# A New Non-Linear Density Fluctuations Stochastic Partial Differential
Equation With a Singular Coefficient of Relevance to Polymer Dynamics and
Rheology: Discussions, Proofs of Solution Existence, Uniqueness, and a
Conjecture

Ludovic Goudenège1 and Liviu Iulian Palade2 111E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we consider an entirely new - previously unstudied to the best
of our knowledge - type of density fluctuations stochastic partial
differential equation with a singular coefficient involving the inverse of a
probability density. The equation was recently introduced by Schieber [36]
while working on a new polymer molecular dynamics approach that pertains to
the generally called polymer reptation (aka tube) theory. The corresponding
probability density is the solution of an evolution equation (a stochastic
transport) defined on a dynamical one-dimensional subspace. A peculiarity of
the here studied equation is its very singular pattern, even though it
exhibits a well-posed structure. As a first step towards furthering the
understanding of this new class of equations we prove the existence and
uniqueness of solutions under suitable conditions. Moreover, capitalizing on
the assumed gradient structure, we prove that the existing solutions converge
to an equilibrium density.

1 CNRS, Fédération de Mathématiques de CentraleSupélec

Bât. Bouygues, 9 rue Joliot Curie, 91190 Gif-sur-Yvette, France.

2 Université de Lyon, CNRS, Institut Camille Jordan UMR 5208, INSA-Lyon

Pôle de Mathématiques, Bât. Leonard de Vinci No. 401, 21 avenue Jean Capelle,
69621 Villeurbanne, France.

Keywords: stochastic partial differential equation with a singular
coefficient; solution existence results.

AMS subject classification: Primary 60H17; Secondary 35B30.

##  1 Introduction

Basically there are two dominant, authoritative theories (together with their
variants) in the realm of polymer melt molecular dynamics which are of
paramount importance to the rheology of those non-Newtonian fluids.
Specifically, that of Bird, Curtiss, Armstrong and Hassager [1] based on early
ideas set-up by Kirkwood [25], and that of Doi and Edwards [19] which
incorporates de Gennes’ reptation concept [18]. The former is known as
kinetical theory for polymer dynamics while the later is commonly referred to
as either the tube model or the reptation theory. In this paper we are
concerned uniquely by the second approach, and for sake of clarity we only
give a succinct overview of the key ideas and borrow some inspiration from a
few of the acclaimed references like e.g. [19], [24], [27], [28], [29], [37],
to which we equally refer the interested reader for detailed presentations.

In a melt polymer material sample, the macromolecular chains are highly
entangled with each other, hence the motion of a single chain is strongly
influenced by the collective presence and dynamics of the surrounding
neighbours. Such an influence is geometrically idealized by considering an
individual chain as being confined to a tube with deformable walls - hence the
name of the theory. Because the tube largely hinders the transverse motion,
the large scale geometric rearrangement of the chain configuration takes place
inside the tube walls via a snake-like reptational motion along its own
length. As the chain is geometrically approximated as a concatenation of
deformable segments (one encounters a many body problem), in the de Gennes -
Doi - Edwards theory a chain segment is subjected to a three-stage consecutive
motion: a sudden deformation immediately followed by a quick process of
retraction to its original length, process continued by a slower reptation
motion of relaxation out of the original tube. Actually the reptational
diffusion also affects the shape of the both tube extremities in this way: the
part of the tube which is emptied as the chain retracts will cease to exist
(thus a constraint is released) while a new tube portion is formed at the
other extremity and surrounds the diffusing chain. Next, all these dominant
interactions are accounted for in a so-called configurational probability
density diffusion equation - typically a parabolic partial differential
equation (which may be deterministic or stochastic in nature). Its solution
constitutes the key ingredient for obtaining the stress tensor which
ultimately enters the momentum balance system of equations describing the
macroscopic fluid motion. For some recent modeling results of the tube model
in applied rheology see [10], [11], [31] and references cited therein.

The reptation theory (together with the kinetical approach) is popular in the
area of polymer rheology. Since it was published in its original form in
1978-79 (see [19]) it has been subjected to a multitude of modifications,
refinements, improvements to make it meet the needs of better predicting an
ever wider variety of rheological experimental data; in turn this however made
modeling necessarily more complex. Such undertakings include - but not
restricted to - tube length diameter fluctuations, constrained release,
fluctuating slip links and slip springs, inhomogeneities associated with
density fluctuations. Many of these theoretical improvements were tested via
advanced Monte Carlo simulations performed on stochastic equations.

For instance, while focusing on the density fluctuations and their relation to
quantities like slip spring distribution (see e.g. [34], [35]), Schieber [36]
has recently authored a density fluctuations stochastic partial differential
equation, a simplified version of which is the focus of this work: see the
below given problem (1.1)-(1.5). All quantities are dimensionless.

Let $\displaystyle\rho(t,s)$ be a stochastic unknown process, with
$\displaystyle s\in\left[L_{-},L_{+}\right]$ being the space variable (related
to a tube length being equal to $\displaystyle 2L$) and $\displaystyle
t\in[0,T]$ being the time variable. Let $\displaystyle\mu(t,s)$ be the
chemical potential, $\displaystyle W(t,s)$ a Wiener process (possibly a space-
time white noise, a time white noise or a finite-dimensional brownian motion).

We are interested in the following formal stochastic partial differential
equation

$\partial_{t}\rho=\dfrac{1}{2}\dfrac{\partial}{\partial
s}\left(\dfrac{1}{\rho}\dfrac{\partial\mu}{\partial
s}\right)\mathrm{d}t+\dfrac{\partial}{\partial
s}\left(\dfrac{\mathrm{d}W}{\sqrt{\rho}}\right),\quad
s\in\left[L_{-},L_{+}\right]$ (1.1)

where the boundary conditions are stochastic in nature:

$\mathrm{d}L_{\pm}=-\dfrac{1}{\rho}\left[\dfrac{1}{2\rho}\left(\dfrac{\partial\mu}{\partial
s}\right)_{\mathcal{T}}\mathrm{d}t+\dfrac{\mathrm{d}W}{\sqrt{\rho}}\right]\Bigg{|}_{s=L_{\pm}}.$
(1.2)

As is common in thermodynamics, the $\displaystyle(\cdots)|_{\mathcal{T}}$
notation indicates parameter (temperature) $\displaystyle\mathcal{T}$ is kept
constant, we will skip this notation later.

The free energy (Hamiltonian) $\displaystyle\mathcal{E}$ of a generic unknown
$\displaystyle\rho$ is given by

$\mathcal{E}(\rho)=\int_{[L_{-},L_{+}]}\left[\rho(\cdot,s)+\dfrac{1}{\rho(\cdot,s)}\right]\mathrm{d}s=\int_{[L_{-},L_{+}]}V(\rho(\cdot,s))\mathrm{d}s,$
(1.3)

with $\displaystyle V$ the obvious potential.

The chemical potential $\displaystyle\mu$ is usually given by the functional
derivative of the free energy

$\mu=\frac{\delta\mathcal{E}}{\delta\rho}=1-\dfrac{1}{\rho^{2}}\quad(+C)$
(1.4)

but the above chemical potential may be defined up to an additive constant
$\displaystyle C$ which is used as a Lagrangian multiplier in various models.
However its presence here is impactless for only the gradient of the chemical
potential will be used.

Equation (1.1) is compatible with total mass conservation law and is also
expected to exhibit an equilibrium probability density (invariant measure)
$\displaystyle P_{eq}$ given by

$\mathrm{d}P_{\text{eq}}(\rho)=\frac{1}{Z}\exp\left[-\displaystyle\mathcal{E}(\rho)\right]\mathds{1}_{\rho\geq
0}\mathrm{d}\gamma(\rho)$ (1.5)

where $\displaystyle\gamma$ is a reference measure in some space, and
$\displaystyle Z$ is a normalization constant.

The originally proposed free energy [36] contained the extra
$\displaystyle\alpha\left(\dfrac{\partial\rho(s)}{\partial s}\right)^{2}$ term
in the integrand of equation (1.3) and the extra
$\displaystyle-2\alpha\dfrac{\partial^{2}\rho}{\partial s^{2}}$ term in
equation (1.4), with $\displaystyle\alpha\geq 0$ being a model parameter. In
the full model, when a gradient term is added to the free energy, the above
introduced singular problem pertains to the Cahn-Hilliard-Cook family of
equations (with a singular mobility, singular potential and singular
multiplicative noise). Specifically:

$\mathcal{E}_{\alpha}(\rho)=\int_{[L_{-},L_{+}]}\left[\rho(\cdot,s)+\dfrac{1}{\rho(\cdot,s)}+\alpha\left(\dfrac{\partial\rho(\cdot,s)}{\partial
s}\right)^{2}\right]\mathrm{d}s,$ (1.6)

$\mu=\frac{\delta\mathcal{E}_{\alpha}}{\delta\rho}=1-\dfrac{1}{\rho^{2}}-2\alpha\dfrac{\partial^{2}\rho}{\partial
s^{2}}\quad(+C).$ (1.7)

In this paper, taking (1.1)-(1.5) as starting point, we undertake to study the
$\displaystyle n$-dimensional generalized problem by considering a regularized
expression of the full problem, using an abstract formulation which encodes
all the difficulties in the classical approach of SPDEs. We detail the
expected concept of solution that could remain unaltered in the singular case.
We also explain why a singular version of the 1-dimensional problem is well-
posed with the extra-$\displaystyle\alpha$-term in (1.6)-(1.7) but under the
assumption of additive noise.

Our main result is the following existence theorem (with the meaning of the
solution concept being detailed later in the text):

######  Theorem 1.1 (Main Result).

Let $\displaystyle\rho_{0}$, defined on a given space domain, be a smooth
enough initial probability density.

  * •

Given a regular abstract formulation (possibly in the $\displaystyle
n$-dimensional space), then, for all $\displaystyle T>0$, there exists a
unique solution $\displaystyle(\rho_{t})_{t\in[0,T]}$ to the equation (1.1).
Moreover, under strict monotonicity, for $\displaystyle T\rightarrow+\infty$,
the distribution of the solution converges to an equilibrium density
(invariant measure).

  * •

Given a singular abstract formulation in the $\displaystyle 1$-dimensional
space but with additive noise and the extra-term with $\displaystyle\alpha>0$,
then, for all $\displaystyle T>0$, there exists a unique solution
$\displaystyle(\rho_{t})_{t\in[0,T]}$ to the equation (1.1). Moreover, the
distribution of the solution converges to the unique equilibrium density
(invariant ergodic measure) given by equation (1.5) with $\displaystyle\gamma$
a reference Gaussian measure.

This paper is organized as following: Section 2 introduces the concept of
solutions based on the general theory of SPDEs. We prove that some a priori
estimates are valid for the introduced solutions. It also contains the problem
under scrutiny together with the appropriate physical considerations. In
Section 3, we introduce a regular abstract formulation to present the
essential difficulties which prevent us of using classical techniques about
the existence of solutions. In Section 4, where the main result Theorem 4.1 is
proved, we introduce a simplified formulation by circumventing selected
difficulties which were presented in the previous section. However, we can
keep a singular potential which constitutes the core of the physical model.
Next, the path-wise uniqueness is given in Section (5) for all the previous
here considered variants of the original model. The existence and convergence
of invariant measures are presented in Section (6). We eventually draw
conclusions for this paper model, introduce a conjecture about the existence
and uniqueness of solutions for a general problem in Section (7).

##  2 Problem Presentation and Main Results

At first sight, this equation could be seen as a Cahn-Hilliard-Cook stochastic
partial differential equation, and in simple cases, the existence and
uniqueness could be obtained by known results. The Cahn-Hilliard-Cook equation
is a model that describes phase separation in a binary alloy (see [5], [6] and
[7]) in the presence of thermal fluctuations (see [12] and [26]). It generally
reads:

$\left\\{\begin{array}[]{ll}\partial_{t}\rho=\frac{1}{2}\nabla\cdot\left(M(\rho)\nabla\mu\right)+\dot{\xi}&\text{
on }\mathcal{D}\subset\mathbb{R}^{n},\\\ \\\
\mu=V^{\prime}(\rho)-2\alpha\Delta\rho,&\text{ on
}\mathcal{D}\subset\mathbb{R}^{n},\\\ \\\
\nabla\rho\cdot\nu=0=\nabla\mu\cdot\nu,&\text{ on }\partial\mathcal{D},\\\
\end{array}\right.$ (2.1)

where $\displaystyle t$ denotes the time variable, $\displaystyle\alpha$ is a
small non-negative parameter, $\displaystyle\nu$ is the outward normal on the
boundary $\displaystyle\partial\mathcal{D}$ of the space domain
$\displaystyle\mathcal{D}$, $\displaystyle\nabla\cdot$ and
$\displaystyle\nabla$ are respectively the divergence and gradient operators
in the space variable $\displaystyle s\in\mathcal{D}$, and
$\displaystyle\Delta$ is the Laplace operator. The variable
$\displaystyle\rho$ represents the ratio between the two species and the noise
term $\displaystyle\dot{\xi}$ accounts for the thermal fluctuations. For this
kind of equation, a key point is that the average
$\displaystyle\int_{\mathcal{D}}\rho\ \mathrm{d}s$ is a conserved quantity.

In the Cahn-Hilliard model, the nonlinear term $\displaystyle V^{\prime}$ is
based on a double-well shaped potential $\displaystyle V$ (hence non-convex),
and the function $\displaystyle M$ represents the mobility (a diffusion
velocity). Usually the mobility is a constant or the function $\displaystyle
M:\rho\mapsto(1-\rho^{2})$. In the deterministic case, the equation is
obtained by taking the gradient of the free energy $\displaystyle\mathcal{E}$
in $\displaystyle\mathrm{H}^{-1}(\mathcal{D})$ but on a fixed domain
$\displaystyle\mathcal{D}$.

Since the gradient operates in $\displaystyle\mathrm{H}^{-1}(\mathcal{D})$, a
fourth order equation is obtained. Thus, if $\displaystyle\alpha>0$, the
presented model is of Cahn-Hilliard-Cook type, but with moving domain,
singular mobility and multiplicative noise. Indeed, in this new model, the
mobility

$M:\rho\mapsto 1/\rho,$ (2.2)

is singular leading also to a singular term in front of the noise. Indeed, by
fluctuation-dissipation theorem, we expect the square root of the mobility in
front of the noise (see Equation (1.1)).

This new model will lead to multiple mathematical difficulties, in particular
concerning the multiplicative noise, which could be a hard question. All of
these difficulties will not be solved in this paper, but we will give hints
for possible direction of improvements. The aim of this article is to give the
first mathematical structure as a starting point for future research in tube
model and reptation theory.

The deterministic Cahn-Hilliard equation where $\displaystyle V$ is a
polynomial function has first been studied in [6], [26] and [30]. The drawback
of a polynomial nonlinearity is that the solution is not constrained to remain
in the physically relevant interval. Singular nonlinear terms such as the
logarithmic or negative power nonlinearity do remedy this problem. Such non
smooth functions $\displaystyle V$ have also been considered (see [4] and
[15]). This stochastic Cahn-Hilliard-Cook equation has also been first studied
in the case of a polynomial nonlinearity: see [2], [3], [8], [9], [14], [21].
Again, with a polynomial nonlinear term the solutions do not in general remain
in the physical space of interest. It actually gets even worse for in the
presence of noise any solution leaves this interval immediately with positive
probability. Of notice, however, in the work of Debussche and Goudenège [17,
22] and that of Debussche and Zambotti [16], the solution is forced to stay in
the physical interval thanks to a singular potential.

In the case considered here, the parameter $\displaystyle\alpha$ may vanish,
the mobility is highly singular, the stochastic term is a singular
multiplicative noise, and the potential $\displaystyle V$ is strictly convex,
thus the equation is not of Cahn-Hilliard-Cook type, and we will need a new
abstract formulation to treat this new non-linear density fluctuations
equation. On the contrary, in the case of additive noise assumption, with the
extra-$\displaystyle\alpha$-term and regular mobility, we can consider a
potential $\displaystyle V(\rho)=\rho+1/\rho$ in the energy
$\displaystyle\mathcal{E}_{\alpha}$ since it is related to the model described
in [22].

It would be preposterous to assume studying such a model is a classical
undertaking. Henceforth conspicuously obtaining a preliminary result with an
abstract formulation or simplified term potential is all the more necessary
and useful. We now refer the reader to the classical work of Da Prato and
Zabczyk [13] for notations, basic notions and concepts.

We describe the concept of solution for a regular potential $\displaystyle V$
(at least of class $\displaystyle\mathcal{C}^{2}$ and convex) and a regular
mobility $\displaystyle M$ (at least of class $\displaystyle\mathcal{C}^{1}$
and non-negative). Also observe the concept of solution (valid for smooth
potential) may remain unaltered in the singular case, but integrability
conditions should be changed to give a clear meaning to all involved terms. In
Section 3, we will simplify some assumptions and keep others unchanged. In
Section 4, we will also simplify some aspects and keep the others unchanged,
and the integrability conditions will be detailed.

######  Definition 2.1.

Let $\displaystyle\ell_{-}$ and $\displaystyle\ell_{+}$ with
$\displaystyle\ell_{-}<\ell_{+}$ be fixed. Let
$\displaystyle\rho_{0}\in\mathcal{C}([\ell_{-},\ell_{+}];\mathbb{R}_{+}^{*})$
be the initial density in the space of probability measures
$\displaystyle\mathcal{P}$. We say that
$\displaystyle\left(L_{-}(t),L_{+}(t),\rho(t),W_{t}\right)_{t\in[0,T]}$,
defined on a filtered complete probability space
$\displaystyle\left(\Omega,\mathbb{P},\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]}\right)$,
is a weak solution to (1.1) on $\displaystyle[0,T]$ for the initial condition
$\displaystyle\rho_{0}$ if we have

  1. (a)

a.s. $\displaystyle L_{\pm}\in\mathcal{C}\left([0,T];\mathbb{R}\right)$ with
$\displaystyle L_{-}(t)<L_{+}(t)$ for all $\displaystyle t\in[0,T]$ and
$\displaystyle L_{\pm}(0)=\ell_{\pm}$,

  2. (b)

$\displaystyle\rho\in\mathrm{L}^{2}\left(\Omega;\mathcal{C}\left([0,T];\mathcal{C}\left([L_{-},L_{+}];\mathbb{R}_{+}^{*}\right)\right)\cap\mathrm{L}^{2}\left([0,T];\mathrm{H}^{1}\left([L_{-},L_{+}];\mathbb{R}_{+}^{*}\right)\right)\right)$
and $\displaystyle\rho(0)=\rho_{0}$,

  3. (c)

$\displaystyle\mu=V^{\prime}(\rho)\in\mathrm{L}^{2}\left(\Omega;\mathcal{C}\left([0,T];\mathcal{C}^{1}\left([L_{-},L_{+}];\mathbb{R}\right)\right)\right)$
and $\displaystyle\mu(0)=V^{\prime}(\rho_{0})$,

  4. (d)

$\displaystyle W$ is a continuous version of Brownian motion,

  5. (e)

the process $\displaystyle\left(\rho(t,\cdot),W_{t}\right)$ is
$\displaystyle(\mathcal{F}_{t})$-adapted,

  6. (f)

the mobility $\displaystyle M$ is such that $\displaystyle M(\rho)\in
L^{2}\left(\Omega;\mathrm{L}^{1}\left([0,T];\mathrm{L}^{1}((L_{-},L_{+});\mathbb{R}_{+})\right)\right)$,

  7. (g)

the operator $\displaystyle Q$ is such that $\displaystyle
Q(\rho)\in\mathrm{L}^{2}\left(\Omega;\mathrm{L}^{2}\left([0,T];\mathrm{L}^{2}((L_{-},L_{+});\mathbb{R}_{+})\right)\right)$,
also we assume that $\displaystyle\mathrm{Tr}(Q^{*}\Delta Q)$ is controlled by
$\displaystyle C_{Q,1}+C_{Q,2}\|\rho\|^{2}$.

  8. (h)

for all $\displaystyle h\in\mathcal{C}^{4}(\mathbb{R};\mathbb{R})$ and for all
$\displaystyle 0\leq t\leq T$ :

$\displaystyle\displaystyle\langle\rho(t,\cdot),h\rangle$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\langle\rho(0,\cdot),h\rangle-\frac{1}{2}\int_{0}^{t}\left\langle
M(\rho(r,\cdot))\nabla\mu,\nabla
h\right\rangle\mathrm{d}r-\int_{0}^{t}\left\langle\nabla
h,Q(\rho(r,\cdot))\mathrm{d}W_{r}\right\rangle$
$\displaystyle\displaystyle-\alpha\int_{0}^{t}\left\langle\rho(r,\cdot),\Delta^{2}h\right\rangle\mathrm{d}r.$

We point out that the continuity of $\displaystyle L_{\pm}$ allows to consider
functions on compact sets which simplifies many technical points, and the
scalar product are defined on $\displaystyle{[L_{-},L_{+}]}$ assuming the
quantity vanishes outside this domain. Since $\displaystyle\rho>0$, then its
infimum is strictly positive, the chemical potential $\displaystyle\mu$ is
well defined even if $\displaystyle V$ has singularities at point
$\displaystyle 0$, and $\displaystyle M(\rho)$ is
$\displaystyle\mathrm{L}^{1}$ for the same reason. In the case of a Wiener
process in abstract space $\displaystyle U$ at point $\displaystyle(d)$, the
condition $\displaystyle(g)$ could be set as: $\displaystyle Q(\rho)$ is
Hilbert-Schmidt from $\displaystyle U$ to the Hilbert space
$\displaystyle\mathrm{H}=\mathrm{L}^{2}(\mathcal{D})$ with maximal domain
$\displaystyle\mathcal{D}$ and inner product
$\displaystyle\langle\cdot,\cdot\rangle$. Actually in Definition (2.1) we
provided details on all the needed properties to make sure all terms in the
last point (h) clearly make sense.

We shall now, step by step, proceed to proving the afore announced main
result. Before engaging in this undertaking, we pause for several short
clarifying comments. On one hand, it is worthy stating the proof may not be
particularly challenging if one considers the equation as being a system
gradient in infinite dimension, by using the Da Prato and Zabczyk [13]
framework and by additionally assuming the noise to be an additive space-time
Wiener process. In this case there will be a unique invariant measure, given
by a Gibbs-like formula, which is compatible with the expected equilibrium
probability density in equation (1.5). On the other hand now, there are
multiple obstacles to be dealt with as one proceeds along those lines. For
instance, one is the gradient structure needing more clarifications. Another
difficulty concerns the moving boundary, since the space domain is not fixed.
This later one seems however not of a hindering nature as the imposed boundary
condition has been constructed to cancel supplementary terms coming from
integration by parts formula.

In order to explore the concept of solutions, we will detail some a priori
estimates and facts based on the existence of smooth solutions. It gives
strong hints that the equation is well-posed and satisfied expected physical
properties.

######  Proposition 2.1.

Consider a solution in the sense of Definition 2.1 up to time $\displaystyle
T>0$, then the solution lives in
$\displaystyle\mathrm{L}^{2}(\Omega;\mathrm{L}^{\infty}([0,T];\mathrm{L}^{2}([L_{-},L_{+}])))$.
Moreover, with $\displaystyle\alpha=0$, for chemical potential
$\displaystyle\mu$ given by (1.4) and mobility $\displaystyle M$ given by
equation (2.2), then the energy $\displaystyle\mathcal{E}$ of the solution is
well defined for all $\displaystyle t\in[0,T]$.

###### Proof.

Let $\displaystyle\alpha=0$, by Itô’s formula, we have

$\displaystyle\displaystyle\frac{1}{2}\mathrm{d}\|\rho\|^{2}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\langle\mathrm{d}\rho,\rho\rangle$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{2}\left\langle
M(\rho)\nabla\mu,\nabla\rho\right\rangle\mathrm{d}t-\left\langle\nabla\rho,Q(\rho)\mathrm{d}W_{t}\right\rangle+\frac{1}{2}\mathrm{Tr}(Q^{*}(-\Delta)Q)\mathrm{d}t$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{2}\left\langle M(\rho)\nabla\rho
V^{\prime\prime}(\rho),\nabla\rho\right\rangle\mathrm{d}t-\left\langle\nabla\rho,Q(\rho)\mathrm{d}W_{t}\right\rangle+\frac{1}{2}\mathrm{Tr}(Q^{*}(-\Delta)Q)\mathrm{d}t$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle-\left\langle\nabla\rho(t,\cdot),Q(\rho(t,\cdot))\mathrm{d}W_{t}\right\rangle+(C_{Q,1}+C_{Q,2}\|\rho\|^{2})\mathrm{d}t$

using that $\displaystyle V$ is a convex potential and $\displaystyle M$ is a
non-negative mobility, and $\displaystyle C_{Q,1}$ and $\displaystyle C_{Q,2}$
are finite constants invoking that the operator $\displaystyle Q$ has enough
boundedness properties (almost linear growing, Lipschitz assumption, etc.).
Here, since the noise is a one-dimensional Brownian motion, the assumption in
point (g) could be simplified. But remark that the classical assumption about
finite trace may be used here to generalize to Hilbert-Schmidt type operators
and space-time Wiener processes. Using that the stochastic term is a
martingale with null expectation, by Gronwall’s Lemma, we obtain a bound for
all $\displaystyle t\in[0,T]$, which leads to a bound for the
$\displaystyle\mathrm{L}^{\infty}$ norm such that

$\displaystyle\mathbb{E}\sup_{t\in[0,T]}\|\rho(t,\cdot)\|^{2}\leq\mathbb{E}\|\rho(0,\cdot)\|^{2}+C_{T}=\|\rho_{0}\|^{2}+C_{T}<+\infty$

for some positive constant $\displaystyle C_{T}$ possibly depending on
$\displaystyle\rho_{0},T,C_{Q,1}$ and $\displaystyle C_{Q,2}$. At this point
it is clear that the initial data $\displaystyle\rho_{0}$ can be randomized
(with bounded second moment) to obtain exactly the same result. If we have
$\displaystyle\alpha>0$, the same computation leads to

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\|\rho(t,\cdot)\|^{2}+2\alpha\int_{0}^{T}\|\Delta\rho(t,\cdot)\|^{2}\mathrm{d}t\right]\leq\|\rho_{0}\|^{2}+C_{T}<+\infty$

which is eventually a stronger result.

Concerning the energy, with $\displaystyle\alpha=0$, using that almost surely
$\displaystyle\rho$ is a continuous non-negative function on compacts spaces,
hence bounded for below by a non-negative constant
$\displaystyle\varepsilon>0$, the Itô’s terms may be bounded, such that all
the terms are well-defined using explicit truncation, and we obtain by Itô
formula

$\displaystyle\displaystyle\mathrm{d}\left(\rho+\frac{1}{\rho}\right)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\left\langle\mathrm{d}\rho,1-\frac{1}{\rho^{2}}\right\rangle=\langle\mathrm{d}\rho,\mu\rangle$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{2}\left\langle
M(\rho)\nabla\mu,\nabla\mu\right\rangle\mathrm{d}t-\left\langle\nabla\mu,Q(\rho)\mathrm{d}W_{t}\right\rangle+\frac{1}{2}Tr(Q^{*}\nabla^{*}H(\rho)\nabla
Q)\mathrm{d}t$

where $\displaystyle H$ is the Hessian operator of the function $\displaystyle
x\mapsto x+1/x$. Using again that the mobility is non-negative, we obtain the
following inequality

$\mathrm{d}\left(\rho+\frac{1}{\rho}\right)\leq-\left\langle\nabla\mu,Q(\rho)\mathrm{d}W_{t}\right\rangle+C(Q,\rho,\varepsilon)\mathrm{d}t$
(2.3)

with a random function $\displaystyle C(Q,\rho,\varepsilon)$ which depends
quadratically on $\displaystyle Q$ and $\displaystyle\rho$, but certainly on
the truncated parameters $\displaystyle\varepsilon$ in unbounded manner. Using
Doob’s inequality for the martingale term, the assumptions about integrability
in Definition 2.1 will be enough to obtain

$\displaystyle\sup_{t\in[0,T]}\mathcal{E}(\rho(t,\cdot))\leq\mathcal{E}(\rho(0,\cdot))+\sup_{t\in[0,T]}\int_{0}^{t}\left|\left\langle\nabla\mu,Q(\rho)\mathrm{d}W_{t}\right\rangle\right|+\int_{0}^{t}\left|C(Q,\varepsilon,\rho)\right|\mathrm{d}t<+\infty.$

∎

Assuming that we can take the expectation in Equation (2.3) to suppress the
martingale term, we obtain a bound in expectation such that

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\mathcal{E}(\rho(t,\cdot))\right]\leq\mathbb{E}\left[\mathcal{E}(\rho_{0})\right]+\mathbb{E}\left[\int_{0}^{T}C(Q,\varepsilon,\rho)\mathrm{d}t\right]$

which is not independent of the truncated parameter
$\displaystyle\varepsilon$, thus potentially infinite when passing to the
limit or taking expectation. Remark again that the initial data can be
randomized (with bounded inverse first moment, i.e. bounded energy) to obtain
the same result.

Finally, in the case $\displaystyle\alpha>0$, the same computation leads to

$\displaystyle\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\mathcal{E}(\rho(t,\cdot))+\frac{1}{2}\inf_{t\in[0,T]}|M(\rho(t,\cdot))|\int_{0}^{T}\|\nabla\mu\|^{2}\mathrm{d}t\right]$
$\displaystyle\displaystyle\leq\mathcal{E}(\rho_{0})+\mathbb{E}\left[\int_{0}^{T}C(Q,\varepsilon,\rho)\mathrm{d}t\right]+\alpha\left|\int_{0}^{T}\langle\Delta\rho,\Delta\mu\rangle\mathrm{d}t\right|$

but the last term is not controlled by integrability condition given in
Definition 2.1, and it seems unreasonable to expect such controlability.

##  3 Regular abstract formulation

In this section, we will assume that the equation can be written in a
classical abstract form

$\displaystyle\mathrm{d}X_{t}+A(X)\mathrm{d}t=B(X)\mathrm{d}W_{t}$

with operators satisfying assumptions that will be detailed below. Considering
this form, the proof of existence and uniqueness of solution is based on
classical theorems the detailed presentations of which are given in [13], [23]
or [33]. Precisely, we assume to have a Gelfand triple
$\displaystyle(\mathrm{V},\mathrm{H},\mathrm{V}^{*})$, where
$\displaystyle\mathrm{V}$ is a reflexive Banach space and
$\displaystyle\mathrm{H}$ is an Hilbert space, and a space of Hilbert-Schmidt
operators $\displaystyle\mathrm{L}_{HS}(\mathrm{U},\mathrm{H})$ on some
abstract space $\displaystyle\mathrm{U}$ to define the Wiener process. We
write the operators on these spaces $\displaystyle
A:\mathrm{V}\rightarrow\mathrm{V}^{*}$ and $\displaystyle
B:\mathrm{V}\rightarrow\mathrm{L}_{HS}(\mathrm{U},\mathrm{H})$. The notation
$\,{}_{\mathrm{V}^{*}}\langle\cdot,\cdot\rangle_{\mathrm{V}}$ is the canonical
notation for the duality bracket between the spaces
$\displaystyle\mathrm{V}^{*}$ and $\displaystyle\mathrm{V}$, compatible with
the inner product $\displaystyle\langle\cdot,\cdot\rangle$ on the space
$\displaystyle\mathrm{H}$. We assume that they satisfy the following
assumptions:

  * •

(Hemicontinuity) For all $\displaystyle u,v,w\in\mathrm{V}$

$\displaystyle\lambda\mapsto 2\,_{\mathrm{V}^{*}}\langle A(u+\lambda
v),w\rangle_{\mathrm{V}}\text{ is continuous}.$

  * •

(Weak Monotonicity) There exists $\displaystyle c\in\mathbb{R}$ such that for
all $\displaystyle u,v\in\mathrm{V}$

$\displaystyle 2\,_{\mathrm{V}^{*}}\langle
A(u)-A(v),v-u\rangle_{\mathrm{V}}+\|B(u)-B(v)\|_{HS}^{2}\leq
c\|u-v\|_{\mathrm{H}}^{2}.$

  * •

(Coercivity) There exist $\displaystyle q\in\ ]1,\infty[$, $\displaystyle
c_{1}\in\mathbb{R}$, $\displaystyle c_{2}\in]0,\infty[$ and an
$\displaystyle\mathcal{F}_{t}$-adapted process $\displaystyle
f\in\mathrm{L}^{1}(0,T)$ such that for all $\displaystyle u\in\mathrm{V}$

$\displaystyle 2\,_{\mathrm{V}^{*}}\langle
A(u),u\rangle_{\mathrm{V}}+\|B(u)\|_{HS}^{2}\leq
c_{1}\|u\|_{\mathrm{H}}^{2}-c_{2}\|u\|^{q}_{\mathrm{V}}+f(t).$

  * •

(Boundedness) There exist $\displaystyle c_{3}\in[0,\infty[$ and an
$\displaystyle\mathcal{F}_{t}$-adapted process $\displaystyle
g\in\mathrm{L}^{\frac{q}{q-1}}(0,T)$ such that for all $\displaystyle
u\in\mathrm{V}$

$\displaystyle\|A(u)\|_{\mathrm{V}^{*}}\leq
c_{3}\|u\|^{q-1}_{\mathrm{V}}+g(t).$

It is important to notice that many problems have been solved assuming that an
abstract formulation actually exists. Indeed, the moving domain
$\displaystyle[L_{-},L_{+}]$ will be certainly encoded in the concept of
solution as $\displaystyle X_{t}$ being $\displaystyle(L_{-},L_{+},\rho_{t})$.
Or, under reformulation, we could assume that the domain is fixed and given by
the initial domain $\displaystyle[\ell_{-},\ell_{+}]$, which solves the point
$\displaystyle(a)$. Moreover, the fiber bundles in points $\displaystyle(b)$
and $\displaystyle(c)$ of Definition 2.1 are now well-defined as classical
spaces. Finally, the mobility $\displaystyle M$ and the operator
$\displaystyle Q$ are included in the operator $\displaystyle A$ and
$\displaystyle B$, which will ensure the points $\displaystyle(f)$ and
$\displaystyle(g)$. Even the concept of chemical potential $\displaystyle\mu$
has been included in the operator $\displaystyle A$. Within this framework,
the concept of solution introduced in 2.1 will be simplified since we can
define a new one.

One has:

######  Definition 3.1.

We say that $\displaystyle\left(X(t),W(t)\right)_{t\in[0,T]}$, defined on a
filtered complete probability space
$\displaystyle\left(\Omega,\mathbb{P},\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]}\right)$,
is a weak solution to (1.1) on $\displaystyle[0,T]$ for the initial condition
$\displaystyle X_{0}$ if we have

  1. (b’)

$\displaystyle
X\in\mathrm{L}^{2}\left(\Omega;\mathcal{C}\left([0,T];\mathrm{H}\right)\cap\mathrm{L}^{2}\left([0,T];\mathrm{V}\right)\right)$
and $\displaystyle X(0)=X_{0}$,

  2. (d’)

$\displaystyle W$ is a continuous version of Brownian motion (or Wiener
process),

  3. (e’)

the process $\displaystyle\left(X(t),W(t)\right)$ is
$\displaystyle(\mathcal{F}_{t})$-adapted,

  4. (h’)

for all $\displaystyle h\in\mathrm{V}^{*}$ and for all $\displaystyle 0\leq
t\leq T$ :

$\displaystyle\displaystyle\langle X(t),h\rangle$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\langle
X(0),h\rangle-\int_{0}^{t}\left\langle
A(X(r)),h\right\rangle\mathrm{d}r+\int_{0}^{t}\left\langle
B(X(r)),h\mathrm{d}W_{r}\right\rangle.$

###  3.1 Existence of solution

One has the following existence Theorem, standing actually for the first part
of Theorem • ‣ 1.1.

######  Theorem 3.1.

Suppose that there exists a regular abstract formulation satisfying classical
assumptions. Given an initial data in the Banach space
$\displaystyle\mathrm{V}$, then, for all $\displaystyle T>0$, there exists a
unique solution $\displaystyle(X_{t})_{t\in[0,T]}$ in the sense of Definition
3.1 up to time $\displaystyle T$.

###### Proof.

We have supposed that the classical assumptions are satisfied, thus we do not
have to verify them. However, we will show how they may be obtained with
assumptions on mobility $\displaystyle M$, potential $\displaystyle V$ and
operator $\displaystyle Q$, or on regularized versions of these objects.

Usually, to prove the existence of a solution of a singular equation, we need
to prove that a regularized version of the equation is well-posed with
estimates being independent of the regularization procedure. The possible
regularized version may be some finite dimensional approximation (like
Galerkin), a truncation of the non-linearity, a Yosida regularization, or a
numerical approximation. Thus, all future approaches will have to use the
following calculations and to prove that the estimates are independent of the
regularization procedure.

In Theorem 3.1, we proved that some solution could exist in a fixed domain and
that the Itô’s formula (being used for a priori estimates) suggests that the
operator $\displaystyle A$ has the form $\displaystyle
A:X\mapsto\frac{1}{2}\langle M(X)V^{\prime\prime}(X)\nabla X,\cdot\rangle$
plus extra additive terms involving Laplace operator when
$\displaystyle\alpha\neq 0$. It also suggests that the operator $\displaystyle
B$ has the form $\displaystyle
B:X\mapsto(w\mapsto\langle\cdot,\nabla^{*}Q(X)w\rangle)$. Denote
$\displaystyle a:=x\mapsto M(x)V^{\prime\prime}(x)$ the non-linearity (or,
more precisely, a regularized version of it) to simplify the notations in the
following computations. It is assumed that it is non-negative, as suggested by
the expected physical properties of the convex potential $\displaystyle V$ and
the non-negative mobility $\displaystyle M$. We now illustrate the
difficulties in obtaining the expected bounds.

Hemicontinuity.

Suppose that the function $\displaystyle a$ is continuous and bounded from
below by non-negative constant (locally on balls of
$\displaystyle\mathrm{V}$). Let $\displaystyle u,v,w\in\mathrm{V}$. Let
$\displaystyle(\lambda_{k})_{k\in\mathbb{N}}$ converging to
$\displaystyle\lambda$ as $\displaystyle k$ grows to infinity, then

$\displaystyle\displaystyle\left|2\,_{\mathrm{V}^{*}}\langle
A(u+\lambda_{k}v)-A(u+\lambda v),w\rangle_{\mathrm{V}}\right|$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\left|\langle
a(u+\lambda_{k}v)\nabla(u+\lambda_{k}v)-a(u+\lambda v)\nabla(u+\lambda
v),\nabla w\rangle\right|$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\left|\langle\left(a(u+\lambda_{k}v)-a(u+\lambda
v)\right)\nabla(u+\lambda_{k}v),\nabla w\rangle\right|+\left|\langle
a(u+\lambda v)(\lambda_{k}-\lambda)\nabla v,\nabla w\rangle\right|.$

Since the sequence $\displaystyle\lambda_{k}$ is bounded, thus the sequence
$\displaystyle u+\lambda_{k}v$ lives in a ball $\displaystyle\mathcal{B}$ in
$\displaystyle\mathrm{V}$. On this ball, the function $\displaystyle a$ is
continuous and bounded. Let $\displaystyle\epsilon>0$ be fixed, there exists
$\displaystyle\delta_{\epsilon}>0$ small enough such that
$\displaystyle|\lambda_{k}-\lambda|<\delta_{\epsilon}$ implies
$\displaystyle|a(u+\lambda_{k}v)-a(u+\lambda v)|<\epsilon$ and thus for large
index $\displaystyle k\in\mathbb{N}$ we have that the first term is bounded by
$\displaystyle\epsilon\sup_{x\in\mathcal{B}}\|x\|_{\mathrm{V}}\|w\|_{\mathrm{V}}$
and the second term is bounded by $\displaystyle\delta_{\epsilon}\
|a(u+\lambda v)|\ \|v\|_{\mathrm{V}}\|w\|_{\mathrm{V}}$, which ensures
convergence to $\displaystyle 0$ and continuity.

Weak Monotonicity.

Let $\displaystyle u,v,w\in\mathrm{V}$, then

$\displaystyle\displaystyle 2\,_{\mathrm{V}^{*}}\langle
A(u)-A(v),v-u\rangle_{V}+\|B(u)-B(v)\|_{HS}^{2}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\langle a(u)\nabla u-a(v)\nabla
v,\nabla(v-u)\rangle+\|\nabla^{*}Q(u)-\nabla^{*}Q(v)\|^{2}_{\mathrm{H}}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\int_{\mathcal{D}}a(u)|\nabla(u-v)|^{2}+\int_{\mathcal{D}}(a(u)-a(v))\nabla
v\cdot\nabla(v-u)+\|\nabla u\ Q^{\prime}(u)-\nabla v\
Q^{\prime}(v)\|^{2}_{\mathrm{H}}.$

At this point, in order to obtain the bound $\displaystyle
c\|u-v\|^{2}_{\mathrm{H}}$, we see that the first term is not a problem since
$\displaystyle a$ is assumed to be non-negative. It can help to control other
problematic terms. For the second term, we certainly need some globally
Lipschitz regularity for the function $\displaystyle a$, but which is not
precisely clear at this point since mobility and potential are singulars. For
the third term, we certainly need the regularity and boundedness of the
derivative of $\displaystyle Q$. Remark that in the additive case
($\displaystyle B$ is a constant), this term does not appear. Remark also that
if it exists an extra term with $\displaystyle\alpha>0$ involving Laplace
operator, then we have an other term with the good sign which can help to
control the problematic terms.

Coercivity.

The coercivity corresponds to a control at infinity of the equation. The
leading term is $\displaystyle c_{1}\|u\|^{2}_{H}$, which should absorb the
operator $\,{}_{\mathrm{V}^{*}}\langle A(u),u\rangle_{\mathrm{V}}$,
$\displaystyle B$ or a supplementary term $\displaystyle
c_{2}\|u\|^{q}_{\mathrm{V}}$.

Consider the equation

$\displaystyle\displaystyle 2\,_{\mathrm{V}^{*}}\langle
A(u),u\rangle_{\mathrm{V}}+\|B(u)\|_{2}^{2}+c_{2}\|u\|^{q}_{\mathrm{V}}=$
$\displaystyle\displaystyle-\langle a(u)\nabla u,\nabla u\rangle+\|\nabla u\
Q^{\prime}(u)\|^{2}_{\mathrm{H}}$
$\displaystyle\displaystyle+c_{2}\langle\nabla u,\nabla
u\rangle^{q}+c_{2}\langle u,u\rangle^{q}.$

We have to bound these terms by $\displaystyle c_{1}\|u\|^{2}_{\mathrm{H}}$
with $\displaystyle c_{1}\in\mathbb{R}$ but remember that $\displaystyle
c_{2}>0$. It is clear that the term $\displaystyle
c_{1}\|u\|^{2}_{\mathrm{H}}$ will not control the terms involving
$\displaystyle\|u\|_{\mathrm{V}}$ since it would imply a Poincaré inequality
in reversed order, thus the only term with the good sign is the first one.
This term should control the term with the constant $\displaystyle c_{2}$
which forces the constant $\displaystyle c_{2}$ not to be large, and
$\displaystyle a$ should be lower bounded by constant $\displaystyle c_{2}>0$.
The second term involves also the gradient of $\displaystyle u$, thus it is a
supplementary term to control. The last term involves only the norm of
$\displaystyle u$ in the Hilbert space $\displaystyle\mathrm{H}$ thus it is
controlled by $\displaystyle c_{2}(1+\|u\|^{2})$ at least when $\displaystyle
1<q\leq 2$.

Boundedness.

Recall that the norm in $\displaystyle\mathrm{V}^{*}$ is defined as the
supremum of the quantity $\,{}_{\mathrm{V}^{*}}\langle
A(u),v\rangle_{\mathrm{V}}$ when $\displaystyle\|v\|_{\mathrm{V}}\leq 1$. Let
$\displaystyle v\in\mathrm{V}$, we are looking for an estimates

$\displaystyle\displaystyle\left|\,{}_{\mathrm{V}^{*}}\langle
A(u),v\rangle_{\mathrm{V}}\right|:=\frac{1}{2}\left|\langle a(u)\nabla
u,\nabla v\rangle\right|\leq c_{3}\|u\|_{\mathrm{V}}^{q-1}+g.$

It is now clear that the constant $\displaystyle c_{3}$ should control the
function $\displaystyle a$ which is expected to be bounded from above. ∎

##  4 Simplified formulation but with singular potential

In this section, we can study other formulations which are related to the
problem introduced earlier. In particular, if we assume to work in dimension 1
in space, the noise is additive and that we have the extra-term containing
$\displaystyle\alpha>0$, we can prove the existence of unique solution even
with a singular potential $\displaystyle V$ of the form
$\displaystyle\rho+1/\rho$. Furthermore, using formal computations, we can
derive an equivalent problem with a modified chemical potential of the form
$\displaystyle-1/\rho^{3}$. This equation could also be solved in the same
previous simplified framework.

Moreover, in both cases, the solution will converge to the unique equilibrium
density (invariant measure) given by the formula (1.5) with
$\displaystyle\gamma$ a reference Gaussian measure.

First, assuming that all the following computations are legitimate, we can
derive an equivalent formulation of the problem to obtain a similar form.

One has:

######  Proposition 4.1.

Assume the solution is non-negative and continuously differentiable. Then the
statement (h) in 2.1 is equivalent, for all $\displaystyle
h\in\mathcal{C}^{4}(\mathbb{R};\mathbb{R})$ and for all $\displaystyle 0\leq
t\leq T$, to

$\displaystyle\displaystyle\langle\rho(t,\cdot),h\rangle$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\langle\rho(0,\cdot),h\rangle-\frac{1}{3}\int_{0}^{t}\left\langle\frac{1}{\rho(r,\cdot)^{3}},\Delta
h\right\rangle\mathrm{d}r-\int_{0}^{t}\left\langle\nabla
h,Q(\rho(r,\cdot))\mathrm{d}W_{r}\right\rangle$
$\displaystyle\displaystyle-\alpha\int_{0}^{t}\left\langle\rho(r,\cdot),\Delta^{2}h\right\rangle\mathrm{d}r.$

###### Proof.

Assuming the solution is non-negative, the quantities $\displaystyle
M(\rho)=\frac{1}{\rho}$ and $\displaystyle\mu=1-\frac{1}{\rho^{2}}$ are well-
defined. It will justify the study of the chemical potential
$\displaystyle\mu=Cst-\frac{1}{\rho^{2}}$ when the mobility $\displaystyle
M(\rho)$ is fixed (for instance by the previous step in a Picard’s fixed-point
iteration procedure). Moreover, assuming regularity, we can derive all these
quantities. Finally, since we assume the differentials are continuous, all the
implied terms are $\displaystyle\mathrm{L}^{\infty}$ and integrable. Let
$\displaystyle h\in\mathcal{C}^{4}(\mathbb{R};\mathbb{R})$; we have the
following

$\displaystyle\displaystyle-\frac{1}{2}\langle M(\rho)\nabla\mu,\nabla
h\rangle$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{2}\left\langle\frac{1}{\rho}\nabla\left(-\frac{1}{\rho^{2}}\right),\nabla
h\right\rangle$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{2}\left\langle\frac{2}{\rho^{4}}\nabla\rho,\nabla
h\right\rangle$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{2}\left\langle\frac{2}{3}\nabla\left(-\frac{1}{\rho^{3}}\right),\nabla
h\right\rangle$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{3}\left\langle\frac{1}{\rho^{3}},\Delta
h\right\rangle.$

Of notice: for the last integration by parts formula (above), it is assumed
that boundary term vanishes, fact which is ensured by selecting appropriate
Neumann boundary conditions, which means the probability flux vanishes on the
domain boundary. Moreover the equation (1.2) is consistent with the fact that
$\displaystyle\nabla h\cdot\vec{n}$ vanishes on the boundary.

We can also stop the calculations at the prior to the last line: in this case
one observes it corresponds to considering a constant mobility $\displaystyle
M=\frac{2}{3}$ but a chemical potential
$\displaystyle\mu:=Cst-\frac{1}{\rho^{3}}$, for all constants $\displaystyle
Cst$. ∎

Next, again based on the definition 2.1, we give the definition of a solution
based on the form suggested by previous Proposition 4.1 for the chemical
potential.

######  Definition 4.1.

We say that $\displaystyle\left(\rho(t),W(t)\right)_{t\in[0,T]}$, defined on a
filtered complete probability space
$\displaystyle\left(\Omega,\mathbb{P},\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]}\right)$,
is a weak solution to (1.1) on $\displaystyle[0,T]$ for the initial condition
$\displaystyle\rho_{0}$, if a.s. we have:

  1. (b’)

$\displaystyle\rho\in\mathcal{C}\left([0,T]\times[0,1];\mathbb{R}_{+}\right)\cap\mathcal{C}\left([0,T];\mathrm{V}_{-1}\right)$
and $\displaystyle X(0)=X_{0}$,

  2. (c’)

the process $\displaystyle\mu=Cst-1/\rho^{3}$ (or
$\displaystyle\mu=Cst-1/\rho^{2}$)
$\displaystyle\in\mathrm{L}^{1}\left([0,T];\mathrm{L}^{1}((0,1);\mathbb{R}_{+})\right)$

  3. (d’)

$\displaystyle W$ is a cylindrical Wiener process in
$\displaystyle\mathrm{L}^{2}(0,1)$,

  4. (f’)

the mobility $\displaystyle M$ is constant,

  5. (g’)

the covariance is constant (additive noise),

  6. (h’)

for all $\displaystyle h\in D(\Delta^{2})$ and for all $\displaystyle 0\leq
t\leq T$ :

$\displaystyle\displaystyle\langle\rho(t),h\rangle$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\langle\rho_{0},h\rangle+\frac{M}{2}\int_{0}^{t}\left\langle\mu(r),\Delta
h\right\rangle\mathrm{d}r-\int_{0}^{t}\left\langle\nabla
h,\mathrm{d}W_{r}\right\rangle$
$\displaystyle\displaystyle-\alpha\int_{0}^{t}\left\langle\rho(r),\Delta^{2}h\right\rangle\mathrm{d}r$

Now we have the following:

######  Theorem 4.1.

Consider the equation (1.1), with constant mobility, and constant additive
noise, with initial density
$\displaystyle\rho_{0}\in\mathcal{C}([0,1];\mathbb{R}^{+})$ (thus nowhere
vanishing) having null Neumann condition on the boundary. Then, for all
$\displaystyle T>0$, there exists a unique solution
$\displaystyle(\rho_{t})_{t\in[0,T]}$ in the sense of Definition 4.1 up to
time $\displaystyle T$.

However, in the case $\displaystyle\mu=Cst-1/\rho^{2}$, an additional
reflection term should be added to force the solution to stay positive. It
takes the form $\displaystyle\langle Ah,\eta\rangle$ for $\displaystyle\eta$ a
time-space measure such that $\displaystyle
supp(\eta)\subset\\{(t,s)\in[0,T]\times(0,1):\rho(t,s)=0\\}$.

###### Proof.

There are two cases to address: $\displaystyle\mu=Cst-1/\rho^{p}$, for
$\displaystyle p=2,3$.

The proof is based on a regularization procedure to define approximated
solutions of the problem. Precisely, we have to regularize the potential
$\displaystyle V(\rho)=Cst\ \rho+1/\rho$ (whose derivative is
$\displaystyle\mu=Cst-1/\rho^{2}$) rendering it a dissipative potential
$\displaystyle V^{n}$ in order to apply the classical results of existence of
solution to gradient system. The same applies to potential $\displaystyle
V(\rho)=Cst\ \rho+1/2\rho^{2}$ whose derivative is
$\displaystyle\mu=Cst-1/\rho^{3}$. For these two approximated problems, the
solutions will exist up to any time $\displaystyle T>0$.

Denoting $\displaystyle\Delta$ the Laplace operator on $\displaystyle(0,1)$
with null Neumann boundary condition and
$\displaystyle\mathrm{V}_{-1}:=D((-\Delta)^{\frac{1}{2}})$ the reference
Hilbert space, we are in the context of gradient systems. Applying Theorem 2.2
of [22], we have the existence of solution up to a reflection measure, which
should force the solution to stay in the space $\displaystyle K$ of positive
solutions.

Moreover, using Theorem 3.1 in [22], in the first case
$\displaystyle\mu:=Cst-1/\rho^{2}$, since the negative power in the associated
chemical potential is less than $\displaystyle 3$, the measure does not
vanish.

However, using Theorem 3.1 in [22], in the second case
$\displaystyle\mu:=Cst-1/\rho^{3}$, since the negative power in the associated
chemical potential is greater than $\displaystyle 3$, the measure vanishes. ∎

##  5 Path-wise Uniqueness

The path-wise uniqueness will be proven in both cases with the same strategy
using Gronwall’s Lemma.

######  Proposition 5.1.

Let $\displaystyle(\rho^{i},W)$, $\displaystyle i=1,2$ be any pair of weak
solutions of (1.1) defined on the same probability space with the same driving
noise $\displaystyle W$, and starting from the same initial data
$\displaystyle\rho_{0}$, then $\displaystyle\rho^{1}(t)=\rho^{2}(t)$ for all
$\displaystyle t\geq 0$.

###### Proof.

In the abstract formulation, we can follow Proposition 4.2.10 of [33] to
obtain that there exists a constant $\displaystyle c\in\mathbb{R}$ such that
we have

$\displaystyle\mathbb{E}\left[\|\rho^{1}(t)-\rho^{2}(t)\|\right]\leq\exp(ct)\mathbb{E}\left[\|\rho^{1}(0)-\rho^{2}(0)\|\right].$

which is valid for a certain $\displaystyle c\in\mathbb{R}$ under the
assumption of weak monotonicity.

Assuming the same initial data
$\displaystyle\rho^{1}(0)=\rho^{2}(0)=\rho_{0}$, the previous result gives the
uniqueness, but also a stronger result about sensitivity with respect to the
initial data.

In the simplified formulation with singular potential, the uniqueness follows
from dissipativity of the non-linearity. However, in Definition 4.1, we only
have continuity in the space $\displaystyle\mathrm{V}_{-1}$, thus the norm
should be taken in $\displaystyle\mathrm{V}_{-1}$ (this space is playing the
same role than the Hilbert space $\displaystyle\mathrm{H}$ of Definition 3.1).
Let $\displaystyle\rho(t)=\rho^{1}(t)-\rho^{2}(t)$ the difference between the
two solutions starting from initial data $\displaystyle\rho_{0}$. Due to the
additive noise assumption, $\displaystyle\rho$ is solution of a random
equation (without stochastic integral)

$\frac{\mathrm{d}\rho}{\mathrm{d}t}=\frac{M}{2}\Delta(\mu^{1}-\mu^{2})-\alpha\Delta^{2}(\rho^{1}-\rho^{2})$

Let now $\displaystyle\rho^{N}=[(-\Delta)^{-1}Proj_{N}]\rho$ with
$\displaystyle Proj_{N}$ being a projector operator on first $\displaystyle N$
eigenvectors of $\displaystyle(-\Delta)$. Taking the scalar product in
$\displaystyle\mathrm{H}$ with $\displaystyle\rho^{N}$ and integrating in
time, we obtain

$|\rho^{N}(t)|^{2}_{-1}=|\rho^{N}(0)|^{2}_{-1}-M\int_{0}^{t}\langle\mu^{1}(r)-\mu^{2}(r),\rho^{N}(r)\rangle\mathrm{d}r-2\alpha\int_{0}^{t}\langle(-\Delta)^{1/2}\rho^{N}(r),(-\Delta)^{1/2}\rho^{N}(r)\rangle\mathrm{d}r.$
(5.1)

Since $\displaystyle\mu^{i}$, $\displaystyle i=1,2$ are in
$\displaystyle\mathrm{L}^{1}$, we have for almost all $\displaystyle
r\in[0,t]$,

$\langle\mu^{1}(r)-\mu^{2}(r),\rho(r)-\rho^{N}(s)\rangle\leq\|\rho(r)-\rho^{N}(r)\|_{\mathrm{L}^{\infty}([0,1])}\|\mu^{1}(r)-\mu^{2}(r)\|_{\mathrm{L}^{1}([0,1])},$
(5.2)

where $\displaystyle\|\cdot\|_{\mathrm{L}^{\infty}([0,1])}$ and
$\displaystyle\|\cdot\|_{\mathrm{L}^{1}([0,1])}$ are the classical Lebesgue
norms on the space $\displaystyle[0,1]$. The latter term converges to zero
since $\displaystyle\rho^{N}(r)$ converges uniformly to $\displaystyle\rho(r)$
on $\displaystyle[0,1]$. Moreover, for both cases
$\displaystyle\mu=Cst-1/\rho^{p}$ with $\displaystyle p=2,3$, we have

$\displaystyle\displaystyle-$
$\displaystyle\displaystyle\langle\mu^{1}(r)-\mu^{2}(r),\rho(r)\rangle=\left\langle\frac{1}{(\rho^{1}(r))^{p}}-\frac{1}{(\rho^{2}(r))^{p}},\rho(r)\right\rangle$
(5.3) $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\left\langle\frac{(\rho^{2}(r)-\rho^{1}(r))((\rho^{2}(r))^{p-1}+(p-2)\rho^{1}(r)\rho^{2}(r)+(\rho^{1}(r))^{p-1})}{(\rho^{1}(r)\rho^{2}(r))^{p}},\rho^{1}(r)-\rho^{2}(r)\right\rangle$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle 0$

Injecting (5.2) into (5.1), using the sign coming from (5.3), taking the limit
in (5.1) as $\displaystyle N$ grows to infinity, we obtain for all
$\displaystyle t\geq 0$:

$\displaystyle|\rho^{1}(t)-\rho^{2}(t)|_{-1}^{2}\leq|\rho^{1}(0)-\rho^{2}(0)|_{-1}^{2}=0,$

thus $\displaystyle\rho^{1}(t)=\rho^{2}(t)$ for all $\displaystyle t\geq 0$. ∎

##  6 Convergence of Invariant Measures

We say that the measure $\displaystyle\Pi$ is an invariant measure for the
stochastic PDE (or the associated semi-group) if, for a random variable
$\displaystyle\rho_{0}$, whose distribution is $\displaystyle\Pi$, the
solution $\displaystyle(\rho(t))_{0\leq t\leq T}$ is such that its
distribution at time $\displaystyle t$ is again $\displaystyle\Pi$. Such an
invariant measure can be easily calculated (for instance in finite dimensional
case) as eigenvector of an adjoint operator $\displaystyle\mathcal{P}^{*}$ of
the semi-group $\displaystyle\mathcal{P}$ of the Markovian evolution.

Usually, we prove that an approximated version (described by some index
$\displaystyle n\in\mathbb{N}$) of the equation possesses a semi-group which
is Strong Feller (see .e.g [32] for details) and irreducible. By Doob theorem
(see [20]) we deduce that there exists a unique and ergodic invariant measure
$\displaystyle\Pi^{n}$ defined on the abstract Hilbert space
$\displaystyle\mathrm{H}$.

Denoting $\displaystyle\rho^{n}(t;x)$ the solution starting form the initial
data $\displaystyle x\in\mathrm{H}$, we can define for all Borel bounded test
functions $\displaystyle\phi\in\mathcal{B}_{b}(\mathrm{H})$ the quantity

$\displaystyle\mathcal{P}^{n}_{t}\phi(x):=\mathbb{E}[\phi(\rho^{n}(t;x))]$

which is the semi-group of the Markovian evolution.

For instance, using Bismut-Elworthy-Li formula (assuming inversibility of
covariance operator and other technical assumptions), it can be seen that
$\displaystyle(\mathcal{P}^{n}_{t})_{t\geq 0}$ satisfies, for all
$\displaystyle\phi\in\mathcal{B}_{b}(\mathrm{H})$, $\displaystyle
n\in\mathbb{N}$ and $\displaystyle t>0$, the following estimate

$|\mathcal{P}^{n}_{t}\phi(x)-\mathcal{P}^{n}_{t}\phi(y)|\leq
C(n)\frac{e^{\lambda
t}}{\sqrt{t}}\|\phi\|_{\infty}\|x-y\|_{\mathrm{H}},\quad\text{for all
}x,y\in\mathrm{H},$ (6.1)

where we expect that the constant $\displaystyle C(n)$ could be independent of
the parameter $\displaystyle n\in\mathbb{N}$.

Furthermore, the irreducibility is usually proven using a control-type
argument, which is quite clear in the case of (non-degenerate) additive noise.

If the variable $\displaystyle\lambda$ is non-positive, then there is
exponential contraction, and the law of the solution converges exponentially
fast to the invariant measure.

###  6.1 Regular abstract formulation

Thanks to the previous Section, we have the existence of a unique solution in
the regular abstract formulation. For the second part of the Theorem • ‣ 1.1
one needs to prove there exists an invariant measure to the equation, and that
the solution converges to this invariant measure as $\displaystyle
T\rightarrow+\infty$. For this, we will need supplementary assumptions
regarding the abstract formulation, precisely a “strict monotonicity”,
described by the following Lemma:

######  Lemma 6.1 (Lemma 4.3.8 from [33] \- strict monotonicity).

Assume that the constant $\displaystyle c=-\lambda$ in the weak monotonicity
assumption for some $\displaystyle\lambda\in(0,+\infty)$. Let
$\displaystyle\eta\in(0,\lambda)$. Then there exists
$\displaystyle\delta_{\eta}\in(0,+\infty)$ such that for all $\displaystyle
v\in V$

$\displaystyle 2\,_{\mathrm{V}^{*}}\langle
A(v),v\rangle_{\mathrm{V}}+\|B(v)\|^{2}_{HS}\leq-\eta\|v\|^{2}_{\mathrm{H}}+\delta_{\eta}.$

The above is enough to obtain the following:

######  Theorem 6.1.

Assume strict monotonicity. Then there exists an invariant measure
$\displaystyle P_{\text{eq}}$ for the semi-group
$\displaystyle\mathcal{P}_{t}$ such that

$\int_{\mathrm{H}}\|x\|^{2}P_{\text{eq}}(\mathrm{d}x)<+\infty.$ (6.2)

Moreover, for any $\displaystyle\phi:\mathrm{H}\rightarrow\mathbb{R}$
Lipschitz, for any initial data $\displaystyle\rho\in\mathrm{H}$ and any
invariant measure $\displaystyle P_{\text{eq}}$ for the semi-group
$\displaystyle\mathcal{P}_{t}$, for all $\displaystyle t\in[0,+\infty)$

$\displaystyle\left|\mathcal{P}_{t}\phi(\rho)-\int_{\mathrm{H}}\phi(x)P_{\text{eq}}(\mathrm{d}x)\right|\leq\exp^{-\frac{\lambda}{2}t}\|\phi\|_{Lip}\int_{\mathrm{H}}\|\rho-x\|_{H}P_{\text{eq}}(\mathrm{d}x)$

In particular, there exists exactly one invariant measure for the semi-group
$\displaystyle\mathcal{P}_{t}$ with the property that
$\displaystyle\int_{\mathrm{H}}\|x\|P_{\text{eq}}(\mathrm{d}x)<+\infty$.

###### Proof.

The proof of this theorem comes from Theorem 4.3.9 of [33]. ∎

###  6.2 Simplified version with singular potential

To prove the existence of invariant measure, first, we need to have a
reference measure in the abstract Hilbert space. It is usually given by the
asymptotic law of the solution of the linear part of the equation. For
instance, if the operators $\displaystyle A$ and $\displaystyle B$ are linear
and constant, we can define explicitly the solution $\displaystyle Z(t;x)$ of
the linear equation with initial data $\displaystyle x\in\mathrm{H}$ which is
given by

$Z(t;x)=e^{-tA/2}x+\int_{0}^{t}e^{-(t-s)A/2}B\mathrm{d}W_{s}.$ (6.3)

As easily seen this process is in
$\displaystyle\mathcal{C}([0,+\infty[;\mathrm{H})$ (see [13]). In particular,
the mean of $\displaystyle Z$ is constant and the law of the process
$\displaystyle Z(t;x)$ is the Gaussian measure on $\displaystyle\mathrm{H}$:

$\displaystyle\gamma(t)=\mathcal{N}\big{(}e^{-tA/2}x,\Sigma_{t}^{2}\big{)},\text{
with }\Sigma_{t}^{2}=\int_{0}^{t}e^{-sA/2}BB^{*}e^{-sA/2}\mathrm{d}s$

with $\displaystyle\mathcal{N}(m,\Sigma^{2})$ the standard notation for the
Normal Law of mean $\displaystyle m$ and covariance operator
$\displaystyle\Sigma^{2}$. If we let $\displaystyle t\rightarrow+\infty$, the
law of $\displaystyle Z(t;x)$ converges to the Gaussian measure on
$\displaystyle\mathrm{H}$:

$\displaystyle\gamma:=\mathcal{N}\left(0,\int_{0}^{\infty}e^{-sA/2}BB^{*}e^{-sA/2}\mathrm{d}s\right).$

When $\displaystyle B$ commutes with $\displaystyle A$, we can obtain an
explicit representation of the covariance operator
$\displaystyle\Sigma^{2}:=B(-A)^{-1}B^{*}$.

What about the invariant measure of the non-linear equation? It is classic
knowledge that if the equation (1.1) may be approximated by a gradient system
in $\displaystyle\mathrm{H}$ which can be rewritten as:

$\left\\{\begin{array}[]{l}\mathrm{d}X_{t}^{n}+\left(\frac{1}{2}A^{n}X_{t}^{n}+\nabla
V^{n}(X_{t}^{n})\right)\mathrm{d}t=B\mathrm{d}W_{t},\\\ \\\
X_{0}^{n}=x\in\mathrm{H},\end{array}\right.$ (6.4)

where $\displaystyle\nabla$ denotes the gradient in the Hilbert space
$\displaystyle\mathrm{H}$, then one deduces there exists an unique and ergodic
Gibbs-type invariant measure $\displaystyle\Pi^{n}$ given by:

$\displaystyle\Pi^{n}(\mathrm{d}x)=\frac{1}{Z^{n}}\exp(-\mathcal{E}^{n}(x))\gamma(\mathrm{d}x),$

where $\displaystyle Z^{n}$ is a normalization constant and
$\displaystyle\mathcal{E}^{n}$ is the energy related to the potential
$\displaystyle V^{n}$ (as in (1.3)). We need finish the proof by showing that
the sequence $\displaystyle(\Pi^{n})_{n\in\mathbb{N}}$ converges to the
measure

$\displaystyle
P_{\text{eq}}(\mathrm{d}x)=\frac{1}{Z}\exp(-\mathcal{E}(x))\mathds{1}_{x\in
K}\gamma(\mathrm{d}x),$

where for all $\displaystyle x\in L^{2}(0,1)$, we have
$\displaystyle\mathcal{E}(x):=\int_{0}^{1}V(x(s))\mathrm{d}s$ (as in (1.3) but
on fixed domain $\displaystyle(0,1)$).

Of course, we need a strong assumption on the limit measure $\displaystyle
P_{\text{eq}}$, precisely we ask that the quantity
$\displaystyle\mathcal{E}(x)$ is well-defined for $\displaystyle\gamma$ almost
all $\displaystyle x\in\mathrm{H}\cap K$ (but possibly infinite with the
convention that $\displaystyle\exp(-\infty)=0$).

In the previous section, one can notice that the integrability condition (6.2)
of the invariant measure $\displaystyle P_{\text{eq}}$ is deduced from the
strict monotonicity.

Naturally, we expect that the energy is well-posed on the space of non-
negative functions

$\displaystyle K=\\{x\in\mathrm{L}^{2}:\;x\geq 0\\}$

assuming we have a non-negative potential $\displaystyle V$.

######  Proposition 6.1.

Assume that $\displaystyle V^{n}$ is non-negative, non-increasing and
converges from below to $\displaystyle V$ on $\displaystyle\mathbb{R}_{+}$,
but converges to $\displaystyle+\infty$ on $\displaystyle\mathbb{R}_{-}$. Then
we have the convergence
$\displaystyle\nu^{n}\mathop{\rightharpoonup}_{n\rightarrow+\infty}\nu.$

###### Proof.

Let $\displaystyle\psi\in\mathcal{C}_{b}(\mathrm{L}^{2}(0,1),\mathbb{R})$. We
want to prove that

$\int_{\mathrm{H}}\psi(x)\exp({-\mathcal{E}^{n}(x)})\gamma(\mathrm{d}x)\quad\mathop{\longrightarrow}_{n\rightarrow+\infty}\quad\int_{\mathrm{H}}\psi(x)\exp({-\mathcal{E}(x)})\mathds{1}_{\rho\in
K}\gamma(\mathrm{d}x).$ (6.5)

We first prove that for ($\displaystyle\gamma$-almost surely) $\displaystyle
x\in\mathrm{H}$ we have

$\exp({-\mathcal{E}^{n}(x)})\quad\mathop{\longrightarrow}_{n\rightarrow+\infty}\quad\exp({-\mathcal{E}(x)})\mathds{1}_{x\in
K}.$ (6.6)

Since the reference measure $\displaystyle\gamma(C([0,1]))=1$, we can restrict
our attention to $\displaystyle x\in C([0,1])$. Then if $\displaystyle x\notin
K$ there exists $\displaystyle\delta_{x}>0$ such that $\displaystyle
Leb(\\{s\in[0,1]:x(s)\leq-\delta_{x}\\})>0$, $\displaystyle Leb$ being the
Lebesgue measure. Then, since $\displaystyle V^{n}(x)$ is non-negative and
non-increasing on $\displaystyle\mathbb{R}_{+}:=(0,\infty)$, we have:

$\displaystyle\displaystyle 0\leq\exp({-\mathcal{E}^{n}(x)})$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\exp\left(-\int_{0}^{1}V^{n}(x(s))1_{\\{x\leq-\delta_{x}\\}}\mathrm{d}s\right)$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\exp\left(-\int_{0}^{1}V^{n}(-\delta_{x})1_{\\{x\leq-\delta_{x}\\}}\mathrm{d}s\right)$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\exp\left(-V^{n}(-\delta_{x})Leb(\\{x\leq-\delta_{x}\\})\right).$

By assumption, this latter term converges to zero as $\displaystyle n$ grows
to infinity.
Now, for $\displaystyle x\in K$, $\displaystyle V^{n}(x(s))$ converges to
$\displaystyle V(x(s))$ almost everywhere as $\displaystyle n$ grows to
infinity. Moreover $\displaystyle 0\leq V^{n}(x(s))\leq V(\sup\|x\|)$, and by
the Dominated Convergence Theorem we deduce (6.6). Finally, (6.5) follows
again by the Dominated Convergence Theorem. ∎

We know (see [16, 22]) that the approach described in the previous section
will also work in the singular context. Moreover the semi-group
$\displaystyle\mathcal{P}_{t}:=\lim_{n\rightarrow+\infty}\mathcal{P}^{n}_{t}$
is Strong Feller, such that for all
$\displaystyle\phi\in\mathcal{B}_{b}(\mathrm{H})$, for all $\displaystyle
x,y\in\mathrm{H}$ and $\displaystyle t>0$:

$|\mathcal{P}_{t}\phi(x)-\mathcal{P}_{t}\phi(y)|\leq\frac{\|\phi\|_{\infty}}{\sqrt{t}}\|x-y\|_{\mathrm{H}}.$
(6.7)

##  7 Concluding With A Conjecture

We took on to proving the existence and uniqueness of solutions to a
previously unstudied class of non-linear stochastic partial differential
equations with a singular coefficient that has recently been introduced [36]
within the framework of polymer dynamics theories geared towards polymer
physics and rheology.

The equations under scrutiny in this paper are simplified versions of the
originally suggested for being studied. We have shown that there exists
solutions to these approximated problems under classical assumptions and
abstract formulation. It appears at this stage that it is plausible to obtain
existence of solution results for the more general singular problem just
alluded to by extracting properly defined convergent sequences of approximated
solutions, and this upon obtaining enough a priori estimates, and passing
afterwards to the limit. For instance, using an $\displaystyle\varepsilon$
approximation (by Yosida regularization, or globally Lipschitz potential, or
dissipative potential), one may achieve the results by compactness related
arguments. Using Picard’s fixed-point iteration procedure, it seems also
plausible to obtain convergence to a solution.

We can state the following conjecture based on the existence of an abstract
formulation which encompasses the space of functions and the non-linearity.
Let $\displaystyle\mathcal{H}$ be this hypothetical, abstract space of
functions and assume there exists a potential derivative $\displaystyle
V^{\prime}$, a mobility $\displaystyle M$ and a covariance operator
$\displaystyle Q$ well-defined on $\displaystyle\mathcal{H}$.

Then the following conjecture remains to be proved:

######  Theorem 7.1 (Conjecture).

Let $\displaystyle\rho_{0}$ be a smooth enough initial data. There exists a
process $\displaystyle\left(\rho(t),W(t)\right)_{t\in[0,T]}$, defined on a
filtered complete probability space
$\displaystyle\left(\Omega,\mathbb{P},\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]}\right)$
which is a solution to (1.1) on $\displaystyle[0,T]$ in the sense that

  1. (b”)

$\displaystyle\rho\in\mathrm{L}^{2}\left(\Omega;\mathcal{H}\right)$ and
$\displaystyle\rho(0)=\rho_{0}$,

  2. (c”)

$\displaystyle\mu:=V^{\prime}(\rho)\in\mathrm{L}^{2}\left(\Omega;\mathcal{H}\right)$,

  3. (d”)

$\displaystyle W$ is a continuous version of Brownian motion,

  4. (e”)

the process $\displaystyle\left(\rho(t),W(t)\right)$ is
$\displaystyle(\mathcal{F}_{t})$-adapted,

  5. (f”)

$\displaystyle M(\rho)\in\mathrm{L}^{2}\left(\Omega;\mathcal{H}\right)$,

  6. (g”)

$\displaystyle Q(\rho)\in\mathrm{L}^{2}\left(\Omega;\mathcal{H}\right)$,

  7. (h”)

for all $\displaystyle 0\leq t\leq T$:

$\displaystyle\displaystyle\rho(t)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\rho_{0}+\frac{1}{2}\int_{0}^{t}\nabla(M(\rho(r))\nabla\mu(r))\mathrm{d}r+\int_{0}^{t}Q(\rho(r))\mathrm{d}W_{r}.$

Moreover, for $\displaystyle T\rightarrow+\infty$, the distribution of the
solution converges to an equilibrium density.

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