4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,.\end{split}$ (3.23)

Now we estimate the nonlocal term as follows:

$\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\leq\int_{0}^{1}(1-s)\int_{B^{-}}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds$

where

$B^{-}:=\\{z\in B\,:\,(\widehat{x-y}\cdot z)\leq 0\\}\,.$

Observe that

$\int_{B^{-}}|z|^{2}\frac{dz}{|z|^{d+\sigma}}\geq C_{d,\sigma}\,(r\wedge
1)^{2-\sigma}>0\,.$

Hence we also estimate, on account of the concavity of $\psi$,

$\begin{split}\psi^{\prime}(r)&\leq\frac{2}{C_{d,\sigma}}\int_{0}^{1}(1-s)\int_{B^{-}}\frac{\psi^{\prime}(r)}{(r\wedge
1)^{2-\sigma}}|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\\\
&\leq\frac{2}{C_{d,\sigma}}\int_{0}^{1}(1-s)\int_{B^{-}}\frac{\psi^{\prime}(r+2s(\widehat{x-y}\cdot
z))}{(r\wedge 1)^{2-\sigma}}|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\\\
&\leq\frac{2}{C_{d,\sigma}}\int_{0}^{1}(1-s)\int_{B^{-}}\frac{\psi^{\prime}(r+2s(\widehat{x-y}\cdot
z))}{(r+2s(\widehat{x-y}\cdot z))\wedge
1)^{2-\sigma}}|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\end{split}$ (3.24)

where we also used, in the last step, that for $z\in B^{-}$ we have
$0<r+2s(\widehat{x-y}\cdot z)\leq r$.

Using (3.24) in (3.23), we obtain

$\begin{split}&-((1+2k_{0})\omega C_{1}+2k_{0}[u_{0}]_{\langle
x\rangle^{k}}+2\varepsilon k_{0}(1+2k_{0}))\\\
&\qquad\leq\int_{0}^{1}(1-s)\int_{B^{-}}\left\\{4\lambda\psi^{\prime\prime}(\xi_{s,z})+C\frac{\psi^{\prime}(\xi_{s,z})}{(\xi_{s,z}\wedge
1)^{2-\sigma}}\right\\}|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,.\end{split}$
(3.25)

where $\xi_{s,z}=r+2s(\widehat{x-y}\cdot z)$, and
$C=\frac{2}{C_{d,\sigma}}(1+2k_{0})(c_{0}+2\sigma_{0}\sigma_{1})$.

Recall that for $z\in B^{-}$ we have $r/2\leq\xi_{s,z}\leq r$. In particular,
we have $\xi_{s,z}\leq r_{1}$ whenever $r<r_{1}$. Using the definition of
$\psi$ we compute

$\displaystyle
4\lambda\psi^{\prime\prime}(\xi)+C\,\frac{\psi^{\prime}(\xi)}{(\xi\wedge
1)^{2-\sigma}}=-C_{1}\,\theta\,C_{2}\,e^{-C_{2}\xi^{\theta}}\xi^{\theta-2}\left\\{4\lambda(1-\theta)+4\lambda\theta
C_{2}\xi^{\theta}-C(\xi^{\sigma-1}\vee\xi)\right\\}\,.$

Now we choose $\theta<\sigma-1$, and we fix $C_{2}$ sufficiently large so that

$4\lambda\theta C_{2}\xi^{\theta}-C(\xi^{\sigma-1}\vee\xi)\geq
0\quad\forall\xi\in(0,r_{1})\,.$

With this choice of $C_{2}$ (which only depends on
$\lambda,\theta,\sigma,C,r_{1}$), we finally estimate

$\displaystyle\forall\xi\leq r\leq r_{1},\qquad
4\lambda\psi^{\prime\prime}(\xi)+C\,\frac{\psi^{\prime}(\xi)}{(\xi\wedge
1)^{2-\sigma}}$ $\displaystyle\leq-
C_{1}\,\theta\,C_{2}\,e^{-C_{2}\xi^{\theta}}\xi^{\theta-2}4\lambda(1-\theta)$
$\displaystyle\leq-C_{1}\,L\,r^{\theta-2}\qquad\forall r<r_{1}$

for some constant $L>0$ independent of $C_{1}$. Then from (3.25) we deduce

$C_{1}\left(c_{d,\sigma}\,L\,r_{1}^{\theta-\sigma}-\omega(1+2k_{0})\right)-2k_{0}[u_{0}]_{\langle
x\rangle^{k}}-2\varepsilon k_{0}(1+2k_{0})\leq 0\,.$

Choosing $\omega$ sufficiently small and $C_{1}$ suitably large ($C_{1}\gtrsim
n([u_{0}]_{\langle x\rangle^{k}}+\varepsilon k_{0})$ for some large $n$) we
get a contradiction. This shows that (3.16) holds true; then letting
$\varepsilon\to 0$ and recalling our choices of the parameters $K,C_{1}$, we
proved that

$\displaystyle u(t,x)-u(t,y)$ $\displaystyle\leq e^{-\omega
t}\left\\{K\,(\varphi(x)+\varphi(y))+\psi(|x-y|)\right\\}$
$\displaystyle\quad\leq e^{-\omega t}\,(K+C_{1})(\varphi(x)+\varphi(y))\leq
e^{-\omega t}\,\tilde{K}\,[u_{0}]_{\langle
x\rangle^{k}}(\varphi(x)+\varphi(y))$

for some constant $\tilde{K}$, which yields (3.14).

(B) Case $\lambda_{0}=0$ and (2.4) with $\sigma\in(0,1]$.  In this case we use
(2.14) in (3.17) and, rather than (3.21), we deduce:

$\begin{split}&-(2Kk_{0}+2\varepsilon k_{0})\leq(c_{0}(r\wedge
1)^{1-\sigma+\delta}+\sigma_{1}(\sigma_{1}r\wedge
2\sigma_{0}))\psi^{\prime}(r)+\omega\,\psi(r)\\\
&\qquad+4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,.\end{split}$ (3.26)

Similarly as in (3.24), we estimate

$\displaystyle(c_{0}(r\wedge 1)^{1-\sigma+\delta}+\sigma_{1}(\sigma_{1}r\wedge
2\sigma_{0}))\psi^{\prime}(r)\leq
C\int_{0}^{1}(1-s)\int_{B^{-}}\frac{\psi^{\prime}(r+2s(\widehat{x-y}\cdot
z))}{(r+2s(\widehat{x-y}\cdot z))\wedge
1)^{1-\delta}}|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds$

for some constant $C$ only depending on
$d,\sigma,\sigma_{0},\sigma_{1},c_{0}$. Hence we get from (3.26), where we
also use $\psi\leq C_{1}$,

$-(2Kk_{0}+2\varepsilon k_{0}+\omega
C_{1})\leq\int_{0}^{1}(1-s)\int_{B^{-}}\left\\{4\lambda\psi^{\prime\prime}(\xi_{s,z})+C\,\frac{\psi^{\prime}(\xi_{s,z})}{(\xi_{s,z}\wedge
1)^{1-\delta}}\right\\}|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds$ (3.27)

where $\xi_{s,z}=r+2s(\widehat{x-y}\cdot z)$. Recall that, for $z\in B^{-}$,
we have $r/2\leq\xi_{s,z}\leq r<r_{1}$. Now we compute

$4\lambda\psi^{\prime\prime}(\xi)+C\,\frac{\psi^{\prime}(\xi)}{(\xi\wedge
1)^{1-\delta}}=-C_{1}\,\theta\,C_{2}\,e^{-C_{2}\xi^{\theta}}\xi^{\theta-2}\left\\{4\lambda(1-\theta)+4\lambda\theta
C_{2}\xi^{\theta}-C(\xi^{\delta}\vee\xi)\right\\}\,.$

and we fix $C_{2}$ sufficiently large so that $4\lambda\theta
C_{2}\xi^{\theta}-C(\xi^{\delta}\vee\xi)\geq 0$ for every $\xi\in(0,r_{1})$.
To this purpose it is enough to choose $\theta<\delta$ and $C_{2}$
sufficiently large depending on $r_{1},C,\lambda,\theta$.

With this choice of $C_{2}$, we finally estimate

$\displaystyle
4\lambda\psi^{\prime\prime}(\xi_{s,z})+C\frac{\psi^{\prime}(\xi_{s,z})}{(\xi_{s,z}\wedge
1)^{1-\delta}}$
$\displaystyle\leq-4\lambda(1-\theta)C_{1}\,\theta\,C_{2}\,e^{-C_{2}\xi_{s,z}^{\theta}}\xi^{\theta-2}$
$\displaystyle\leq-4\lambda(1-\theta)C_{1}\,\theta\,C_{2}\,e^{-C_{2}r_{1}^{\theta}}r^{\theta-2}\qquad\forall
r<r_{1}\,.$

Then from (3.27) we deduce

$-(2Kk_{0}+2\varepsilon k_{0}+\omega C_{1})\leq-
c_{d,\sigma}\,4\lambda(1-\theta)C_{1}\,\theta\,C_{2}\,e^{-C_{2}r_{1}^{\theta}}r_{1}^{\theta-\sigma}$

Recall from (3.20), and the definition of $\psi$, that we have

$K=\omega
C_{1}+(c_{0}+2\sigma_{0}\sigma_{1})C_{1}C_{2}\,e^{-C_{2}r_{1}^{\theta}}\theta\,r_{1}^{\theta-1}+2\varepsilon
k_{0}+[u_{0}]_{\langle x\rangle^{k}}$

so that we can rephrase what obtained as

$\begin{split}&C_{1}\left([4\lambda(1-\theta)c_{d,\sigma}\,r_{1}^{1-\sigma}-2k_{0}(c_{0}+2\sigma_{0}\sigma_{1})]C_{2}\,e^{-C_{2}r_{1}^{\theta}}\theta\,r_{1}^{\theta-1}-\omega(1+2k_{0})\right)\\\
&\qquad\qquad\qquad\leq 2k_{0}[u_{0}]_{\langle x\rangle^{k}}+2\varepsilon
k_{0}(1+2k_{0})\,.\end{split}$ (3.28)

Suppose now that $\sigma<1$; then we first choose $r_{1}$ sufficiently large
so that
$4\lambda(1-\theta)c_{d,\sigma}\,r_{1}^{1-\sigma}-2k_{0}(c_{0}+2\sigma_{0}\sigma_{1})>0$.
Next we choose $C_{2}$ as we said, depending on $r_{1}$, and finally, choosing
$\omega$ sufficiently small and $C_{1}$ suitably large ($C_{1}\gtrsim
n(1+2k_{0})([u_{0}]_{\langle x\rangle^{k}}+2\varepsilon k_{0})$) we get a
contradiction. This shows that (3.16) holds true; then letting $\varepsilon\to
0$ and recalling our choices of the parameters $K,C_{1}$, we proved (3.14).

Finally, we are left with the case $\sigma=1$. To handle this case we observe
that, with the strategy used above, we have
$4\lambda\psi^{\prime\prime}(\xi)+C\frac{\psi^{\prime}(\xi)}{(\xi\wedge
1)^{1-\delta}}\leq 0$, hence

$\displaystyle\int_{0}^{1}(1-s)\int_{B^{-}}\left\\{4\lambda\psi^{\prime\prime}(\xi_{s,z})+C\frac{\psi^{\prime}(\xi_{s,z})}{(\xi_{s,z}\wedge
1)^{1-\delta}}\right\\}|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds$
$\displaystyle\leq\int_{0}^{1}(1-s)\int_{B^{-}}\left\\{4\lambda\psi^{\prime\prime}(\xi_{s,z})+C\frac{\psi^{\prime}(\xi_{s,z})}{(\xi_{s,z}\wedge
1)^{1-\delta}}\right\\}|z|^{2}\frac{dz}{|z|^{d+\sigma-\eta}}ds$

for any $\eta>0$. This allows us to follow the same steps as before, replacing
$r_{1}^{1-\sigma}$ with $r_{1}^{\eta}$ in (3.28).

###### Remark 3.7

We point out that the conditions assumed on the drift $b(t,x)$ can be slightly
relaxed, at the expense of a mild restriction on the power $\langle
x\rangle^{k}$ which is used in the weighted seminorm. To be precise, we can
replace assumptions (2.13) with

$\exists\,\,c_{0}>0\,:\,\quad(b(t,x)-b(t,y))\cdot(x-y)\geq-
c_{0}|x-y|(1\vee|x-y|)^{q}\qquad\forall x,y\in{\mathbb{R}}^{d}\,,\,\forall
t>0\,,$ (3.29)

for some $c_{0}>0,q>0$, and respectively (2.14) with

$\sigma\in(0,1]\,\hbox{and $\exists\,\delta\in(0,1)$: \,
$(b(t,x)-b(t,y))\cdot(x-y)\geq-c_{0}|x-y|(|x-y|\wedge
1)^{1-\sigma+\delta}(1\vee|x-y|)^{q}$.}$ (3.30)

Then the conclusion of Theorems 3.5 and Theorem 3.6 remain valid for any $k$
such that $\gamma+k-2\geq q$ (notice that the above Theorems correspond to
$q=0$, where any $k>0,\gamma\geq 2$ satisfy this condition).

To provide with this generalization, the only difference in the proof occurs
for large values of $|x-y|$. This is a straightforward variation, since using

$\mathcal{L}[\varphi](x)+b(t,x)\cdot
D\varphi(x)-\omega\varphi(x)\geq\frac{\alpha
k}{2}|x|^{\gamma-2}\varphi(x)\qquad\hbox{for large $|x|$}$

one can change (3.19) into the following estimate

$K\,\frac{\alpha\,k}{2}\left(\frac{r}{2}\right)^{\gamma-2+k}-2\varepsilon
k_{0}-(c_{0}\,r^{q}+2\sigma_{0}\sigma_{1})\psi^{\prime}(r_{1})-\omega
C_{1}<0\qquad\forall r\geq r_{1}.$

which, whenever $\gamma+k-2\geq q$, still implies a choice of $K$ of the same
order as (3.20).

This remark shows that the classical condition which is mostly used in the
probabilistic approach, say that

$(b(t,x)-b(t,y))\cdot(x-y)\geq-c_{0}|x-y|^{2}\qquad\forall
x,y\in{\mathbb{R}}^{d}\,,\,\forall t>0$

is not only allowed in our setting, but can be improved e.g. into

$(b(t,x)-b(t,y))\cdot(x-y)\geq-c_{0}(|x-y|\vee|x-y|^{2})\qquad\forall
x,y\in{\mathbb{R}}^{d}\,,\,\forall t>0\,.$

###  3.3 Sub-exponential rate

We now come at the slowly confining case, where, for Lyapunov functions
$\varphi$ of power type, we have $\mathcal{L}^{b}(\varphi)\to\infty$ but
$\frac{\mathcal{L}^{b}(\varphi)}{\varphi}\to 0$. This occurs when $\gamma<2$
in (2.12). In this case one can prove a sub-exponential decay rate (so-called
sub-geometrical decay, from discrete models). We start with a general
statement of polynomial decay of the weighted oscillation. A similar strategy
as in Theorem 3.6 will work, but provided the weight is sufficiently strong.
As in Section 3.2, we give first a separate statement for the purely local
case: indeed, if the diffusion is local the full range of $\gamma\in(0,2)$ can
be exploited and we have estimates even in the regime of degenerate confining,
say when $\gamma\in(0,1)$ in (2.12).

###### Theorem 3.8

Let ${\mathcal{L}}_{0}$ be defined by (2.1) where $\Sigma$ satisfies (2.2) and
$\lambda_{0}>0$. Assume that $b$ satisfies (2.12)-(2.13) with
$\gamma\in(0,2)$. Let $u$ be a (viscosity) solution of

$\begin{cases}\partial_{t}u+{\mathcal{L}}_{0}[u]+b(t,x)\cdot Du=0&\text{ in
}(0,T)\times{\mathbb{R}}^{d},\\\ u(0,x)=u_{0}(x),\qquad\,&\text{ in
}{\mathbb{R}}^{d}\end{cases}$ (3.31)

such that $|u(t,x)|\leq C(1+|x|)^{m}$ for some $m>0$ and $C>0$ (uniformly in
time).

Then, for any $k>2-\gamma$ and $u_{0}\in L^{\infty}(\langle x\rangle^{-k})$,
and for any $\bar{k}>k$, there exists a positive constant $M>0$ (depending on
$\alpha,\gamma,c_{0},d,\lambda_{0},\sigma_{0},\sigma_{1},k,\bar{k}$) such that
$u$ satisfies

$[u(t)]_{\langle x\rangle^{\bar{k}}}\leq M\,(1+t)^{-q}\,[u_{0}]_{\langle
x\rangle^{k}}\qquad\hbox{where $q=\frac{\bar{k}-k}{2-\gamma}$.}$ (3.32)

For nonlocal diffusions we have a similar result, although the presence of the
jump part (in particular, the behavior of the kernel at infinity) restricts
the result to the case that $1<\gamma<2$, as observed in Remark 3.2.

###### Theorem 3.9

Let ${\mathcal{L}}$ be defined by (2.5) where $\Sigma$ satisfies (2.2), the
Levy measure in ${\mathcal{I}}$ satisfies (2.4), and $\lambda_{0}+\lambda>0$.
Assume that $b$ satisfies (2.12) with $\gamma\in(2-\sigma,2)$ such that
$\gamma>1$, and that either of the following conditions hold:

(i) $\lambda_{0}>0$ or $\sigma\in(1,2)$, and (2.13) holds true.

(ii) $\lambda_{0}=0$ and $\sigma\in(0,1]$, and (2.14) holds true.

Let $u$ be a (viscosity) solution of (2.7) such that $|u(t,x)|\leq
C(1+|x|)^{m}$ for some $m\in(0,\sigma)$ and $C>0$ (uniformly in time).

Then, for any $k\in(0,\sigma)$ with $k>2-\gamma$ and $u_{0}\in
L^{\infty}(\langle x\rangle^{-k})$, and for any $\bar{k}>k$ with
$k<\bar{k}<\sigma$, there exists a positive constant $M>0$ (depending on
$\alpha,\gamma,\sigma,(\lambda+\lambda_{0})^{-1},\sigma_{0},\sigma_{1},\Lambda,c_{0},d,k,\bar{k}$)
such that $u$ satisfies

$[u(t)]_{\langle x\rangle^{\bar{k}}}\leq M\,(1+t)^{-q}\,[u_{0}]_{\langle
x\rangle^{k}}\qquad\hbox{where $q=\frac{\bar{k}-k}{2-\gamma}$.}$ (3.33)

Proof of Theorem 3.8 and Theorem 3.9. We give the proof for the nonlocal case,
the other being similar, except for the range of $\gamma$ which is allowed in
the Lyapunonv function (from (3.1)).

Let us set $\varphi(x)=\langle x\rangle^{k}$ ($k<\sigma$), $\Phi(x)=\langle
x\rangle^{p}$ ($m<p<\sigma$). With the same notations of Theorem 3.6, and
$\psi$ given by (3.15), we aim at proving that

$u(t,x)-u(t,y)\leq(1+t)^{-q}\left\\{K\,(\varphi(x)+\varphi(y))+\psi(|x-y|)+\varepsilon[\Phi(t,x)+\Phi(t,y)]\right\\}+\frac{\varepsilon}{T-t}$
(3.34)

for every $x,y\in{\mathbb{R}}^{d},t>0$, and for every $\varepsilon>0$
sufficiently small.

As usual, we observe that if (3.34) were not true, then the function

$u(t,x)-u(t,y)-(1+t)^{-q}\underbrace{\left\\{K[\varphi(x)+\varphi(y)]+\psi(|x-y|)+\varepsilon[\Phi(t,x)+\Phi(t,y)]\right\\}}_{\zeta(x,y)}-\frac{\varepsilon}{T-t}$

admits a point $(t,x,y)$ of global maximum with $x\neq y,t<T$ and $t>0$,
provided $K\geq[u_{0}]_{\langle x\rangle^{k}}$.

Then we use Theorem 2.3, writing the viscosity inequalities for $u$, and we
proceed as in Theorem 3.6 using Lemma 3.3 and Lemma 3.4. We end up with a
similar kind of inequality as (3.17):

$\begin{split}&\frac{\varepsilon}{(T-t)^{2}}+(1+t)^{-q}\,K\left\\{\mathcal{L}^{b}[\varphi](x)-\frac{q}{(1+t)}\varphi(x)+\mathcal{L}^{b}[\varphi](y)-\frac{q}{(1+t)}\varphi(y)\right\\}\\\
&+(1+t)^{-q}\,\varepsilon\left(\mathcal{L}^{b}[\Phi](x)-\frac{q}{(1+t)}\Phi(x)+\mathcal{L}^{b}[\Phi](y)-\frac{q}{(1+t)}\Phi(y)\right)\\\
&+(1+t)^{-q}\left((b(t,x)-b(t,y))\cdot\widehat{x-y}\right)\psi^{\prime}(r)-q(1+t)^{-q-1}\psi(r)\\\
&\qquad\leq(1+t)^{-q}\left\\{4\lambda_{0}\psi^{\prime\prime}(r)+(2\sigma_{0}\wedge\sigma_{1}r)\sigma_{1}\,\psi^{\prime}(r)\right\\}\\\
&\qquad+(1+t)^{-q}4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,.\end{split}$ (3.35)

where $r=|x-y|$. We start by considering the case $q=0$. As we did in Theorem
3.6, we first suppose that $r=|x-y|\geq r_{1}$ for a sufficiently large
$r_{1}>1$; in this range assumptions (2.13) and (2.14) reduce to the same.
Moreover, since one between $|x|,|y|$ is larger than $r_{1}/2$, and since
$k>2-\gamma$ implies ${\mathcal{L}}^{b}[\varphi]\to+\infty$ as $|x|\to\infty$
(from Lemma 3.1), we can choose $r_{1}>1$ such that

$\mathcal{L}^{b}[\varphi](x)+\mathcal{L}^{b}[\varphi](y)\geq\frac{\alpha
k}{2}\left(\langle x\rangle\vee\langle y\rangle\right)^{k+\gamma-2}\geq 1\,.$

Moreover, there exists $k_{0}$ such that $\mathcal{L}^{b}[\varphi]\geq-k_{0}$,
and so is for $\Phi$. Therefore, from (3.35) written with $q=0$ we deduce
(using (2.13) or (2.14) and dropping the terms with $\psi^{\prime\prime}$)

$K-2\varepsilon
k_{0}\leq[c_{0}+2\sigma_{0}\sigma_{1}]\,\psi^{\prime}(r)\qquad\hbox{if $r\geq
r_{1}>1$}\,.$

Since, by concavity, $\psi^{\prime}(r)\leq\psi^{\prime}(r_{1})$ for $r\geq
r_{1}$, the above inequality cannot hold if we choose

$K=2\varepsilon
k_{0}+(c_{0}+2\sigma_{0}\sigma_{1})\psi^{\prime}(r_{1})+[u_{0}]_{\langle
x\rangle^{k}}\,.$

On the other hand, for $r<r_{1}$, just using $\mathcal{L}^{b}[\varphi]\geq-
k_{0},\mathcal{L}^{b}[\phi]\geq-k_{0}$, we get

$\displaystyle\left((b(t,x)-b(t,y))\cdot\widehat{x-y}\right)\psi^{\prime}(r)<4\lambda_{0}\psi^{\prime\prime}(r)+(2\sigma_{0}\wedge\sigma_{1}r)\sigma_{1}\,\psi^{\prime}(r)$
$\displaystyle\qquad+4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds+2Kk_{0}+2\varepsilon k_{0}\,.$

Henceforth, using $\lambda_{0}+\lambda>0$, we argue as in Theorem 3.6 in order
to get at a contradiction with suitable choices of $C_{2}$ (depending on
$r_{1}$) and $C_{1}\gtrsim[u_{0}]_{\langle x\rangle^{k}}$. This proves (3.34)
for the case $q=0$. After letting $\varepsilon\to 0$, we deduce that, for any
$k>2-\gamma$, we have

$u(t,x)-u(t,y)\leq K\,(\langle x\rangle^{k}+\langle
y\rangle^{k})+\psi(|x-y|)\leq K\,(\langle x\rangle^{k}+\langle
y\rangle^{k})+C_{1}$

which implies (after the choices of $K,C_{1}$)

$[u(t)]_{\langle x\rangle^{k}}\leq C\,[u_{0}]_{\langle
x\rangle^{k}}\qquad\forall t>0\,,$ (3.36)

where $C$ is a constant only depending on
$\alpha,\gamma,(\lambda+\lambda_{0})^{-1},c_{0},d,\sigma_{0},\sigma_{1},\sigma,\delta$
and $k$. In particular, we have proved that, for all weights $\langle
x\rangle^{k}$ with $2-\gamma<k<\sigma$, the weighted oscillation of $u$ at
time $t$ is controlled by the initial one.

Now we improve the estimate, by proving (3.34) with $q\neq 0$. To this
purpose, we observe that, on the maximum point $(t,x,y)$, we have

$(1+t)^{-q}[\varphi(x)+\varphi(y)]K\leq u(t,x)-u(t,y)\leq[\langle
x\rangle^{k}+\langle y\rangle^{k}][u(t)]_{\langle x\rangle^{k}}\,.$

Choosing $\varphi(x)=\langle x\rangle^{\bar{k}}$, with $k<\bar{k}<\sigma$, due
to (3.36) we estimate

$(1+t)^{-q}K\leq C\,\frac{[\langle x\rangle^{k}+\langle
y\rangle^{k}]}{[\langle x\rangle^{\bar{k}}+\langle
y\rangle^{\bar{k}}]}[u_{0}]_{\langle x\rangle^{k}}$

and choosing $q=\frac{\bar{k}-k}{2-\gamma}$ we get

$\frac{1}{1+t}\leq C\,\left(\frac{[u_{0}]_{\langle
x\rangle^{k}}}{K}\right)^{\frac{1}{q}}\frac{1}{[\langle
x\rangle^{2-\gamma}+\langle y\rangle^{2-\gamma}]}$ (3.37)

for a possibly different $C$ depending on the same constants as before, as
well as on $q$. Thanks to (3.37), there exists a sufficiently large constant
$L$ such that if we choose $K>L\,[u_{0}]_{\langle x\rangle^{k}}$ then we have

$\mathcal{L}^{b}[\varphi](x)-\frac{q}{(1+t)}\varphi(x)\to+\infty\qquad\hbox{as
$|x|\to\infty$,}$

where we used Lemma 3.1 (and the fact that $\bar{k}>2-\gamma$). In particular
we have $\mathcal{L}^{b}[\varphi](x)-\frac{q}{(1+t)}\varphi(x)\geq-k_{0}$ for
some $k_{0}>0$. Similarly we may reason for $\Phi(x)=\langle x\rangle^{p}$,
where we can assume, without loss of generality, that $p>2-\gamma$. Therefore,
we can choose $r_{1}$ sufficiently large so that, whenever $r=|x-y|\geq r_{1}$
we have

$\mathcal{L}^{b}[\varphi](x)-\frac{q}{(1+t)}\varphi(x)+\mathcal{L}^{b}[\varphi](y)-\frac{q}{(1+t)}\varphi(y)\geq
1\,\quad\hbox{and}\,\quad\mathcal{L}^{b}[\Phi](x)-\frac{q}{(1+t)}\Phi(x)\geq-
k_{0}\,.$

Hence we deduce from (3.35) (using as before the concavity of $\psi$)

$\begin{split}K&<2\varepsilon
k_{0}+[c_{0}+2\sigma_{0}\sigma_{1}]\psi^{\prime}(r)+\psi(r)\,\frac{q}{1+t}\\\
&<2\varepsilon
k_{0}+[c_{0}+2\sigma_{0}\sigma_{1}]\psi^{\prime}(r_{1})+C_{1}\,C\left(\frac{[u_{0}]_{\langle
x\rangle^{k}}}{K}\right)^{\frac{1}{q}}\qquad\forall r\geq r_{1}\,,\end{split}$
(3.38)

where we used estimate (3.37), and that $\psi(r)\leq
C_{1},\psi^{\prime}(r)\leq\psi^{\prime}(r_{1})$ for $r>r_{1}$. Here $C$ is a
generic constant independent of $r_{1},K,C_{1},C_{2},u_{0}$. Henceforth, we
fix

$C_{1}:=n[u_{0}]_{\langle x\rangle^{k}}+n^{\frac{1}{q+1}}\,\varepsilon
k_{0}(1+4k_{0})$ (3.39)

where $n$ is a sufficiently large number to be chosen later. Moreover, we also
fix

$K:=L\,(1+n^{\frac{q}{q+1}})[u_{0}]_{\langle x\rangle^{k}}+4\varepsilon
k_{0}+[c_{0}+2\sigma_{0}\sigma_{1}]\psi^{\prime}(r_{1})$ (3.40)

where $L$ is a sufficiently large constant. With this choice of $C_{1},K$, we
have

$C_{1}\,C\left(\frac{[u_{0}]_{\langle
x\rangle^{k}}}{K}\right)^{\frac{1}{q}}\leq
C_{1}\,C\frac{1}{\left(L\,(1+n^{\frac{q}{q+1}})\right)^{\frac{1}{q}}}\leq\frac{C}{L^{\frac{1}{q}}}\left\\{n^{\frac{q}{q+1}}[u_{0}]_{\langle
x\rangle^{k}}+\varepsilon k_{0}(1+4k_{0})\right\\}$ (3.41)

and (3.38) implies

$K\leq 2\varepsilon
k_{0}+[c_{0}+2\sigma_{0}\sigma_{1}]\psi^{\prime}(r_{1})+\frac{C}{L^{\frac{1}{q}}}\left\\{n^{\frac{q}{q+1}}[u_{0}]_{\langle
x\rangle^{k}}+\varepsilon k_{0}(1+4k_{0})\right\\}\,.$

On account of (3.40), this inequality cannot hold if $L$ is chosen suitably
large (only depending on the constant $C$ and $k_{0}$).

We are left to analyse the case that $r<r_{1}$, which we split in the two
ranges of $\sigma$. (A) Case $\lambda_{0}>0$ or (2.4) with $\sigma\in(1,2)$.

In this case we use (2.13) and we deduce from (3.35)

$\begin{split}&-(2Kk_{0}+2\varepsilon k_{0})\leq
4\lambda_{0}\psi^{\prime\prime}(r)+[c_{0}+(2\sigma_{0}\wedge\sigma_{1}r)\sigma_{1}]\psi^{\prime}(r)+\frac{q}{(1+t)}\psi(r)\\\
&\qquad+4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,.\end{split}$ (3.42)

We consider the case that $\lambda_{0}=0$ (otherwise, the proof would follow
easily by dropping the term with $\lambda$). Then, using
$\psi^{\prime}(r_{1})\leq\psi^{\prime}(r)$ (because $r<r_{1}$) we estimate $K$
from (3.40) as

$K\leq L\,(1+n^{\frac{q}{q+1}})[u_{0}]_{\langle x\rangle^{k}}+4\varepsilon
k_{0}+[c_{0}+2\sigma_{0}\sigma_{1}]\psi^{\prime}(r)$

and we insert this estimate in (3.42) obtaining

$\begin{split}&-\left\\{2\varepsilon
k_{0}(1+4k_{0})+2k_{0}\,L\,(1+n^{\frac{q}{q+1}})[u_{0}]_{\langle
x\rangle^{k}}\right\\}\leq(1+2k_{0})[c_{0}+2\sigma_{0}\sigma_{1}]\psi^{\prime}(r)+\frac{q}{(1+t)}\psi(r)\\\
&\qquad+4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,,\qquad\forall r<r_{1}\,.\end{split}$
(3.43)

Now we observe that (3.37), (3.39) and (3.40), together with $\psi(r)\leq
C_{1}$, imply

$\displaystyle\frac{q}{1+t}\psi\leq q\,C\,C_{1}\left(\frac{[u_{0}]_{\langle
x\rangle^{k}}}{K}\right)^{\frac{1}{q}}$ $\displaystyle\leq
q\,\,\frac{C}{L^{\frac{1}{q}}}\left\\{n^{\frac{q}{q+1}}[u_{0}]_{\langle
x\rangle^{k}}+\varepsilon k_{0}(1+4k_{0})\right\\}$ $\displaystyle\leq
2k_{0}\,L\,(1+n^{\frac{q}{q+1}})[u_{0}]_{\langle x\rangle^{k}}+2\varepsilon
k_{0}(1+4k_{0})$

where last inequality follows by choosing $L$ large enough. Using this
estimate, we deduce from (3.43)

$\begin{split}&-4\varepsilon
k_{0}(1+4k_{0})-4k_{0}\,L\,(1+n^{\frac{q}{q+1}})[u_{0}]_{\langle
x\rangle^{k}}\leq(1+2k_{0})[c_{0}+2\sigma_{0}\sigma_{1}]\psi^{\prime}(r)\\\
&\qquad\qquad\quad+4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\qquad\forall r<r_{1}\,.\end{split}$
(3.44)

We handle the right-hand side exactly as in Theorem 3.6. In particular, with a
convenient choice of $C_{2}$ we estimate

$\displaystyle(1+2k_{0})[c_{0}+2\sigma_{0}\sigma_{1}]\psi^{\prime}(r)$
$\displaystyle+4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds$
$\displaystyle\leq-\tilde{C}\,C_{1}\,r_{1}^{\theta-\sigma}\qquad\forall
r<r_{1}$

for some constant $\tilde{C}$ independent of $C_{1},K$. Inserting this
inequality in (3.44), we obtain

$-4\varepsilon
k_{0}(1+4k_{0})-4k_{0}\,L\,(1+n^{\frac{q}{q+1}})[u_{0}]_{\langle
x\rangle^{k}}\leq-C_{1}\,\tilde{C}\,r_{1}^{\theta-\sigma}\,,\qquad\forall
r<r_{1}\,.$ (3.45)

Recalling the value of $C_{1}$ from (3.39), we see that the above inequality
cannot hold, if we finally choose $n$ sufficiently large (depending on all
previous parameters).

Finally, we proved that (3.34) holds true, where letting $\varepsilon\to 0$
and recalling (3.39)–(3.40), we deduce that

$\begin{split}u(t,x)-u(t,y)&\leq(1+t)^{-q}\left\\{K\,(\varphi(x)+\varphi(y))+C_{1}\right\\}\\\
&\leq(1+t)^{-q}M\,[u_{0}]_{\langle x\rangle^{k}}[\langle
x\rangle^{\bar{k}}+\langle y\rangle^{\bar{k}}]\qquad\forall
x,y\in{\mathbb{R}}^{d}\,,\,t>0\,.\end{split}$ (3.46)

Hence (3.33) is proved.

(B) Case $\lambda_{0}=0$ and (2.4) with $\sigma\in(0,1]$.  If
$\sigma\in(0,1)$, we use (2.14) and we get from (3.35)

$\displaystyle-(2Kk_{0}+2\varepsilon k_{0})\leq(c_{0}(r\wedge
1)^{1-\sigma+\delta}+(2\sigma_{0}\wedge\sigma_{1}r)\sigma_{1})\psi^{\prime}(r)+\frac{q}{(1+t)}\psi(r)$
$\displaystyle\qquad+4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,.$

With the same arguments as in Theorem 3.6, with a convenient choice of $C_{2}$
we estimate

$\displaystyle(c_{0}(r\wedge
1)^{1-\sigma+\delta}+(2\sigma_{0}\wedge\sigma_{1}r)\sigma_{1})\psi^{\prime}(r)$
$\displaystyle+4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds$
$\displaystyle\leq-4\lambda(1-\theta)c_{d,\sigma}\,C_{1}\,\theta\,C_{2}\,e^{-C_{2}r_{1}^{\theta}}r_{1}^{\theta-\sigma}\qquad\forall
r<r_{1}\,.$

Hence we get

$-(2Kk_{0}+2\varepsilon
k_{0})\leq\frac{q}{(1+t)}\psi(r)-4\lambda(1-\theta)c_{d,\sigma}\,C_{1}\,\theta\,C_{2}\,e^{-C_{2}r_{1}^{\theta}}r_{1}^{\theta-\sigma}\,.$
(3.47)

Here we recall the explicit choice of $K$ (3.40), which (using the explicit
value of $\psi^{\prime}$) reads as

$K=L\,(1+n^{\frac{q}{q+1}})[u_{0}]_{\langle x\rangle^{k}}+4\varepsilon
k_{0}+[c_{0}+2\sigma_{0}\sigma_{1}]C_{1}\,C_{2}\,\theta\,e^{-C_{2}r_{1}^{\theta}}r_{1}^{\theta-1}\,.$

Moreover, we estimate as before $\frac{q}{(1+t)}\psi$ using (3.37) and (3.41),
and we obtain from (3.47)

$\begin{split}&C_{1}\left\\{\theta
C_{2}e^{-C_{2}r_{1}^{\theta}}\,r_{1}^{\theta-1}[4\lambda(1-\theta)c_{d,\sigma}r_{1}^{1-\sigma}-2k_{0}(c_{0}+2\sigma_{0}\sigma_{1})]-\frac{\tilde{C}}{L^{\frac{1}{q}}(1+n^{\frac{1}{q+1}})}\right\\}\\\
&\qquad\leq 2\varepsilon
k_{0}(1+4k_{0})+2k_{0}\,L\,(1+n^{\frac{q}{q+1}})[u_{0}]_{\langle
x\rangle^{k}}\end{split}$ (3.48)

for some $\tilde{C}$ independent of $C_{1},C_{2},r_{1}$. Now we observe that
$r_{1}$ can be chosen sufficiently large so that

$4\lambda(1-\theta)c_{d,\sigma}r_{1}^{1-\sigma}-2k_{0}(c_{0}+2\sigma_{0}\sigma_{1})>0\,.$

Finally, having fixed $r_{1}$, and then $C_{2}$ accordingly, and recalling the
choice of $C_{1}$ from (3.39), we see that (3.48) cannot hold for $n$
sufficiently large. This concludes the argument and proves that (3.34) holds
true. From that, letting $\varepsilon\to 0$, we deduce (3.46), and then
(3.33).

Finally, we only mention that the case $\sigma=1$ is dealt with as in Theorem
3.6 obtaining a slight variation of (3.48) where $1-\sigma$ is replaced by
$\eta$ for some small $\eta>0$. This is enough to conclude exactly as before.

###### Remark 3.10

We point out that the same generalization as mentioned in Remark 3.7 on the
growth condition of the drift also applies for the content of Theorem 3.6.

##  4 Decay of Fokker-Planck equations

We deduce now the decay estimates for the general Fokker-Planck equation
(2.10). Let us recall that, in the setting of our assumptions, problem (2.10)
admits a unique solution, according to Theorem 2.5.

As a first step, we recall the following characterization of weighted
seminorms. The following lemma is taken from [27]; we give a slightly
different statement for our purposes (and a proof for the reader’s
convenience).

###### Lemma 4.1

Let $\varphi$ be a continuous function such that $\varphi(x)\geq 1$ for all
$x\in{\mathbb{R}}^{d}$. For any function $u$ such that $u\in
L^{\infty}(\varphi^{-1})$, we have

$\sup_{x,y\in{\mathbb{R}}^{2d}}\,\,\frac{|u(x)-u(y)|}{\varphi(x)+\varphi(y)}=\inf_{c\in{\mathbb{R}}}\,\,\|u+c\|_{L^{\infty}(\varphi^{-1})}$

Proof. Let us call
$M:=\sup\limits_{x,y\in{\mathbb{R}}^{2d}}\,\,\frac{|u(x)-u(y)|}{\varphi(x)+\varphi(y)}$
and $m:=\inf\limits_{c\in{\mathbb{R}}}\,\,\|u+c\|_{L^{\infty}(\varphi^{-1})}$.
Since

$\displaystyle|u(x)-u(y)|$ $\displaystyle\leq|u(x)+c|+|u(y)+c|$
$\displaystyle\leq\varphi(x)\|u+c\|_{L^{\infty}(\varphi^{-1})}+\varphi(y)\|u+c\|_{L^{\infty}(\varphi^{-1})}$

we immediately deduce that $M\leq m$.

For the reverse inequality, define $c:=\inf_{{\mathbb{R}}^{d}}\,[M\varphi-u]$.
Notice that $c$ is finite because $M\varphi(x)-u(x)\geq-M\,\varphi(y)-u(y)$
for any $y\in{\mathbb{R}}^{d}$. By definition, we have

$\frac{u(x)+c}{\varphi(x)}\leq M\,.$

We also estimate, using the definition of $M$ and $c$,

$\displaystyle\frac{u(x)+c}{\varphi(x)}$
$\displaystyle=\inf_{y\in{\mathbb{R}}^{d}}\,\left[\frac{u(x)+M\varphi(y)-u(y)}{\varphi(x)}\right]$
$\displaystyle\geq\inf_{y\in{\mathbb{R}}^{d}}\left[\frac{M\varphi(y)}{\varphi(x)}-M\left(\frac{\varphi(x)+\varphi(y)}{\varphi(x)}\right)\right]=-M\,.$

Hence we deduce that $\|u+c\|_{L^{\infty}(\varphi^{-1})}\leq M$, which yields
$m\leq M$.

Thanks to Lemma 4.1, the decay estimates follow by duality using the results
of the previous Section.

###### Theorem 4.2

Let ${\mathcal{L}}$ be defined by (2.5) where $\Sigma$ satisfies (2.2), the
Levy measure in ${\mathcal{I}}$ satisfies (2.4), and $\lambda_{0}+\lambda>0$.
Assume that $b\in C^{0}(Q_{T})$ satisfies (2.12) and either of the following
conditions: (i) $\lambda_{0}>0$ or $\sigma\in(1,2)$, and (2.13) holds true.
(ii) $\lambda_{0}=0$, $\sigma\in(0,1]$ and (2.14) holds true. Assume that
$m_{0}\in{\mathcal{M}}_{k}({\mathbb{R}}^{d})$, for some $0<k<\sigma$, and that
${\int_{{\mathbb{R}}^{d}}}dm_{0}=0$, and let $m$ be the unique solution of
(2.10) (in the sense of Definition 2.4). Then we have:

  * (a)

If $\gamma\geq 2$ in (2.12), then

$\|m(t)\|_{{\mathcal{M}}_{k}}\leq K\,e^{-\omega
t}\,\|m_{0}\|_{{\mathcal{M}}_{k}}$ (4.1)

for some constant $K,\omega$ only depending on
$\alpha,\sigma,(\lambda_{0}+\lambda)^{-1},\sigma_{0},\sigma_{1},\Lambda,\delta,d,c_{0},k$.

  * (b)

If $\gamma\in(\max((2-\sigma),1),2)$ in (2.12), then for any
$k<\bar{k}<\sigma$ such that $m_{0}\in{\mathcal{M}}_{\bar{k}}$ we have

$\|m(t)\|_{{\mathcal{M}}_{k}}\leq
K\,(1+t)^{-q}\,\|m_{0}\|_{{\mathcal{M}}_{\bar{k}}}\qquad\hbox{where
$q=\frac{\bar{k}-k}{2-\gamma}$}$ (4.2)

for some $K$ depending on
$\alpha,\gamma,\sigma,(\lambda_{0}+\lambda)^{-1},\sigma_{0},\sigma_{1},\Lambda,\delta,d,c_{0},k,\bar{k}$.

Proof. For $t>0$ we consider the unique viscosity solution of the adjoint
equation

$\begin{cases}-\partial_{t}u+{\mathcal{L}}[u]+b(t,x)\cdot Du=0&\text{ in
}(0,t)\times{\mathbb{R}}^{d},\\\ u(t,x)=\xi,\qquad\,&\text{ in
}{\mathbb{R}}^{d}\end{cases}$ (4.3)

where $\xi\in C({\mathbb{R}}^{d})\cap L^{\infty}(\langle x\rangle^{-k})$. By
Definition 2.4 $m$ satisfies

$\displaystyle\int_{{\mathbb{R}}^{d}}\xi\,dm(t)=\int_{{\mathbb{R}}^{d}}u(0)\,dm_{0}$
$\displaystyle=\int_{{\mathbb{R}}^{d}}(u(0)+c)\,dm_{0}$
$\displaystyle\leq\|m_{0}\|_{{\mathcal{M}}_{k}}\,\|u(0)+c\|_{L^{\infty}(\langle
x\rangle^{-k})}$

where we used that $m_{0}$ has zero average. Taking the infimum with respect
to $c\in{\mathbb{R}}$ and using Lemma 4.1 we get

$\int_{{\mathbb{R}}^{d}}\xi\,dm(t)\leq\|m_{0}\|_{{\mathcal{M}}_{k}}\,[u(0)]_{\langle
x\rangle^{k}}\,.$ (4.4)

If $\gamma\geq 2$, we estimate $[u(0)]_{\langle x\rangle^{k}}$ from Theorem
3.6 and we obtain

$\displaystyle\int_{{\mathbb{R}}^{d}}\xi\,dm(t)$ $\displaystyle\leq
K\,e^{-\omega t}\|m_{0}\|_{{\mathcal{M}}_{k}}\,[\xi]_{\langle x\rangle^{k}}$
$\displaystyle\leq K\,e^{-\omega
t}\|m_{0}\|_{{\mathcal{M}}_{k}}\,\|\xi\|_{L^{\infty}(\langle
x\rangle^{-k})}\,.$

Since $\xi$ is arbitrary, we deduce (4.1). If $\gamma\in(1,2)$ with
$\gamma>2-\sigma$, we argue similarly but rather than (4.4) we estimate as
follows:

$\int_{{\mathbb{R}}^{d}}\xi\,dm(t)\leq\|m_{0}\|_{{\mathcal{M}}_{\bar{k}}}\,[u(0)]_{\langle
x\rangle^{\bar{k}}}\,$

for $\bar{k}>k>2-\gamma$ (notice that $u\in L^{\infty}(\langle x\rangle^{-k})$
implies $u\in L^{\infty}(\langle x\rangle^{-\bar{k}})$). Applying now Theorem
3.9 we obtain

$\int_{{\mathbb{R}}^{d}}\xi\,dm(t)\leq\|m_{0}\|_{{\mathcal{M}}_{\bar{k}}}\,K\,(1+t)^{-q}\,\|\xi\|_{L^{\infty}(\langle
x\rangle^{-k})}\qquad\hbox{where $q=\frac{\bar{k}-k}{2-\gamma}$}$

which yields (4.2).

###### Remark 4.3

The previous result for Levy operators includes the particular case stated for
the fractional Laplacian in the Introduction (Theorem 1.2). We only need to
observe that, if the drift $b$ is locally Lipschitz in $x$ (or alternatively,
under milder conditions involving only ${\rm div}(b)$), the weak solution is
unique (so it coincides with the unique solution given by Theorem 2.5) and
belongs to $L^{1}$ (see e.g. [16], [31], [42]).

In the same spirit of the above remark, the solutions are $L^{1}$ whenever the
operator is second order, as well. In addition, if the operator is local, one
can consider the full range of $\gamma$, including $\gamma\in(0,1)$. The next
result extends Theorem 1.1 which was stated in the Introduction for the case
of Laplace operator. The proof is the same as above, using duality and Theorem
3.5 or Theorem 3.8. This extends results as [39, Thm 3], or [30, Thm 1.2].

###### Theorem 4.4

Let $m_{0}\in L^{1}({\mathbb{R}}^{d},\langle x\rangle^{k})$ be such that
${\int_{{\mathbb{R}}^{d}}}m_{0}\,dx=0$. Let $m$ be a weak solution to the
Cauchy problem

$\begin{cases}\partial_{t}m+{\mathcal{L}}_{0}^{*}[m]-{\rm
div}(b(t,x)m)=0&\text{ in }(0,T)\times{\mathbb{R}}^{d},\\\
m(0,x)=m_{0}(x),\qquad\,&\text{ in }{\mathbb{R}}^{d}\end{cases}$ (4.5)

where we assume that $\lambda_{0}>0$ and (2.2) holds true, and $b(t,x)$ is a
continuous function satisfying (2.12) and (2.13). Then we have:

  * (a)

If $\gamma\geq 2$ in (2.12), then

$\|m(t)\|_{L^{1}(\langle x\rangle^{k})}\leq K\,e^{-\omega
t}\,\|m_{0}\|_{L^{1}({\langle x\rangle^{k}})}$ (4.6)

for some constant $K,\omega$ only depending on
$\alpha,\lambda_{0}^{-1},\sigma_{0},\sigma_{1},d,c_{0},k$.

  * (b)

If $\gamma\in(0,2)$ in (2.12), then for any $k<\bar{k}<\sigma$ such that
$m_{0}\in L^{1}({\mathbb{R}}^{d},\langle x\rangle^{\bar{k}})$ we have

$\|m(t)\|_{L^{1}(\langle x\rangle^{k})}\leq
K\,(1+t)^{-q}\,\|m_{0}\|_{L^{1}({\langle
x\rangle^{\bar{k}}})}\qquad\hbox{where $q=\frac{\bar{k}-k}{2-\gamma}$.}$ (4.7)

for some $K$ depending on
$\alpha,\gamma,\lambda_{0}^{-1},\sigma_{0},\sigma_{1},d,c_{0},k,\bar{k}$.

##  5 Generalizations and further results

We presented so far the results that can be obtained using Lyapunov functions
of power type, resulting in moments decay of measure-valued solutions. Here we
review our strategy in a more abstract form, showing that it can apply to
different choices of Lyapunov functions. The Lyapunov function will determine
the weighted space for the oscillation estimate and, what is more important,
the rate of convergence is consequential to the weighted norm which is used.

There are two main settings that we consider, corresponding to exponential or
sub-exponential decay. Here and below, $\varphi$ denotes a generic positive
smooth function such that $\varphi(x)\to\infty$ as $|x|\to\infty$. There is no
loss of generality in assuming that $\varphi(x)\geq 1$ in ${\mathbb{R}}^{d}$.
We first consider the simplest case of exponential decay in some weighted
norm, namely when some $\varphi$ exists satisfying

$\exists\,\,\omega_{0}>0\,:\quad\liminf_{|x|\to\infty}\,\frac{{\mathcal{L}}^{b}[\varphi]}{\varphi}\geq\omega_{0}\,.$
(H1)

We denote

$\Lambda_{1}:=\\{\varphi\in
C^{2}({\mathbb{R}}^{d}):\,\,\varphi(x)\mathop{\longrightarrow}^{|x|\to\infty}\infty,\,\,\hbox{${\mathcal{I}}(x,[\varphi])$
is well defined and $\varphi$ satisfies (\ref{H1})}\\}.$

Clearly, for every $\varphi\in\Lambda_{1}$, one can choose $\omega$
sufficiently small so that

${\mathcal{L}}^{b}[\varphi]-\omega\varphi\geq\frac{\omega_{0}}{2}\varphi-
k_{0}\qquad\forall x\in{\mathbb{R}}^{d}$

for some $k_{0}>0$. Using the above property, the proof of Theorem 3.6 follows
replacing $\langle x\rangle^{k}$ with $\varphi(x)$, so that we can generalize
the exponential decay estimate for the oscillation, as follows.

###### Theorem 5.1

Let ${\mathcal{L}}$ be defined by (2.5) where $\Sigma$ satisfies (2.2), the
Levy measure in ${\mathcal{I}}$ satisfies (2.4), and $\lambda_{0}+\lambda>0$.
Assume that $b$ satisfies (2.12) and either of the following conditions:

(i) $\lambda_{0}>0$ or $\sigma\in(1,2)$, and (2.13) holds true.

(ii) $\lambda_{0}=0$ and $\sigma\in(0,1]$, and (2.14) holds true.

Assume that there exists $\varphi\in\Lambda_{1}$. Let $u$ be a (viscosity)
solution of (2.7) such that, for some $\Phi\in\Lambda$, one has
$|u(t,x)|=o(\Phi)$ as $|x|\to\infty$, uniformly for $t\in(0,T)$.

Then there exist positive constants $K,\omega>0$ (depending on
$\alpha,\gamma,\sigma,(\lambda+\lambda_{0})^{-1},\sigma_{0},\sigma_{1},c_{0},d,\varphi$)
such that $u$ satisfies

$[u(t)]_{\varphi}\leq K\,e^{-\omega t}\,[u_{0}]_{\varphi}.$ (5.1)

The case of sub-exponential growth is more subtle. This corresponds to the
case that the Lyapunov function $\varphi$ satisfies a condition as follows:

$\begin{split}{\mathcal{L}}^{b}[\varphi]\mathop{\to}^{|x|\to\infty}\infty\,,\quad&\hbox{and
$\,\,\exists\,\,$ a decreasing function
$h:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$, with
$h(r)\mathop{\to}\limits^{r\to\infty}0$\,\,:}\quad\,\\\
&{\mathcal{L}}^{b}[\varphi]\geq h(\varphi)\varphi\qquad\hbox{as
$|x|\to\infty$\,.}\end{split}$ (H2)

Correspondingly, we set

$\Lambda_{2}:=\\{\varphi\in
C^{2}({\mathbb{R}}^{d}):\,\,\varphi(x)\mathop{\longrightarrow}^{|x|\to\infty}\infty,\,\,\hbox{${\mathcal{I}}(x,[\varphi])$
is well defined and $\varphi$ satisfies (\ref{H2})}\\}.$

As in the estimate (3.33), the sub-exponential decay is observed only when
using different weights in the seminorms of $u(t)$ and $u(0)$. One possible
choice, among others, is to use different powers of $\varphi$, as we did in
Theorem 3.9.

###### Theorem 5.2

Let ${\mathcal{L}}$ be defined by (2.5) where $\Sigma$ satisfies (2.2), the
Levy measure in ${\mathcal{I}}$ satisfies (2.4), and $\lambda_{0}+\lambda>0$.
Assume that $b$ satisfies (2.12) with $\gamma\in(0,2)$, and either of the
following conditions:

(i) $\lambda_{0}>0$ or $\sigma\in(1,2)$, and (2.13) holds true.

(ii) $\lambda_{0}=0$ and $\sigma\in(0,1]$, and (2.14) holds true.

Assume that there exists $\varphi$ satisfying (H2). Let $u$ be a (viscosity)
solution of (2.7) such that $|u(t,x)|=o(\Phi)$ as $|x|\to\infty$ (uniformly
for $t\in(0,T)$) for some $\Phi\in C^{2}({\mathbb{R}}^{d})$ such that
$\Phi(x)\mathop{\to}\limits^{|x|\to\infty}\infty$,
${\mathcal{L}}^{b}[\Phi]\mathop{\to}\limits^{|x|\to\infty}\infty$ and
$\liminf_{|x|\to\infty}[{\mathcal{L}}^{b}[\Phi]-\kappa\,h(\varphi)\Phi]>-\infty$
for some $\kappa>0$.

Then, for any $\theta<1$ such that $h(\varphi)\varphi^{\theta}\to\infty$ as
$|x|\to\infty$, there exist positive constants $L,M>0$ (depending on
$\alpha,\gamma,\sigma,(\lambda_{0}+\lambda)^{-1},\sigma_{0},\sigma_{1},c_{0},d,\varphi,\theta$)
such that $u$ satisfies

$[u(t)]_{\varphi}\leq
M\,\varpi(t)\,[u_{0}]_{\varphi^{\theta}}\qquad\hbox{where $\varpi(t)$ is
implicitly defined as
$\int_{\varpi}^{1}\frac{ds}{sh(L\,s^{-\frac{1}{1-\theta}})}=\frac{t}{2}$,}$
(5.2)

with the function $h$ given by (H2).

###### Remark 5.3

The interesting part of (5.2) is that the decay rate $\varpi(t)$ is given in
terms of the condition (H2) involving the Lyapunov function $\varphi$. Notice
that $\varpi$ is sub-exponential precisely because $h(r)\to 0$ as $r\to\infty$
($h$ gives a quantification of the fact that
$\frac{{\mathcal{L}}^{b}[\varphi]}{\varphi}\mathop{\to}\limits^{|x|\to\infty}0$).
The implicit condition for $\varpi(t)$ comes from the requirement that
$\varpi$ satisfies the ODE

$\begin{cases}\varpi^{\prime}=-\frac{1}{2}\varpi\,h(L\varpi^{-\frac{1}{1-\theta}})\,,\,\,t>0&\\\
\varpi(0)=1\end{cases}$ (5.3)

Here, we did not look for sharp conditions in terms of $\theta$, and the
choice of the constant $L$ is not meant to be optimal, either. Thus, there
could be room for sharper conditions. However, the rate function which stems
from (5.3) looks consistent with all previous results; in particular, in the
range of power functions $\varphi=\langle x\rangle^{k}$, the above statement
reduces to Theorem 3.9.

###### Remark 5.4

Despite the technical assumption appearing in the statement, the growth of
solutions $u$ plays a minor role in the above result. One could readily use
the statement with $\Phi=\varphi$, and apply the estimate to any bounded
solution $u$. This is how we are going to use this result later.

Proof. The proof is a (slightly more abstract) version of Theorem 3.9.

As a first step, we observe that, for all functions $\varphi\in
C^{2}({\mathbb{R}}^{d})$ such that ${\mathcal{L}}^{b}[\varphi]\to\infty$ as
$|x|\to\infty$, it holds

$[u(t)]_{\varphi}\leq C\,[u_{0}]_{\varphi}$

for a suitable constant $C>0$. This is obtained exactly as (3.36) in Theorem
3.9. Now, if $\theta<1$, we have

$D^{2}\varphi^{\theta}=\theta\varphi^{\theta-1}\left(D^{2}\varphi+(\theta-1)\frac{D\varphi\otimes
D\varphi}{\varphi}\right)\leq\theta\varphi^{\theta-1}D^{2}\varphi\ $

and, by concavity,

$\displaystyle{\mathcal{I}}(x,[\varphi^{\theta}])$
$\displaystyle=\int_{|z|>1}\\{\varphi^{\theta}(x+z)-\varphi^{\theta}(x)\\}\nu(dz)+\int_{|z|\leq
1}\\{\varphi^{\theta}(x+z)-\varphi^{\theta}(x)-\theta\varphi^{\theta-1}(D\varphi(x)\cdot
z)\\}\nu(dz)$
$\displaystyle\leq\theta\varphi(x)^{\theta-1}{\mathcal{I}}(x,[\varphi])\,.$

Hence we have

${\mathcal{L}}^{b}[\varphi^{\theta}]\geq\theta\varphi^{\theta-1}{\mathcal{L}}^{b}[\varphi]\geq\theta\varphi^{\theta}\,h(\varphi)\to\infty\qquad\hbox{as
$|x|\to\infty$}$

by assumption. This implies that $\varphi^{\theta}$ is itself a Lyapunov
function and we have, by what proved before, $[u(t)]_{\varphi^{\theta}}\leq
C\,[u_{0}]_{\varphi^{\theta}}$.

Now we go back to the maximum point of

$u(t,x)-u(t,y)-\varpi(t)\underbrace{\left\\{K[\varphi(x)+\varphi(y)]+\psi(|x-y|)+\varepsilon[\Phi(x)+\Phi(y)]\right\\}}_{\zeta(x,y)}-\frac{\varepsilon}{T-t}$

where $K\geq[u_{0}]_{\varphi^{\theta}}\geq[u_{0}]_{\varphi}$ and $\varpi$
satisfies (5.3). On the maximum point we have, similarly as in (3.35):

$\displaystyle\frac{\varepsilon}{(T-t)^{2}}+\varpi(t)\,K\left\\{\mathcal{L}^{b}[\varphi](x)+\frac{\varpi^{\prime}}{\varpi}\varphi(x)+\mathcal{L}^{b}[\varphi](y)+\frac{\varpi^{\prime}}{\varpi}\varphi(y)\right\\}$
$\displaystyle+\varpi(t)\,\varepsilon\left(\mathcal{L}^{b}[\Phi](x)+\frac{\varpi^{\prime}}{\varpi}\Phi(x)+\mathcal{L}^{b}[\Phi](y)+\frac{\varpi^{\prime}}{\varpi}\Phi(y)\right)$
$\displaystyle+\varpi(t)\left((b(t,x)-b(t,y))\cdot\widehat{x-y}\right)\psi^{\prime}(r)+\varpi^{\prime}(t)\psi(r)$
$\displaystyle\qquad\leq\varpi(t)\left\\{4\lambda_{0}\psi^{\prime\prime}(r)+(2\sigma_{0}\wedge\sigma_{1}r)\sigma_{1}\,\psi^{\prime}(r)\right\\}$
$\displaystyle\qquad+\varpi(t)4\lambda\int_{0}^{1}(1-s)\int_{B}\psi^{\prime\prime}(r+2s(\widehat{x-y}\cdot
z))|z|^{2}\frac{dz}{|z|^{d+\sigma}}ds\,$

where $r=|x-y|$. We consider the case that $r=|x-y|>r_{1}$, for a sufficiently
large $r_{1}$. In particular, we have $|x|\vee|y|>\frac{r_{1}}{2}$. We drop
the terms with $\psi^{\prime\prime}$, we use (2.13) (or (2.14)) for the drift
term, and we use (H2). So we obtain

$\displaystyle
K\left\\{\left(h(\varphi)+\frac{\varpi^{\prime}}{\varpi}\right)\varphi(x)+\left(h(\varphi)+\frac{\varpi^{\prime}}{\varpi}\right)\varphi(y)\right\\}+\varepsilon\left(\mathcal{L}^{b}[\Phi](x)+\frac{\varpi^{\prime}}{\varpi}\Phi(x)+\mathcal{L}^{b}[\Phi](y)+\frac{\varpi^{\prime}}{\varpi}\Phi(y)\right)$
$\displaystyle\qquad+\frac{\varpi^{\prime}}{\varpi}\psi(r)<(c_{0}+2\sigma_{0}\sigma_{1})\,\psi^{\prime}(r)\,.$

Using $\psi\leq C_{1}$ and $\psi^{\prime}(r)\leq\psi^{\prime}(r_{1})$ we get

$\displaystyle
K\left\\{\left(h(\varphi)+\frac{\varpi^{\prime}}{\varpi}\right)\varphi(x)+\left(h(\varphi)+\frac{\varpi^{\prime}}{\varpi}\right)\varphi(y)\right\\}+\varepsilon\left(\mathcal{L}^{b}[\Phi](x)+\frac{\varpi^{\prime}}{\varpi}\Phi(x)+\mathcal{L}^{b}[\Phi](y)+\frac{\varpi^{\prime}}{\varpi}\Phi(y)\right)$
$\displaystyle\qquad<(c_{0}+2\sigma_{0}\sigma_{1})\,\psi^{\prime}(r_{1})-\frac{\varpi^{\prime}}{\varpi}C_{1}\,.$

We observe that, since the maximum is positive, we have

$K\varpi(t)[\varphi(x)+\varphi(y)]\leq
u(t,x)-u(t,y)\leq[\varphi^{\theta}(x)+\varphi^{\theta}(y)][u(t)]_{\varphi^{\theta}}\leq
C\,[\varphi^{\theta}(x)+\varphi^{\theta}(y)]\,[u_{0}]_{\varphi^{\theta}}$

which yields

$\varpi(t)\leq
C\frac{[u_{0}]_{\varphi^{\theta}}}{K}\,\frac{\varphi^{\theta}(x)+\varphi^{\theta}(y)}{\varphi(x)+\varphi(y)}\leq
C\frac{[u_{0}]_{\varphi^{\theta}}}{K}\,\frac{1}{(\varphi(x)\vee\varphi(y))^{1-\theta}}\,.$

In particular, since $h$ is decreasing, we have

$h(\varphi)\geq
h\left(\left(C\frac{[u_{0}]_{\varphi^{\theta}}}{K}\right)^{\frac{1}{1-\theta}}\varpi^{-\frac{1}{1-\theta}}\right)\,.$

For a constant $L$ to be chosen below, such that
$L\geq\left(C\frac{[u_{0}]_{\varphi^{\theta}}}{K}\right)^{\frac{1}{1-\theta}}$,
we fix $\varpi$ as the solution of (5.3). Then

$\frac{\varpi^{\prime}}{\varpi}=-\frac{1}{2}h\left(L\varpi^{-\frac{1}{1-\theta}}\right)\geq-\frac{1}{2}h(\varphi)$

so we deduce that

$\begin{split}&K\left\\{\frac{1}{2}h(\varphi)\varphi(x)+\frac{1}{2}h(\varphi)\varphi(y)\right\\}+\varepsilon\left(\mathcal{L}^{b}[\Phi](x)-\frac{1}{2}h(\varphi(x))\Phi(x)+\mathcal{L}^{b}[\Phi](y)-\frac{1}{2}h(\varphi(y))\Phi(y)\right)\\\
&\qquad\qquad\qquad<(c_{0}+2\sigma_{0}\sigma_{1})\,\psi^{\prime}(r_{1})-\frac{\varpi^{\prime}}{\varpi}C_{1}\end{split}$
(5.4)

Notice that $\varpi\leq 1$ implies
$h\left(L\varpi^{-\frac{1}{1-\theta}}\right)\leq h(L)$, hence
$-\frac{\varpi^{\prime}}{\varpi}C_{1}\leq\frac{1}{2}h(L)C_{1}$. As for the
function $\Phi$, we know by assumption that
$\liminf_{|x|\to\infty}{\mathcal{L}}^{b}[\Phi]-\kappa h(\varphi)\Phi>-\infty$,
for some $\kappa>0$ which we can assume, without loss of generality, such that
$\kappa\geq\frac{1}{2}$. Hence
$\mathcal{L}^{b}[\Phi](x)-\frac{1}{2}h(\varphi(x))\Phi(x)\geq-k_{0}$ for some
constant $k_{0}>0$. Then, since $h(\varphi)\varphi\to\infty$ as
$|x|\to\infty$, choosing $r_{1}$ sufficiently large the inequality (5.4)
implies

$K<2k_{0}\varepsilon+(c_{0}+2\sigma_{0}\sigma_{1})\,\psi^{\prime}(r_{1})+\frac{1}{2}h(L)C_{1}\,.$

We conclude that the maximum point cannot be reached with $r>r_{1}$ provided

$K\geq
2k_{0}\varepsilon+(c_{0}+2\sigma_{0}\sigma_{1})\,\psi^{\prime}(r_{1})+\frac{1}{2}h(L)C_{1}\,.$

This replaces (3.40) in Theorem 3.9. Once $K$ is chosen, the rest of the proof
is the same, by simply using that
$\mathcal{L}^{b}[\varphi],\mathcal{L}^{b}[\Phi]\geq-k_{0}$ for some $k_{0}>0$,
and the conclusion is obtained by choosing $L$ and $C_{1}$ sufficiently large.

As in Section 4, we deduce by duality the decay estimate for Fokker-Planck
equations.

###### Theorem 5.5

Let ${\mathcal{L}}$ be defined by (2.5) where $\Sigma$ satisfies (2.2), the
Levy measure in ${\mathcal{I}}$ satisfies (2.4), and $\lambda_{0}+\lambda>0$.
Assume that $b$ satisfies (2.12) and either of the following conditions:

(i) $\lambda_{0}>0$ or $\sigma\in(1,2)$, and (2.13) holds true.

(ii) $\lambda_{0}=0$ and $\sigma\in(0,1]$, and (2.14) holds true.

For $m_{0}\in{\mathcal{M}}({\mathbb{R}}^{d})$ with
${\int_{{\mathbb{R}}^{d}}}\,dm_{0}=0$, let $m$ be the solution of (2.10). Then
we have:

  * (a)

If $\varphi\in\Lambda_{1}$ and $m_{0}\in{\mathcal{M}}_{\varphi}$, then

$\|m(t)\|_{{\mathcal{M}}_{\varphi}}\leq
K\,e^{-\omega\,t}\|m_{0}\|_{{\mathcal{M}}_{\varphi}}$

for some $K,\omega$ depending on
$\alpha,\gamma,(\lambda_{0}+\lambda)^{-1},\sigma,\sigma_{0},\sigma_{1},c_{0},\delta,\varphi,d$.

  * (b)

If $\varphi\in\Lambda_{2}$ and $m_{0}\in{\mathcal{M}}_{\varphi}$, then for all
$\theta\in(0,1)$ such that $h(\varphi)\varphi^{\theta}\to\infty$ as
$|x|\to\infty$ we have

$\|m(t)\|_{{\mathcal{M}}_{\varphi^{\theta}}}\leq
K\,\varpi(t)\|m_{0}\|_{{\mathcal{M}}_{\varphi}}$

where $\varpi$ is defined as in (5.2), for some $K,L$ depending on
$\theta,\alpha,\gamma,(\lambda_{0}+\lambda)^{-1},\sigma,\sigma_{0},\sigma_{1},c_{0},\delta,\varphi,d$.

As an example of application of Theorem 5.5, when we consider the Lyapunov
function

$\varphi=e^{\mu\langle x\rangle^{k}}$ (5.5)

with parameters $\mu,k>0$, we extend to the local operator ${\mathcal{L}}_{0}$
similar results obtained in the recent paper [30] for the Laplace operator.

###### Corollary 5.6

Let ${\mathcal{L}}_{0}$ be defined by (2.1), where we assume $\lambda_{0}>0$
and (2.2). Assume that $b(t,x)$ is a continuous function satisfying (2.12) and
(2.13).

Let $\varphi$ be defined by (5.5), where we assume that either $k<\gamma$ (and
$\mu>0$ is arbitrary) or $k=\gamma$ and $\mu<\frac{\alpha}{\gamma}$ ($\alpha$
appears in (2.12)). Let $m_{0}\in L^{1}(\varphi)$ be such that
${\int_{{\mathbb{R}}^{d}}}m_{0}\,dx=0$, and let $m$ be the solution to the
Cauchy problem (4.5). Then we have:

  * (a)

If $\gamma\geq 2$, then

$\|m(t)\|_{L^{1}(\varphi)}\leq K\,e^{-\omega t}\,\|m_{0}\|_{L^{1}(\varphi)}$
(5.6)

for some constant $K,\omega$ only depending on
$\alpha,\lambda_{0},\sigma_{0},\sigma_{1},d,c_{0},k,\mu$.

  * (b)

If $\gamma\in[1,2)$, then the exponential decay (5.6) holds for $k\geq
2-\gamma$.

  * (c)

If $\gamma\in(0,1)$, for any $\theta\in(0,1)$ we have

$\|m(t)\|_{L^{1}(\varphi^{\theta})}\leq
K\,e^{-C\,t^{\frac{k}{2-\gamma}}}\,\|m_{0}\|_{L^{1}(\varphi)}\,$ (5.7)

for some $K,C$ depending on
$\theta,\alpha,\gamma,\lambda_{0},\sigma_{0},\sigma_{1},d,c_{0},k,\mu$.

Proof. We observe the following, through standard computations:

(i) $\lim\limits_{|x|\to\infty}{\mathcal{L}}_{0}[\varphi]=\infty$ provided
$k<\gamma$ or $k=\gamma$ and $\mu<\alpha/\gamma$ ($\alpha$ appears in (2.12))

(ii) if $\gamma\geq 2$, we have
$\liminf\limits_{|x|\to\infty}\frac{{\mathcal{L}}_{0}[\varphi]}{\varphi}>0$.
Hence $\|m(t)\|_{L^{1}(\varphi)}$ decays exponentially, according to Theorem
5.5.

(iii) if $\gamma\in[1,2)$,
$\liminf\limits_{|x|\to\infty}\frac{{\mathcal{L}}_{0}[\varphi]}{\varphi}>0$
provided $k\in[2-\gamma,\gamma)$ (or even $k=\gamma$ if
$\mu<\frac{\alpha}{\gamma}$)

(iv) If $\gamma\in(0,1)$, then we have ${\mathcal{L}}_{0}[\varphi]\to\infty$
but ${\mathcal{L}}_{0}[\varphi]\simeq h(\varphi)\varphi$ for
$h=O\left(\frac{1}{(\log\varphi)^{\frac{2-\gamma}{k}-1}}\right)$. Following
the recipe of Theorem 5.5, we deduce that, for any $\theta\in(0,1)$, there
exists some $C,K$ (depending on $\theta$) such that (5.7) holds true.

In particular, the above result shows that, in the local case, there is
exponential decay of solutions, even for $\gamma\in[1,2)$, provided one uses a
suitable weighted norm. Even for a milder confining property of the drift,
compared to Theorem 4.4, we have proved that the solution decays much faster
than polynomially when considered in the above weighted norm.

Let us observe that, in view of assumption (2.4), the Lyapunov function (5.5)
grows too fast at infinity for the nonlocal part, this is why the above result
was restricted to the case of local operator. However, by simply modifying the
condition at infinity on the Levy measure $\nu$, the reader can readily
generalize the above statement to other classes of Levy operators, with fast
decay of the kernel at infinity.

###  5.1 Convergence to stationary solutions

We stress that the above results can also be used to provide existence of
stationary solutions, namely invariant measures of Fokker-Planck equations. We
give here a sample of this application.

###### Theorem 5.7

Under the assumptions of Theorem 5.5, assume that either
$\varphi\in\Lambda_{1}$, or $\varphi\in\Lambda_{2}$ and $\varpi$ defined in
(5.2) satisfies $\varpi\in L^{1}(0,\infty)$. Assume that $b=b(x)$ is a
continuous, time-independent vector field. Then there exists a unique
stationary measure $\bar{m}\in{\mathcal{P}}({\mathbb{R}}^{d})$, which is a
solution of

${\mathcal{L}}^{*}[\bar{m}]-{\rm div}(b(x)\bar{m})=0\qquad\text{ in
}{\mathbb{R}}^{d},$ (5.8)

and such that, if
$m_{0}\in{\mathcal{P}}({\mathbb{R}}^{d})\cap{\mathcal{M}}_{\varphi}$ and $m$
is the solution of (2.10), we have:

$\displaystyle\hbox{if $\varphi\in\Lambda_{1}$,
then}\quad\|m(t)-\bar{m}\|_{{\mathcal{M}}_{\varphi}}\leq K\,e^{-\omega t}\,$
(5.9) $\displaystyle\hbox{if $\varphi\in\Lambda_{2}$,
then}\quad\|m(t)-\bar{m}\|_{{\mathcal{M}}_{\varphi^{\theta}}}\leq
K\,\varpi(t)\,$ (5.10)

for any $\theta<1$ such that $h(\varphi)\varphi^{\theta}\to\infty$ as
$|x|\to\infty$.

Proof. First of all we observe that, if $m$ is the solution of (2.10) (in the
sense of Definition 2.4), then $m_{0}\in{\mathcal{M}}_{\varphi}$ implies
$m(t)\in{\mathcal{M}}_{\varphi}$ (reasoning exactly as in (6.6) for the
function $\langle x\rangle^{k}$).

Suppose now that $\varphi\in\Lambda_{1}$, or $\varphi\in\Lambda_{2}$ is such
that $\varpi$ defined in (5.2) satisfies $\varpi\in L^{1}(0,\infty)$. For any
$s>0$, we can apply Theorem 5.5 to $m(t+s)-m(t)$, which yields

$\begin{split}\hbox{if $\varphi\in\Lambda_{1}$,
then}\,\,\|m(t+s)-m(t)\|_{{\mathcal{M}}_{\varphi}}&\leq K\,e^{-\omega
t}\,\|m(s)-m_{0}\|_{{\mathcal{M}}_{\varphi}}\,,\\\ \hbox{if
$\varphi\in\Lambda_{2}$,
then}\,\,\|m(t+s)-m(t)\|_{{\mathcal{M}}_{\varphi^{\theta}}}&\leq
K\,\varpi(t)\,\|m(s)-m_{0}\|_{{\mathcal{M}}_{\varphi}}\,.\end{split}$ (5.11)

In particular, setting $m_{n}(x):=m(n,x)$, we have

$m_{n}=m_{0}+\sum_{k=0}^{n-1}[m_{k+1}-m_{k}]$ (5.12)

where last series is strongly convergent in ${\mathcal{M}}_{\varphi}$ (or in
${\mathcal{M}}_{\varphi^{\theta}}$ if $\varphi\in\Lambda_{2}$), due to (5.11).
Hence there exists a measure $\bar{m}\in{\mathcal{P}}({\mathbb{R}}^{d})$ such
that $m_{n}\to\bar{m}$ in ${\mathcal{M}}_{\varphi}$ (or in
${\mathcal{M}}_{\varphi^{\theta}}$ if $\varphi\in\Lambda_{2}$). In fact,
$\bar{m}$ is itself a probability measure. Now we observe that, for any
stationary $\xi$, we have

${\int_{{\mathbb{R}}^{d}}}\xi\,\,d(m_{n}-m_{n-1})+\int_{n-1}^{n}{\int_{{\mathbb{R}}^{d}}}{\mathcal{L}}^{b}[\xi]\,dm(t)\,dt=0$

which implies

${\int_{{\mathbb{R}}^{d}}}{\mathcal{L}}^{b}[\xi]\,dm_{n}=\int_{n-1}^{n}{\int_{{\mathbb{R}}^{d}}}{\mathcal{L}}^{b}[\xi]\,d(m_{n}-m(t))\,dt-{\int_{{\mathbb{R}}^{d}}}\xi\,d(m_{n}-m_{n-1})\,.$
(5.13)

If $\varphi\in\Lambda_{1}$, by (5.11) we have
$\|m_{n}-m_{n-1}\|_{{\mathcal{M}}_{\varphi}}\to 0$. The first integral in the
right-hand side also goes to zero; indeed, again by (5.11) we have

$\left|\int_{n-1}^{n}{\int_{{\mathbb{R}}^{d}}}{\mathcal{L}}^{b}[\xi]\,d(m(t)-m_{n})\,dt\right|\leq\|{\mathcal{L}}^{b}[\xi]\|_{L^{\infty}(\varphi^{-1})}\int_{n-1}^{n}\|m(t)-m_{n}\|_{{\mathcal{M}}_{\varphi}}dt\leq
C\,\int_{n-1}^{n}e^{-\omega t}dt\to 0\,.$

If $\varphi\in\Lambda_{2}$, the same argument applies in
${\mathcal{M}}_{\varphi^{\theta}}$ and replacing $e^{-\omega t}$ with
$\varpi(t)$, according to (5.11). Therefore, passing to the limit in (5.13),
recalling the convergence of $m_{n}$, we conclude that $\bar{m}$ is a
stationary solution. Finally, we can now apply Theorem 5.5 to $m-\bar{m}$, and
we deduce that the whole sequence $m(t)$ converges to $\bar{m}$. Let us point
out that, in case $\varphi\in\Lambda_{2}$, one should apply Theorem 5.5
between the spaces ${\mathcal{M}}_{\varphi^{\theta}}$ and
${\mathcal{M}}_{\varphi^{\theta+\varepsilon}}$, since a priori one only knows
that $\bar{m}\in{\mathcal{M}}_{\varphi^{\theta}}$ for any $\theta<1$.

We point out that, even in the case when the rate function $\varpi(t)$ in
(5.2) is not integrable at infinity, the above argument produces the existence
of a stationary solution $\bar{m}\in{\mathcal{P}}({\mathbb{R}}^{d})$. Indeed,
in the inequality (5.13) it is enough to use that $\varpi(t)\to 0$ as
$t\to\infty$. By the compactness of ${\mathcal{P}}({\mathbb{R}}^{d})$, the
sequence $m_{n}$ will converge to a stationary solution. What is less obvious,
in this case, is to show that the stationary solution enjoys the further
regularity $\bar{m}\in{\mathcal{M}}_{\varphi^{\theta}}$ for any $\theta<1$. In
fact, we don’t know that the sum (5.12) is converging in
${\mathcal{M}}_{\varphi^{\theta}}$, if $\varpi$ is no longer in
$L^{1}(0,\infty)$. In specific cases, one should use extra arguments to show
first that $m(t)$ is bounded in ${\mathcal{M}}_{\varphi^{\theta}}$ (uniformly
in time). We mention that, for the case of power weights $\varphi=\langle
x\rangle^{k}$ and fractional Laplace operator, one can use e.g. [31, Thm 1.3]
to deduce that $\bar{m}\in L^{1}(\langle x\rangle^{\bar{k}})$ for all
${\bar{k}}\leq 1$; this allows for instance to apply to $m(t)-\bar{m}$ the
decay estimate (4.2) for some $k\in(2-\gamma,1\wedge\sigma)$.

##  6 Appendix: existence and uniqueness of solutions

In this section we give a proof of the existence and uniqueness of solutions
of the Fokker-Planck equation (2.10). Once more, this can be essentially
deduced by duality from the well-posedness of viscosity solutions of the
adjoint problem. This is why we start by giving a proof of the comparison
principle, and uniqueness, of viscosity solutions

###### Proposition 6.1

Let $u_{0}\in C({\mathbb{R}}^{d})\cap L^{\infty}(\langle x\rangle^{-k})$,
$f\in C(Q_{T})\cap L^{\infty}(0,T;L^{\infty}(\langle x\rangle^{-k}))$, for
some $k<\sigma$. Let ${\mathcal{L}}$ be defined by (2.5) where $\Sigma$
satisfies (2.2), the Levy measure in ${\mathcal{I}}$ satisfies (2.4), and
$\lambda_{0}+\lambda>0$. Assume that $b\in C(Q_{T})$ is such that $b(t,x)\cdot
x$ is bounded below and either of the following conditions hold: (i)
$\lambda_{0}>0$ or $\sigma\in(1,2)$, and (2.13) holds true whenever $|x-y|\leq
1$. (ii) $\lambda_{0}=0$, $\sigma\in(0,1]$ and (2.14) holds true whenever
$|x-y|\leq 1$. Then there exists a unique viscosity solution of

$\begin{cases}\partial_{t}u+{\mathcal{L}}[u]+b(t,x)\cdot Du=f&\text{ in
}(0,T)\times{\mathbb{R}}^{d},\\\ u(0,x)=u_{0}(x),\qquad\,&\text{ in
}{\mathbb{R}}^{d}\end{cases}$ (6.1)

such that $|u(t,x)|\leq C(1+|x|)^{m}$ for some $m<\sigma$.

Proof. We prove first the uniqueness of solutions. Let us take some
$\beta<\sigma$ such that $\beta\geq k\vee m$. We first remark that (see the
computations in Lemma 3.1) ${\mathcal{L}}^{b}[\langle
x\rangle^{\beta}]\geq\beta\langle x\rangle^{\beta-2}(b\cdot
x)-c_{\beta}(1+\langle x\rangle^{\beta-1})$; hence there exists $K_{0}>0$ such
that

$\langle x\rangle^{\beta}+{\mathcal{L}}^{b}[\langle
x\rangle^{\beta}]\geq\frac{1}{2}\langle x\rangle^{\beta}-K_{0}$

where $K_{0}$ only depends on $\sigma,d,\|\Sigma(x)\|_{\infty},\Lambda$ and
the lower bound of $b\cdot x$. We deduce that, choosing $K$ large enough, the
function $W:=e^{t}\langle x\rangle^{\beta}+Kt+K_{1}+\frac{1}{T-t}$ is a
supersolution, provided $\langle x\rangle^{\beta}+K_{1}\geq|u_{0}|$. Let now
$u,v$ be, respectively, a sub and super-solution of (6.1). By linearity, we
have that $u_{\varepsilon}:=u-\varepsilon W$ and
$v_{\varepsilon}:=v+\varepsilon W$ are still sub/supersolutions, and, if
$\beta>m$, there exists a compact set $K_{\varepsilon}\subset{\mathbb{R}}^{d}$
such that $u_{\varepsilon}\leq 0\leq v_{\varepsilon}$ for $t\in(0,T)$,
$x\in{\mathbb{R}}^{d}\setminus K_{\varepsilon}$. Of course, if we prove the
comparison for $u_{\varepsilon},v_{\varepsilon}$, then we get the comparison
for $u,v$ by letting $\varepsilon\to 0$. Therefore, up to replacing $u,v$ with
$u_{\varepsilon},v_{\varepsilon}$, we can assume that $u\leq 0\leq v$ outside
a compact set $K$. Now, suppose that $M:=\sup_{Q_{T}}(u-v)>0$, which means
that $M$ is attained at some point in $(0,T)\times K$, with $t>0$. Doubling
variables, we consider

$M_{\varepsilon}:=\sup_{(x,y)}\,\,\\{u(t,x)-v(t,y)-\frac{|x-y|^{2}}{2\varepsilon}-A_{\varepsilon}t^{\eta}\\}$

for some $\eta,A_{\varepsilon}>0$ to be chosen, with $A_{\varepsilon}\to 0$.
Since $M_{\varepsilon}\geq M-A_{\varepsilon}T^{\eta}$, then $M_{\varepsilon}$
is positive for $\varepsilon$ small. Recalling that $u\leq 0\leq v$ outside
$K$, this implies that $M_{\varepsilon}$ is assumed at points
$(x_{\varepsilon},y_{\varepsilon})$ which lie in a compact set, which is
actually an $\varepsilon$-neighborhood of $K$, since
$\frac{|x_{\varepsilon}-y_{\varepsilon}|^{2}}{2\varepsilon}\leq\hat{M}=\sup
u-\inf v<\infty$. In particular we have

$|x_{\varepsilon}-y_{\varepsilon}|\leq\sqrt{2\varepsilon\,\hat{M}}$ (6.2)

and by usual arguments, since $M_{\varepsilon}\to M$ as $\varepsilon\to 0$, we
also get $\frac{|x_{\varepsilon}-y_{\varepsilon}|^{2}}{\varepsilon}\to 0$.
Using Theorem 2.3, the Lipschitz character of $\Sigma(x)$ and standard
arguments in the doubling variable method, we get

$\eta
A_{\varepsilon}\,t^{\eta-1}+(b(t,x_{\varepsilon})-b(t,y_{\varepsilon}))\cdot\frac{(x_{\varepsilon}-y_{\varepsilon})}{\varepsilon}\leq
f(t,x_{\varepsilon})-f(t,y_{\varepsilon})+C\,\sigma_{1}\frac{|x_{\varepsilon}-y_{\varepsilon}|^{2}}{\varepsilon}\,.$
(6.3)

At this stage we use the following regularity result, which can be obtained
with similar arguments as used in Section 3 (see the Appendix in [22]): using
conditions (2.13), or (2.14), for $|x-y|\leq 1$, every viscosity solution of
(6.1) satisfies

$\begin{split}&|u(t,x)-u(t,y)|\leq C_{T}\frac{[\langle x\rangle^{k}+\langle
y\rangle^{k}]}{t^{\frac{1}{\sigma-\kappa}}}\,|x-y|\,,\quad\forall
x,y\,,\qquad\hbox{if $\sigma\in(1,2)$, $\kappa>0$}\\\ &|u(t,x)-u(t,y)|\leq
C_{T}\frac{[\langle x\rangle^{k}+\langle
y\rangle^{k}]}{t^{\frac{\theta}{\sigma}}}\,|x-y|^{\theta}\,,\quad\forall
x,y\,,\qquad\hbox{if $\sigma\in(0,1]$, $\theta<\sigma$.}\end{split}$ (6.4)

Therefore, assuming that at least one between $u,v$ is a solution, estimate
(6.2) can be improved accordingly. Recalling that the points
$(x_{\varepsilon},y_{\varepsilon})$ lie in a compact set, we deduce that they
satisfy, if $\sigma>1$,

$\frac{|x-y|^{2}}{2\varepsilon}\leq\frac{C_{T,K}}{t^{\frac{1}{\sigma-\kappa}}}|x-y|\leq
C\frac{\sqrt{\varepsilon}}{t^{\frac{1}{\sigma-\kappa}}}\,.$

Notice that this constant $C_{T,K}$ only depends on the local bounds of $u,v$,
and in particular is determined independently of $A_{\varepsilon}$. Hence,
(6.3) implies

$\displaystyle\eta A_{\varepsilon}\,t^{\eta-1}$
$\displaystyle\leq\frac{2C_{T,K}}{t^{\frac{1}{\sigma-\kappa}}}\,|b(t,x_{\varepsilon})-b(t,y_{\varepsilon})|+f(t,x_{\varepsilon})-f(t,y_{\varepsilon})+C\,\sigma_{1}\frac{|x_{\varepsilon}-y_{\varepsilon}|^{2}}{\varepsilon}$
$\displaystyle\leq
C\,\omega(\varepsilon)\left(\frac{1}{t^{\frac{1}{\sigma-\kappa}}}+1\right)+\frac{C}{t^{\frac{1}{\sigma-\kappa}}}\,\sqrt{\varepsilon}$

for some $C>0$ and some modulus of continuity $\omega(\cdot)$ depending on
$b,f$. Choosing $\kappa$ small so that $\sigma-\kappa>1$, and then
$\eta=1-\frac{1}{\sigma-\kappa}$, if
$A_{\varepsilon}=L\,(\omega(\varepsilon)+\sqrt{\varepsilon})$ we obtain a
contradiction for $L$ sufficiently large.

If $\sigma\in(0,1]$, we estimate

$\frac{|x-y|^{2}}{2\varepsilon}\leq\frac{C}{t^{\frac{\theta}{\sigma}}}|x-y|^{\theta}$

then we use (2.14) and we get

$\displaystyle\eta A_{\varepsilon}\,t^{\eta-1}$ $\displaystyle\leq
c_{0}\,|x_{\varepsilon}-y_{\varepsilon}|^{1-\sigma+\delta}\,\frac{|x_{\varepsilon}-y_{\varepsilon}|}{\varepsilon}+f(t,x_{\varepsilon})-f(t,y_{\varepsilon})+C\,\sigma_{1}\frac{|x_{\varepsilon}-y_{\varepsilon}|^{2}}{\varepsilon}$
$\displaystyle\leq\frac{C}{t^{\frac{\theta}{\sigma}}}\,|x_{\varepsilon}-y_{\varepsilon}|^{\theta-\sigma+\delta}+f(t,x_{\varepsilon})-f(t,y_{\varepsilon})+\sigma_{1}\frac{C}{t^{\frac{\theta}{\sigma}}}\,|x_{\varepsilon}-y_{\varepsilon}|^{\theta}\,.$

Choosing $\theta=\sigma-\frac{\delta}{2}$ and $\eta=1-\frac{\theta}{\sigma}$,
we conclude as before with a suitable choice of $A_{\varepsilon}$. Having
obtained a contradiction, we infer that $\sup_{Q_{T}}(u-v)\leq 0$, i.e. $u\leq
v$. Reversing the role, we get the desired uniqueness. The existence of
solutions can be proved through the vanishing viscosity method (using (6.4) to
infer local compactness, see also the proof of Theorem 2.5 below).

The following lemma is a classical statement in viscosity solutions theory to
ensure equi-continuity in time from continuity in space (see [4, Lemma 9.1]).
We give a proof for the reader’s convenience, since the setting of our
assumptions does not fall into previous versions.

###### Lemma 6.2

Let $K\subset{\mathbb{R}}^{d}$ be a compact set, and let $u$ be a viscosity
solution of (6.1), where $b,f,u_{0}$ are continuous. For any $\varepsilon>0$,
there exists $\delta>0$ such that

$\|u(t)-u_{0}\|_{L^{\infty}(K)}\leq\varepsilon\qquad\forall t<\delta\,,$

where $\delta$ depends only on the bounds of $b,f,u$ and on the modulus of
continuity of $u_{0}$, for $(t,x)$ in a neighborhood of $\\{0\\}\times K$ (and
of course on $\sigma,d,\|\Sigma(x)\|_{\infty},\Lambda$ as well).

Proof. For $\eta>0$ small enough, $y\in B_{R}$, $x\in B_{2R}$, let us consider
the function

$\Phi(x):=u_{0}(y)+C\,\zeta(|x-y|)+kt+\eta$

where $\zeta(s)={\mathcal{T}}_{R}(s^{2})$; here ${\mathcal{T}}_{R}(\cdot)$
denotes a $C^{2}$, bounded and concave function such that
${\mathcal{T}}_{R}(s)=s$ for $|s|<(3R)^{2}$. By concavity of
${\mathcal{T}}_{R}$, we have

$\displaystyle{\mathcal{I}}(x,[\zeta(|x-y|)])$
$\displaystyle\leq\int_{|z|>1}\\{\zeta(|x+z-y|)-\zeta(|x-y|)\\}\nu(dz)$
$\displaystyle\qquad+{\mathcal{T}}_{R}^{\prime}(|x-y|^{2})\int_{|z|\leq
1}\\{|x+z-y|^{2}-|x-y|^{2}-2((x-y)\cdot z)\mathds{1}_{|z|\leq 1}\\}\nu(dz)$
$\displaystyle\leq M_{R}\,.$

In the local terms, we just observe that $\zeta(|x-y|)=|x-y|^{2}$; so, using
the local bound of $b$, we have

${\mathcal{L}}_{0}^{b}(\zeta)\geq-M_{R}$

for a possibly different constant, only depending on $R$, and local bounds of
$Q(x)$, $b$. Hence we have

$\Phi_{t}+{\mathcal{L}}^{b}([\Phi])-f(x)\geq
k-C\,M_{R}-\|f\|_{L^{\infty}(B_{2R})}$

which implies that $\Phi$ is a supersolution for $x\in B_{2R}$, up to choosing
$k\gtrsim CM_{R}+\|f\|_{L^{\infty}(B_{2R})}$. If $x\in\partial B_{2R}$, we
have $|x-y|>R$, and $\Phi\geq u$ provided
$C\,R^{2}+u_{0}(y)\geq\|u(t)\|_{L^{\infty}(B_{2R})}$; in particular, this
holds if

$C\,R^{2}\geq 2\|u\|_{L^{\infty}((0,T)\times B_{2R})}\,.$

At $t=0$, we have $u_{0}\leq\Phi$ if $u_{0}(x)-u_{0}(y)\leq C|x-y|^{2}+\eta$.
This holds if $\omega(|x-y|)-\eta\leq C|x-y|^{2}$, where $\omega(\cdot)$ is a
modulus of continuity of $u_{0}$ in $B_{2R}$. Without loss of generality, we
can assume that $\omega(r)\geq L\,r$ near the origin, so there exists
$\eta_{C}:=\sup_{r\in(0,3R)}[\omega(r)-Cr^{2}]>0$ and $\eta_{C}\to 0$ as
$C\to\infty$. Choosing $\eta=\eta_{C}$, we have the comparison between $u$ and
$\Phi$ and we conclude that $u(t,x)\leq\Phi(t,x)$ for $x\in B_{2R}$. Choosing
$x=y$ gives

$u(t,y)\leq u_{0}(y)+C\,M_{R}\,t+\|f\|_{L^{\infty}(B_{2R})}\,t+\eta_{C}\,.$

Using $\eta_{C}\to 0$ as $C\to\infty$, we deduce that

$\|u(t)-u_{0}\|_{L^{\infty}(B_{R})}\mathop{\to}^{t\to 0}0\,,$

and this limit is uniform for all solutions $u$ which are uniformly bounded in
$B_{2R}$.

As a Corollary of the previous lemma, we infer a compactness principle for
solutions of the Fokker-Planck equation.

###### Lemma 6.3

Let $m\in C^{0}([0,T];{\mathcal{P}}({\mathbb{R}}^{d}))$ be a solution of
(2.10) in the sense of Definition 2.4, where
$m_{0}\in{\mathcal{P}}({\mathbb{R}}^{d})\cap{\mathcal{M}}_{k}({\mathbb{R}}^{d})$,
and ${\mathcal{L}},b$ satisfy the assumptions of Theorem 2.5. Then, for every
$\varepsilon>0$ there exists $\delta>0$ such that

$d_{1}(m(t),m(s))\leq\varepsilon\qquad\forall t,s\,,\,|t-s|\leq\delta$ (6.5)

where $\delta$ only depends on $\|m_{0}\|_{{\mathcal{M}}_{k}}$, on local
uniform bounds of $b$, on lower bound of $b(t,x)\cdot x$, as well as on the
upper bounds $\|\Sigma(x)\|_{\infty},\Lambda$.

In particular, any sequence $\\{m_{n}\\}\subset
C^{0}([0,T];{\mathcal{P}}({\mathbb{R}}^{d}))$ of solutions of (2.10),
corresponding to $b_{n},m_{0n}$, is relatively compact in