# pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact
manifolds

Liang Cheng, Yongjia Zhang  School of Mathematics and Statistics $\&$ Hubei
Key Laboratory of Mathematical Sciences, Central China Normal University,
Wuhan, 430079, P.R.China<EMAIL_ADDRESS>School of Mathematical
Sciences, Shanghai Jiao Tong University, Shanghai, 200240, P.R.China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove a pseudolocality-type theorem for
$\mathcal{L}$-complete noncompact Ricci flow which may not have bounded
sectional curvature; with the help of it we study the uniqueness of the Ricci
flow on noncompact manifolds. In particular, we prove the strong uniqueness
theorem for the $\mathcal{L}$-complete Ricci flow on the Euclidean space. This
partially answers a question proposed by B-L. Chen [5].

###### Key words and phrases:

pseudo-locality theorem, strong uniquess, noncompact Ricci flow

###### 2020 Mathematics Subject Classification:

Primary 53E20; Secondary 35K45.

Liang Cheng’s research is partially supported by Natural Science Foundation of
China 12171180.

Yongjia Zhang’s research is partially supported by the starting-up grant of
Shanghai Jiao Tong University.

##  1\. Introduction

The Ricci flow is a geometric evolution equation introduced by Hamilton [13],
which deforms a Riemannian manifold by the Ricci curvature, namely,

$\frac{\partial}{\partial t}g_{t}=-2\mathrm{Ric}_{g_{t}}.$

On the one hand, the Ricci flow equation is nonlinear, which implies that
almost certainly a Ricci flow develops singularity and does not exist for all
time; see [15]. By a thorough analysis of the singularities in dimension three
[19], Perelman successfully overcame the obstacles in solving the Poincaré and
geometrization conjuectures [19, 20, 21] using Hamilton’s program [15]. On the
other hand, the Ricci flow equaiton is parabolic, and consequently, many
problems related to parabolic partial differential equations arise in this
field. For instance, the short-time existence problem (c.f. [11, 14, 23]), the
long-time existence problem (c.f. [8, 15, 22, 25, 26]), and the uniqueness
problem (c.f. [6, 17, 18]).

Another interesting problem in the study of parabolic equations is that of the
strong uniqueness. In general, the solution to the initial value problem of a
parabolic equation on an unbounded region is not unique. For instance, the
$1$-dimensional linear heat equation $u_{t}-u_{xx}=0$ on $\mathbb{R}^{1}$
admits a nontrivial solution with the initial value $u(\cdot,0)\equiv 0$,
namely, the well-known Tychonoff example

$u(x,t)=\sum_{k=0}^{\infty}\frac{x^{2k}}{(2k)!}\frac{d^{k}}{dt^{k}}\exp\left(-\tfrac{1}{t^{2}}\right).$

So it is natural to ask whether or not the solution to the Ricci flow equation
is always unique provided that the initial data is good.

In this respect, B-L. Chen [5] proved a strong uniqueness result in dimension
three, namely, starting from a $3$-dimensional complete Riemannian manifold
with bounded and nonnegative sectional curvature, two Ricci flows must be
identical to each other for a short time. As a consequence, a Ricci flow
starting from the standard $3$-dimensional Euclidean space must always remain
Euclidean. It is therefore interesting to ask whether such property is true
for the Euclidean space in higher dimensions. We shall give an affirmative
answer to this question with some additional assumptions.

At the end of his paper [5], Chen mentioned that the strong uniqueness theorem
can be obtained from a (more general) pseudolocality theorem, which states
that the Ricci flow cannot evolve an almost Euclidean region immediately into
a high-curvature region. Let us recall Perelman’s well-known pseudolocality
theorem [19] (c.f. [4]).

######  Theorem 1.1 (Perelman’s pseudolocality theorem).

For any $\alpha>0$ there exist $\delta>0$ and $\epsilon_{0}>0$ depending on
$\alpha$ and $n$ with the following preperty. Let
$(M^{n},g_{t})_{t\in[0,\epsilon^{2}]}$ be a complete Ricci flow on a
noncompact manifold with bounded sectional curvature at each time on
$(0,\epsilon^{2})$, where $\epsilon\in(0,\epsilon_{0}]$, such that

(1.1) $R_{g_{0}}\geq-1\quad\text{ on }\quad B_{g_{0}}(x_{0},1),$

and

(1.2)
$\left(\operatorname{Area}_{g_{0}}(\partial\Omega)\right)^{n}\geq(1-\delta)n^{n}\omega_{n}\left(\operatorname{Vol}_{g_{0}}(\Omega)\right)^{n-1}$

for any regular domain $\Omega\subset B_{g_{0}}(x_{0},1)$. Then we have

$|Rm|(x,t)\leq\frac{\alpha}{t}+\frac{1}{\epsilon^{2}},$

for any $d_{g_{t}}(x_{0},x)<\epsilon$ and $t\in(0,\epsilon^{2}]$.

The original version of Perelman’s pseudolocality theorem [19] was proved
under the assumption of the manifold being closed. In the complete and
noncompact case, this result was verified by Chau, Tam and Yu [4]. Later, Tian
and Wang [24] proved another version of Perelman’s pseudolocality theorem in
which they showed that the conditions (1.1) and (1.2) in Theorem 1.1 can be
replaced by small Ricci curvature and almost Euclidean volume ratio. Wang [27]
improved both Perelman’s and Tian-Wang’s pseudolocality theorems and proved
that (1.1) and (1.2) in Theorem 1.1 can be replaced by the smallness of the
localized version of Perelman’s $\mu$-functional. One may also see [3] for
another proof of the pseudolocality theorem based on Bamler’s
$\epsilon$-regularity theorem [1].

It is to be noted that the above pseudolocality theorems require bounded
sectional curvature on the whole manifold within each compact time-interval,
under which condition the uniqueness of the Ricci flow is already established
by Chen-Zhu [7]. Hence we cannot use the previous pseudolocality theorems to
study the uniquess problems of the Ricci flow. In this paper we are bound to
find a pseudolocality theorem valid in a more general setting, or at least
with weaker regularity requirements after the initial time. It turns out that
the $\mathcal{L}$-complete condition is a notion that fits our purpose.

######  Definition 1.2.

A complete smooth Ricci flow $(M^{n},g_{t})_{t\in[0,T]}$ is called
$\mathcal{L}$-complete on $M^{n}\times[0,T]$ if the following holds. For any
$s,\,t\in[0,T]$ with $s<t$ and any compact set $E\subset M$, there exists
another compact set $F\subset M$ that contains $E$, such that if $x,\,y\in E$,
then there exists at least one minimal $\mathcal{L}$-geodesic connecting
$(x,t)$ and $(y,s)$, and _all_ possible minimal $\mathcal{L}$-geodesics
connecting $(x,t)$ and $(y,s)$ are contained (spatially) in $F$.

First of all, we will show that under certain conditions the Ricci flow is
$\mathcal{L}$-complete.

######  Theorem 1.3.

Let $(M^{n},g_{t})_{t\in[0,T]}$ be a Ricci flow such that either one of the
following is true.

  1. (1)

$R_{g_{t}}(x)\geq-k\quad\text{for all}\quad(x,t)\in M\times[0,T]$, where $k$
is a nonnegative constant, and there exist a positive constant $c$ and a
complete smooth Riemannian metric $\bar{g}$ on $M$, such that

$\displaystyle g_{t}\geq c\bar{g}\quad\text{ for all }\quad t\in[0,T].$

  2. (2)

$R_{g_{t}}(x)\geq-k\quad\text{for all}\quad(x,t)\in M\times[0,T]$, where $k$
is a nonnegative constant, and there exist constants $\alpha>0$, $\beta>0$,
$\gamma\in(0,1)$, and a fixed point $x_{0}\in M$, such that

$\displaystyle\mathrm{Ric}_{g_{t}}(x)\geq-\alpha
t^{-\gamma}\ln\left(\operatorname{dist}_{g_{t}}(x_{0},x)+\beta\right)g_{t}\quad\text{
for all }\quad(x,t)\in M\times[0,T].$

Moreover, $T\leq\overline{T}(\alpha,\gamma)$, where
$\overline{T}(\alpha,\gamma)$ signifies a positive constant depending only on
$\alpha$ and $\gamma$.

Then $(M^{n},g_{t})_{t\in[0,T]}$ is $\mathcal{L}$-complete on $[0,T]$.

On the other hand, the significance of the $\mathcal{L}$-complete assumption
is that with it a version of Perelman’s pseudolocality theorem can be proved.
Indeed, it is well-known that Perelman’s entropy and reduced geometry reflect
the same geometric property of the Ricci flow from differently perspectives.
While the bounded curvature condition enables the estimates for the entropy,
through which the pseudolocality theorem was proved, the
$\mathcal{L}$-complete assumption leads to a similar conclusion via the
reduced geometry. In this way, we prove the following pseudolocality-type
theorem for $\mathcal{L}$-complete Ricci flows.

######  Theorem 1.4.

For any $\alpha>0$ there exist $\delta>0$ and $\epsilon_{0}>0$ depending on
$\alpha$, and $n$ with the following property. Let
$(M^{n},g_{t})_{t\in[0,\epsilon^{2}]}$ be a smooth $\mathcal{L}$-complete
Ricci flow on a noncompact manifold, where $\epsilon\in(0,\epsilon_{0}]$, such
that

$R(g_{0})\geq-1,$

and

$\left(\operatorname{Area}_{g_{0}}(\partial\Omega)\right)^{n}\geq(1-\delta)n^{n}\omega_{n}\left(\operatorname{Vol}_{g_{0}}(\Omega)\right)^{n-1}$

for any regular domain $\Omega\subset M$. Then we have

$|Rm|(x,t)\leq\frac{\alpha}{t}\qquad\text{for all}\qquad(x,t)\in
M\times(0,\epsilon^{2}].$

Applying the above pseudolocality theorem, we can study the strong uniqueness
of the $\mathcal{L}$-complete Ricci flow in higher dimensions. In particular,
we show the strong uniqueness theorem for the $\mathcal{L}$-complete Ricci
flow on Euclidean space in higher dimensions, which partially answers a
question proposed by B-L. Chen [5].

######  Theorem 1.5.

Let $(M^{n},g_{t})_{t\in[0,T]}$ be a smooth $\mathcal{L}$-complete Ricci flow
on a noncompact manifold $M$ such that

$|\mathrm{Rm}_{g_{0}}|\leq C,$

and

$\left(\operatorname{Area}_{g_{0}}(\partial\Omega)\right)^{n}\geq(1-\delta)n^{n}\omega_{n}\left(\operatorname{Vol}_{g_{0}}(\Omega)\right)^{n-1}$

for any regular domain $\Omega\subset M$. Then we have

$|\mathrm{Rm}_{g_{t}}|\leq C\qquad\text{for all}\qquad t\in[0,T].$

Consequently, the solution $g_{t}$, $t\in[0,T]$, is uniquely determined by
$g_{0}$. In particular, two $\mathcal{L}$-complete Ricci flows starting from a
Riemannian manifold satisfying the assumptions of the theorem must always be
isometric to each other.

Combining Theorem 1.3(1)(2) and Theorem 1.5, we obtain the following result.

######  Theorem 1.6.

Let $(\mathbb{R}^{n},g_{t})_{t\in[0,T]}$ be a complete and smooth solution to
the Ricci flow with $g_{0}=g_{E}$, where $g_{E}$ is the standard Euclidean
metric. Assume that either one of the following is true.

  1. (1)

there exists a positive constant $c$, such that

$\displaystyle g_{t}\geq c\bar{g}\quad\text{ for all }\quad t\in[0,T],$

where $\bar{g}$ is a complete smooth Riemannian metric on $\mathbb{R}^{n}$

  2. (2)

there exist constants $\alpha>0$, $\beta>0$, $\gamma\in(0,1)$, and a fixed
point $x_{0}\in\mathbb{R}^{n}$, such that

$\displaystyle\mathrm{Ric}_{g_{t}}(x)\geq-\alpha
t^{-\gamma}\cdot\ln\left(\operatorname{dist}_{g_{t}}(x_{0},x)+\beta\right)g_{t}\quad\text{
for all }\quad(x,t)\in\mathbb{R}^{n}\times[0,T].$

Then we have $g_{t}\equiv g_{E}$ for all $t\in[0,T]$.

Acknowledgement. The authors would like to thank Professor Jiaping Wang for
some helpful suggestions.

##  2\. Preliminaries

In this section, we collect some well-known results applied in the proofs of
our main theorems. These results are chiefly about Perelman’s reduced distance
and reduced volume; most of them can be found in [19, 28]. Let
$(M^{n},g(\tau))_{\tau\in[0,T]}$ be a solution to the backward Ricci flow
equation

$\displaystyle\frac{\partial}{\partial\tau}g(\tau)=2\mathrm{Ric}_{g(\tau)},$

where $\tau$ stands for the backward time. In fact, if
$(M^{n},g_{t})_{t\in[0,T]}$ is a Ricci flow, then $g(\tau):=g_{T-\tau}$ solves
the backward Ricci flow equation. In the remaining part of the paper, since we
often shift between Ricci flow and backward Ricci flow, we remind the reader
that if the “time variable” is a subindex, such as $g_{t}$, then the notation
stands for a Ricci flow; if the “time variable” is placed in a pair of
parentheses, such as $g(\tau)$, then the notation represents a backward Ricci
flow.

Let $\gamma(s):[0,\tau]\to M$, where $\tau\in(0,T]$, be a piecewise $C^{1}$
curve, then Perelman’s $\mathcal{L}$-energy is defined as

(2.1)
$\displaystyle\mathcal{L}(\gamma)=\int^{\tau}_{0}\sqrt{s}\left(R_{g(s)}(\gamma(s))+|\gamma^{\prime}(s)|_{g(s)}^{2}\right)ds.$

One may view $\gamma$ as a curve in the (backward) Ricci flow space-time from
$(\gamma(0),0)$ to $(\gamma(\tau),\tau)\in M\times[0,T]$, satisfying
$(\gamma(s),s)\in M\times\\{s\\}$ for all $s\in[0,\tau]$. Let $x_{0}\in M$ be
a fixed base point. For any $(x,\tau)\in M\times(0,T]$, we define

(2.2) $\displaystyle
L(x,\tau)=\inf\limits_{\gamma}\mathcal{L}(\gamma(s)),\quad\ell(x,\tau)=\frac{L(x,\tau)}{2\sqrt{\tau}},$

where the infimum is taken over all piecewise $C^{1}$ curves
$\gamma(s):[0,\tau]\to M$ satisfying $\gamma(0)=x_{0}$ and $\gamma(\tau)=x$. A
minimizer of $L$ (should it exist) is called a _minimal
$\mathcal{L}$-geodesic_, and $\ell(\cdot,\cdot):M\times(0,T]\to\mathbb{R}$ is
called Perelman’s reduced distance function or the $\ell$-function based at
$(x_{0},0)$.

######  Remark 2.1.

When considering a forward Ricci flow $(M,g_{t})_{t\in[0,T]}$, then a minimal
$\mathcal{L}$-geodesic is said to connect $(x,t)$ and $(y,s)$, where $s<t$, if
it minimizes (2.1) with respect to the backward Ricci flow
$(M,g(\tau):=g_{t-\tau})_{\tau\in[0,t]}$, starts at $(x,0)$, and ends at
$(y,t-s)$.

For $v\in T_{x_{0}}M,$ let $\gamma_{v}$ denote the $\mathcal{L}$-geodesic
satisfying $\lim\limits_{s\to 0}\sqrt{s}\gamma^{\prime}(s)=v$. If $\gamma_{v}$
exists on $[0,\tau]$, then the $\mathcal{L}$-exponential map
$\mathcal{L}\text{exp}^{\tau}_{x_{0}}:T_{x_{0}}M\to M$ at time $\tau$ is
defined as

$\mathcal{L}\text{exp}^{\tau}_{x_{0}}(v)=\gamma_{v}(\tau).$

Let $U(\tau)\subset T_{x_{0}}M\cong\mathbb{R}^{n}$ be the maximal domain of
$\mathcal{L}\text{exp}^{\tau}_{x_{0}}$. By applying the basic ODE theory to
the $\mathcal{L}$-geodesic equation—a linear ODE depending only on the
geometry near the $\mathcal{L}$-geodesic—one may conclude that $U(\tau)$ is an
open set and that $\mathcal{L}\text{exp}^{\tau}_{x_{0}}$ is a smooth map from
$U(\tau)$ to $M$. The injectivity domain at time $\tau$ is defined as

$\displaystyle\Omega^{T_{x_{0}}M}(\tau)=\Big{\\{}v\in U(\tau)\ \Big{|}\
\gamma_{v}|_{[0,\tau]}:[0,\tau]\to M\text{ is the unique minimal
}\mathcal{L}\text{-geodesic}$ $\displaystyle\text{\ from $(x_{0},0)$ to
$(\gamma_{v}(\tau),\tau)$ };\text{$\gamma_{v}(\tau)$ is not conjugate to
}x_{0}\text{ along\ }\gamma_{v}.\Big{\\}}.$

It is easily seen that

(2.3)
$\displaystyle\Omega^{T_{x_{0}}M}(\tau_{2})\subset\Omega^{T_{x_{0}}M}(\tau_{1})\qquad\text{
if }\quad\tau_{1}\leq\tau_{2}.$

Correspondingly we also define

$\displaystyle\Omega(\tau)=\Big{\\{}q\in M\ \Big{|}\ \text{There is a unique
minimal }\mathcal{L}-\text{geodesic }\gamma:[0,\tau]\to M\text{\ with\ }$
$\displaystyle\gamma(0)=x_{0},\gamma(\tau)=q;q\text{ is not conjugate to
}x_{0}\text{ along\ }\gamma.\Big{\\}}.$

Obvisously,

$\Omega(\tau)=\mathcal{L}\text{exp}^{\tau}_{x_{0}}\left(\Omega^{T_{x_{0}}M}(\tau)\right),$

and Perelman’s reduced volume based at $(x_{0},0)$ is defined as

(2.4)
$\displaystyle\mathcal{V}(\tau)=\int_{\Omega(\tau)}(4\pi\tau)^{-\frac{n}{2}}e^{-\ell(\cdot,\tau)}d\mu(\tau).$

The $\mathcal{L}$-cut-locus is defined as

$C(\tau)=M\backslash\Omega(\tau).$

Let $J_{i}^{V}(\tau),i=1,\cdots,n,$ be $\mathcal{L}$-Jacobi fields along
$\gamma_{V}(\tau)$ with $J_{i}^{V}(0)=0,(\nabla_{V}J_{i}^{V})(0)=E^{0}_{i},$
where $\\{E^{0}_{i}\\}^{n}_{i=1}$ is an orthonormal basis on $T_{x_{0}}M$ with
respect to $g(0)$. Then
$D(\mathcal{L}exp^{\tau})\big{|}_{V}(E^{0}_{i})=J^{V}_{i}(\tau)$, and the
reduced volume can also be computed as

(2.5)
$\displaystyle\mathcal{V}(\tau)=\int_{\Omega^{T_{x_{0}}M}(\tau)}(4\pi\tau)^{-\frac{n}{2}}e^{-l(\gamma_{V}(\tau),\tau)}\mathcal{L}J_{V}(\tau)dx_{g(0)}(V),$

where
$\mathcal{L}J_{V}(\tau)=\sqrt{det\left(\left<J^{V}_{i}(\tau),J^{V}_{j}(\tau)\right>\right)}$
and $dx_{g(0)}$ is the standard Euclidean volume form on
$(T_{x_{0}}M,g(x_{0},0))$.

To end this section, we recall the famous logarithmic Sobolev inequality due
to Gross [12]. This inequality is important in the proofs of our main
theorems, since it is strongly related to Perelman’s monotonicity formulas.
The particular form of the following theorem can be found in [10, Theorem
22.5].

######  Theorem 2.2 (Logarithmic Sobolev inequality).

Let $(M^{n},g)$ be a complete Riemannian manifold which satisfies

$\left(\operatorname{Area}_{g}(\partial\Omega)\right)^{n}\geq
I_{n}\left(\operatorname{Vol}_{g}(\Omega)\right)^{n-1}$

for any regular compact domain $\Omega\subset M^{n}$. Then for any $W^{1,2}$
function $\varphi$ on $M^{n}$, we have

$\displaystyle\int_{M^{n}}\left(2|\nabla\varphi|^{2}-\varphi^{2}\log\varphi^{2}\right)d\mu+\log\left(\int_{M^{n}}\varphi^{2}d\mu\right)\int_{M^{n}}\varphi^{2}d\mu\geq\left(\frac{n}{2}\log(2\pi)+n+\log\left(\frac{I_{n}}{c_{n}}\right)\right)\int_{M^{n}}\varphi^{2}d\mu,$

where $c_{n}=n^{n}\omega_{n}$ is the isoperimetric constant of Euclidean
space.

##  3\. Perelman’s $\mathcal{L}$-geometry on $\mathcal{L}$-complete Ricci flow

In this section, we shall first show that Perelman’s theory of
$\mathcal{L}$-geometry can be extended to the case of $\mathcal{L}$-complete
Ricci flows, and then show that the Ricci flow is $\mathcal{L}$-complete under
the assumptions made in Theorem 1.3.

###  3.1. $\mathcal{L}$-complete Ricci flow

In Perelman’s study [19] of the $\mathcal{L}$-geometry, a general (although
implicit) assumption is bounded sectional curvature. However, Ye [28] studied
the properties of the $\ell$-function and the reduced volume assuming only a
lower bound of the Ricci curvature. With a little more observation, Perelman’s
theory of $\mathcal{L}$-geometry can be extended to $\mathcal{L}$-complete
Ricci flows. We will leave it to the reader to check that most proofs in [28]
are valid, and we summarize some important results below.

######  Theorem 3.1 ([19], see also [28, Proposition 2.14, Lemma 2.22, Theorem
2.23]).

Let $(M^{n},g(\tau))_{\tau\in[0,T]}$ be a smooth $\mathcal{L}$-complete
backward Ricci flow. Let $\ell$ be the reduced distance function based at
$(x_{0},0)$. Then $\ell$ is locally Lipschitz on $M\times(0,T]$ and the
following equation and inequalities hold almost everywhere in the smooth sense
on $M\times(0,T]$.

(3.1) $\displaystyle 2\frac{\partial\ell}{\partial\tau}+|\nabla
l|^{2}-R+\frac{\ell}{\tau}=0,$ (3.2)
$\displaystyle\frac{\partial}{\partial\tau}\ell-\Delta\ell+|\nabla\ell|^{2}-R+\frac{n}{2\tau}\geq
0,$ (3.3) $\displaystyle
2\Delta\ell-|\nabla\ell|^{2}+R+\frac{\ell-n}{\tau}\leq 0.$

Furthermore, (3.2) and (3.3) both hold on $M\times(0,T]$ in the sense of
distribution. That is to say, for any $0<\tau_{1}<\tau_{2}\leq T$ and for any
nonnegative Lipshcitz function $\phi$ compactly supported on
$M\times[\tau_{1},\tau_{2}]$, it holds that

(3.4)
$\displaystyle\int_{\tau_{1}}^{\tau_{2}}\int_{M}\left(\nabla\ell\cdot\nabla\phi+\left(\frac{\partial}{\partial\tau}\ell+|\nabla\ell|^{2}-R+\frac{n}{2\tau}\right)\phi\right)dg(\tau)d\tau\geq
0,$

and, for any $\tau\in(0,T]$ and any nonnegative Lipshcitz function $\phi$
compactly supported on $M$, it holds that

(3.5)
$\displaystyle\int_{M}\left(-2\nabla\ell\cdot\nabla\phi+\left(-|\nabla\ell|^{2}+R+\frac{\ell-n}{\tau}\right)\phi\right)dg(\tau)\leq
0,$

Moreover, the integrand in (2.5) is pointwise monotonically non-increasing,
namely,

(3.6)
$\displaystyle\frac{d}{d\tau}\left((4\pi\tau)^{-\frac{n}{2}}e^{-l(\gamma_{V}(\tau),\tau)}\mathcal{L}J_{V}(\tau)\right)\leq
0,$

for any $V\in\Omega^{T_{x_{0}}M}(\tau)$.

###### Proof.

By virtue of our definition of the $\mathcal{L}$-complete Ricci flow
(Definition 1.2), when analyzing the local properties of the
$\mathcal{L}$-geodesics or the reduced distance, only local geometry is
involved. Thus the proofs in [28] can be wholly adopted in our case.

We emphasize that although according to Ye’s formulation, (3.5) and (3.6) hold
only on $M\times(0,T)$, yet this restriction stems from the application of
Shi’s estimates when obtaining the local curvature derivative bounds. However,
in our assumption, the backward Ricci flow is assumed to be smooth on
$M\times[0,T]$, namely, each curvature derivative is bounded locally. Thus
(3.5) and (3.6) are also valid on $M\times(0,T]$. ∎

######  Theorem 3.2 ([19], see also [28, Theorem 4.3, Theorem 4.5]).

Let $(M^{n},g(\tau))_{\tau\in[0,T]}$ be a $\mathcal{L}$-complete backward
Ricci flow. Let $\mathcal{V}(\tau)$ be the reduced volume based at
$(x_{0},0)$. Under the same assumptions as in the above theorem, we have

  1. (1)

$0<\mathcal{V}(\tau)\leq 1$ for all $\tau\in(0,T]$;

  2. (2)

$\mathcal{V}(\tau)$ is non-increasing in $\tau$;

  3. (3)

if $\mathcal{V}(\tau)=1$ for some $\tau\in(0,T]$, then $M^{n}=\mathbb{R}^{n}$
and $g(s)\equiv g_{E}$ for all $s\in[0,\tau]$, where $g_{E}$ is the standard
Euclidean metric.

###### Proof.

According to the definition of the reduced volume (2.5), this theorem is a
consequence of (2.3) and (3.6). The details can be found in [28]. ∎

###  3.2. Sufficient conditions of $\mathcal{L}$-completeness

In this subsection, we shall prove Theorem 1.3. To begin with, we prove the
following straightforward auxilliary lemma.

######  Lemma 3.3 (bounded $\mathcal{L}$-energy in compact set).

Let $(M,g(\tau))_{\tau\in[0,T]}$ be a complete backward Ricci flow. For any
$\bar{\tau}\in(0,T]$ and any compact set $E\in M$, there exists a positive
number $L$, such that for any $x_{0}$, $x\in E$, there is a picewise $C^{1}$
curve $\gamma:[0,\bar{\tau}]$ with $\gamma(0)=x_{0}$ and
$\gamma(\bar{\tau})=x$, satisfying

$\mathcal{L}(\gamma)\leq L.$

###### Proof.

Since $E$ is compact and the backward Ricci flow is complete, we can find a
$y_{0}\in M$ and fix $A>0$ and $K>0$, such that

$\displaystyle E\subset
B_{g(0)}(y_{0},A)\qquad\text{and}\qquad\left|\mathrm{Ric}\right|\leq K\ \text{
on }\ B_{g(0)}(y_{0},3A)\times[0,\bar{\tau}].$

For any $x_{0}$, $x\in E$, let $\gamma:[0,\bar{\tau}]\to M$ a normalized
shortest $g(0)$-geodesic with $\gamma(0)=x_{0}$ and $\gamma(\bar{\tau})=x$.
Then it is obvious that

$\displaystyle\gamma(\tau)\in B_{g(0)}(y_{0},3A)\ \ \text{ and }\ \
|\gamma^{\prime}(\tau)|_{g(0)}=\frac{\operatorname{dist}_{g(0)}(x_{0},x)}{\bar{\tau}}\leq\frac{2A}{\bar{\tau}},\
\ \text{ for all }\ \tau\in[0,\bar{\tau}].$

Thus, we compute

$\displaystyle\mathcal{L}(\gamma)$
$\displaystyle=\int_{0}^{\bar{\tau}}\sqrt{\tau}\left(R_{g(\tau)}(\gamma(\tau))+\left|\gamma^{\prime}(\tau)\right|^{2}_{g(\tau)}\right)\,d\tau$
$\displaystyle\leq\frac{2}{3}\sqrt{n}K\bar{\tau}^{\frac{3}{2}}+e^{K\bar{\tau}}\int_{0}^{\bar{\tau}}\sqrt{\tau}\left|\gamma^{\prime}(\tau)\right|^{2}_{g(0)}\,d\tau$
$\displaystyle\leq\frac{2}{3}\sqrt{n}K\bar{\tau}^{\frac{3}{2}}+\frac{8A^{2}e^{K\bar{\tau}}}{3\sqrt{\bar{\tau}}}.$

Letting $L$ be the constant on the last line above finishes the proof of the
lemma. ∎

####  3.2.1. Lower uniform equivalence

We prove Theorem 1.3(1). Consider a backward Ricci flow
$(M,g(\tau))_{\tau\in[0,T]}$ satisfying

(3.7) $\displaystyle R_{g(\tau)}(x)\geq-k\quad$ $\displaystyle\text{for
all}\quad(x,\tau)\in M\times[0,T],$ (3.8) $\displaystyle g(\tau)\geq
c\bar{g}\quad$ $\displaystyle\text{for all}\quad\tau\in[0,T].$

where $k\geq 0$, $c>0$, and $\bar{g}$ is a complete smooth Riemannian metric
$\bar{g}$ on $M$. The central idea of proving the $\mathcal{L}$-completeness
is to show that any space-time curve with bounded $\mathcal{L}$-energy lies in
a space-time compact set; this idea is also used in the following part of the
subsection.

######  Proposition 3.4.

Under the assumptions (3.7) and (3.8), the following holds. For any
$(x_{0},0)$ and $(x,\tau)\in M\times(0,T]$, there exists a minimal
$\mathcal{L}$-geodesic $\gamma:[0,\tau]\to M$ satisfying $\gamma(0)=x_{0}$ and
$\gamma(\tau)=x$. Furthermore, the backward Ricci flow is
$\mathcal{L}$-complete on $M\times[0,T]$.

###### Proof.

Let us consider any piecewise $C^{1}$ curve $\gamma:[0,\tau]\to M$ satisfying
$\gamma(0)=x_{0}$ and $\mathcal{L}(\gamma)\leq L$. Then, by (3.7) and (3.8),
we may estimate

$\displaystyle L\geq\mathcal{L}(\gamma)$
$\displaystyle=\int_{0}^{\tau}\sqrt{s}\left(R_{g(s)}(\gamma(s))+|\gamma^{\prime}(s)|^{2}_{g(s)}\right)ds$
$\displaystyle\geq-\frac{2k}{3}\tau^{\frac{3}{2}}+\int_{0}^{\tau}\sqrt{s}|\gamma^{\prime}(s)|^{2}_{g(s)}ds$
$\displaystyle\geq-\frac{2k}{3}\tau^{\frac{3}{2}}+c\int_{0}^{\tau}\sqrt{s}|\gamma^{\prime}(s)|^{2}_{\bar{g}}ds$
$\displaystyle=-\frac{2k}{3}\tau^{\frac{3}{2}}+\frac{c}{2}\int_{0}^{\sqrt{\tau}}|\zeta^{\prime}(\sigma)|^{2}_{\bar{g}}d\sigma,$

where we have applied the change of variable
$\zeta(\sigma):=\gamma(\sigma^{2})$. Then, by the Cauchy-Schwarz inequality,
we have

$\displaystyle 2c^{-1}L$
$\displaystyle\geq-\frac{4k}{3c}\tau^{\frac{3}{2}}+\int_{0}^{\sqrt{\tau}}|\zeta^{\prime}(\sigma)|^{2}_{\bar{g}}d\sigma$
$\displaystyle\geq-\frac{4k}{3c}\tau^{\frac{3}{2}}+\frac{\left(\operatorname{dist}_{\bar{g}}(x_{0},\gamma(s))\right)^{2}}{\sqrt{T}}$

for all $s\in[0,\tau]$. It follows that for any $L>0$, there exists
$A=A(L,c,T)$, such that if $\gamma:[0,\tau]\to M$ satisfies
$\mathcal{L}(\gamma)\leq L$, then

$\gamma(s)\in B_{\bar{g}}(x_{0},A)\quad\text{ for all }\quad s\in[0,\tau].$

In particular, $A$ is independent of the choice of $x_{0}$

For any $x_{0}$ and $x$, let us fix an arbitrary piecewise $C^{1}$ curve
$\beta:[0,\tau]\to M$ satisfying $\beta(0)=x_{0}$ and $\beta(\tau)=x$. Let
$L=\mathcal{L}(\beta)$. Then we have that every minimizing sequence
$\\{\gamma_{i}\\}_{i=1}^{\infty}$ of the $\mathcal{L}$-energy with
$\mathcal{L}(\gamma_{i})\leq L$ from $(x_{0},0)$ to $(x,\tau)$ is contained in
$\overline{B}_{g(0)}(x_{0},A)\times[0,\tau]$, which is a compact set in space-
time. It follows that there exists at least one minimal $\mathcal{L}$-geodesic
from $(x_{0},0)$ to $(x,\tau)$.

Finally, we show that the backward Ricci flow is $\mathcal{L}$-complete. Let
$E\in M$ be an arbitrary compact set and fix a $\tau\in(0,T]$. By Lemma 3.3,
there is a constant $L(E,\tau)>0$, such that for any $x$, $y\in E$, there is a
piecewise $C^{1}$-curve $\gamma:[0,\tau]\to M$ with $\gamma(0)=x$,
$\gamma(\tau)=y$, and $\mathcal{L}(\gamma)$ not exceeding $L(E,\tau)$. The
argument above shows that there exists at least one minimal
$\mathcal{L}$-geodesic connecting $(x,0)$ and $(y,\tau)$, and all such minimal
$\mathcal{L}$-geodesics are contained in a compact set. ∎

####  3.2.2. Logarithmic lower bound for the Ricci curvature

Let us move on to the proof of Theorem 1.3(2). We assume that the backward
Ricci flow in question has a logarithmic lower bound for the Ricci curvature.
In particular, we let $k>0$, $\alpha>0$, $\beta>0$, $\gamma\in(0,1)$ be
constants and $x_{0}$ be a fixed point, such that

(3.9) $R_{g(\tau)}(x)\geq-k\quad\text{for all}\quad(x,\tau)\in M\times[0,T]$

and

(3.10)
$\displaystyle\mathrm{Ric}_{g(\tau)}(x)\geq-\alpha(T-\tau)^{-\gamma}\cdot\ln\left(\operatorname{dist}_{g(\tau)}(x_{0},x)+\beta\right)g(\tau)\quad\text{for
all}\quad(x,\tau)\in M\times[0,T].$

As in the previous subsection, we show the existence of a minimal
$\mathcal{L}$-geodesic from $(x_{0},0)$ to $(x,\tau)\in M\times(0,T]$,
provided that $T$ is small enough.

######  Lemma 3.5 (Distance distortion).

For any $\tau_{1}$ and $\tau_{2}\in[0,T]$ with $\tau_{1}\leq\tau_{2}$ and for
any $x\in M$, we have

$\displaystyle\operatorname{dist}_{g(\tau_{1})}(x_{0},x)+\beta\leq\left(\operatorname{dist}_{g(\tau_{2})}(x_{0},x)+\beta\right)^{\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}.$

###### Proof.

For any $\tau\in[0,T]$, let us define
$r(\tau):=\operatorname{dist}_{g(\tau)}(x_{0},x)$ and let
$\gamma:[0,r(\tau)]\to M$ be a $g(\tau)$-geodesic connecting $x_{0}$ and $x$
with unit speed. We may apply (3.10) to compute

$\displaystyle\frac{dr}{d\tau}$
$\displaystyle=\int_{0}^{r}\mathrm{Ric}_{g(\tau)}(\gamma^{\prime}(s),\gamma^{\prime}(s))ds$
$\displaystyle\geq-\alpha(T-\tau)^{-\gamma}\cdot r\ln(r+\beta)$
$\displaystyle\geq-\alpha(T-\tau)^{-\gamma}\cdot(r+\beta)\ln(r+\beta),$

and equivalently

$\frac{d}{d\tau}\ln\left(\ln\left(r+\beta\right)\right)\geq-\alpha(T-\tau)^{-\gamma}.$

Integrating the above inequality from $\tau_{1}$ to $\tau_{2}$, we obtain the
lemma. ∎

Next, we prove that a minimizing sequence of the $\mathcal{L}$-energy is
always contained in a compact set in space-time.

######  Lemma 3.6.

There exists a positive number $\overline{T}=\overline{T}(\alpha,\gamma)>0$
with the following property. If $T<\overline{T}$, then for any $r>0$ and
$L>0$, there is a positive number $A=A(r,L,T,\alpha,\beta,\gamma)$, such that
the following holds. For any $(x,\tau)\in M\times(0,T]$ with

$\operatorname{dist}_{g(\tau)}(x_{0},x)\leq r,$

if $\gamma:[0,\tau]\to M$ is a piecewise $C^{1}$ curve satisfying
$\gamma(0)=x_{0}$, $\gamma(\tau)=x$, and

$\displaystyle\mathcal{L}(\gamma)\leq L$

Then we have

$\displaystyle\gamma(s)\in\overline{B}_{g(s)}(x_{0},A)\quad\text{ for all
}\quad s\in[0,\tau].$

###### Proof.

We shall argue by contradiction. Let $A\gg r$ be a large number to be fixed.
Define

$\displaystyle\bar{\tau}:=\inf\left\\{\tau^{\prime}\,\left|\,\gamma(s)\in
B_{g(s)}(x_{0},A)\text{ for all
}s\in[\tau^{\prime},\tau]\right\\}\right.\in[0,\tau).$

We assume that $\bar{\tau}>0$ and show that there is a contradiction when $A$
is large enough. By this contradictory assumption, we have

(3.11)
$\displaystyle\operatorname{dist}_{g(\bar{\tau})}(x_{0},\gamma(\bar{\tau}))=A,$
$\displaystyle\operatorname{dist}_{g(s)}(x_{0},\gamma(s))<A\quad\text{ for all
}\quad s\in[\bar{\tau},\tau].$

By Lemma 3.5, we have

$\displaystyle\operatorname{dist}_{g(\tau_{1})}(x_{0},\gamma(\tau_{2}))\leq\left(\operatorname{dist}_{g(\tau_{2})}(x_{0},\gamma(\tau_{2}))+\beta\right)^{\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}\quad\text{
for all }\quad 0\leq\tau_{1}\leq\tau_{2}\leq\tau,$

and consequently

(3.12)
$\displaystyle\operatorname{dist}_{g(\tau_{1})}(x_{0},\gamma(\tau_{2}))\leq\left(A+\beta\right)^{\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}\quad\text{
for all }\quad\bar{\tau}\leq\tau_{1}\leq\tau_{2}\leq\tau$ (3.13)
$\displaystyle\operatorname{dist}_{g(\bar{\tau})}(x_{0},x)\leq(r+\beta)^{\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}.$

By (3.10) and (3.12), we may estimate

$\displaystyle\frac{d}{d{\tau_{1}}}|\gamma^{\prime}(\tau_{2})|^{2}_{g(\tau_{1})}$
$\displaystyle=2\mathrm{Ric}_{g(\tau_{1})}(\gamma^{\prime}(\tau_{2}),\gamma^{\prime}(\tau_{2}))$
$\displaystyle\geq-2\alpha(T-\tau_{1})^{-\gamma}\cdot\ln\left((A+\beta)^{\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}+\beta\right)|\gamma^{\prime}(\tau_{2})|^{2}_{g(\tau_{1})},$

for all $\bar{\tau}\leq\tau_{1}\leq\tau_{2}\leq\tau$. Integrating over
$\tau_{1}$, we have

(3.14) $\displaystyle|\gamma^{\prime}(\tau_{2})|^{2}_{g(\tau_{2})}$
$\displaystyle\geq\left((A+\beta)^{\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}+\beta\right)^{-\frac{2\alpha}{1-\gamma}T^{1-\gamma}}|\gamma^{\prime}(\tau_{2})|^{2}_{g(\bar{\tau})}$
$\displaystyle\geq(2A)^{-\frac{2\alpha}{1-\gamma}T^{1-\gamma}\cdot\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}|\gamma^{\prime}(\tau_{2})|^{2}_{g(\bar{\tau})}\quad\text{
for all }\quad\bar{\tau}\leq\tau_{2}\leq\tau,$

if $A\geq\underline{A}(\alpha,\beta,\gamma,T)$ for some large positive
constant $\underline{A}(\alpha,\beta,\gamma,T)$. By the definition of the
$\mathcal{L}$-energy and the assumption (3.7), we have

$\displaystyle L$
$\displaystyle\geq\mathcal{L}(\gamma)\geq-\frac{2k}{3}\tau^{\frac{3}{2}}+\int_{\bar{\tau}}^{\tau}\sqrt{s}|\gamma^{\prime}(s)|^{2}_{g(s)}ds\geq-\frac{2k}{3}\tau^{\frac{3}{2}}+(2A)^{-\frac{2\alpha}{1-\gamma}T^{1-\gamma}\cdot\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}\int_{\bar{\tau}}^{\tau}\sqrt{s}|\gamma^{\prime}(s)|^{2}_{g(\bar{\tau})}ds$

Letting $\zeta(\sigma):=\gamma(\sigma^{2})$, we have

$\displaystyle L$
$\displaystyle\geq-\frac{2k}{3}T^{\frac{3}{2}}+(2A)^{-\frac{2\alpha}{1-\gamma}T^{1-\gamma}\cdot\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}\int_{\sqrt{\bar{\tau}}}^{\sqrt{\tau}}\tfrac{1}{2}|\zeta^{\prime}(\sigma)|^{2}_{g(\bar{\tau})}d\sigma$
$\displaystyle\geq-\frac{2k}{3}T^{\frac{3}{2}}+(2A)^{-\frac{2\alpha}{1-\gamma}T^{1-\gamma}\cdot\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}\cdot\frac{\left(\operatorname{dist}_{g(\bar{\tau})}(x,\gamma(\bar{\tau}))\right)^{2}}{2\left(\sqrt{\tau}-\sqrt{\bar{\tau}}\right)}$
$\displaystyle\geq-\frac{2k}{3}T^{\frac{3}{2}}+(2A)^{-\frac{2\alpha}{1-\gamma}T^{1-\gamma}\cdot\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}\cdot\frac{\left(A-(r+\beta)^{\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}\right)^{2}}{2\sqrt{T}},$

where we have applied (3.11) and (3.13). Finally, if we take
$T<\overline{T}(\alpha,\gamma)$ such that
$\frac{2\alpha}{1-\gamma}T^{1-\gamma}\cdot\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)\leq
1$, then we have

$\displaystyle
L\geq\frac{\left(A-(r+\beta)^{\exp\left(\frac{\alpha}{1-\gamma}T^{1-\gamma}\right)}\right)^{2}}{4A\sqrt{T}}-\frac{2k}{3}T^{\frac{3}{2}},$

and this obviously is a contradiction if
$A>\underline{A}(r,L,T,\alpha,\beta,\gamma)$. ∎

######  Proposition 3.7 (Existence of minimal $\mathcal{L}$-geodesic).

If $T<\overline{T}$, where $\overline{T}$ is the constant in the statement of
Lemma 3.6, then for any $(x,\tau)\in M\times(0,T]$, there is a minimal
$\mathcal{L}$-geodesic from $(x_{0},0)$ to $(x,\tau)$. In particular, the
backward Ricci flow is $\mathcal{L}$-complete on $[0,T]$.

###### Proof.

We fix an arbitrary $(x,\tau)\in M\times(0,T]$ and a piecewise $C^{1}$ curve
$\gamma:[0,\tau]\to M$ with $\gamma(0)=x_{0}$ and $\gamma(\tau)=x$. Let

$\displaystyle r$ $\displaystyle:=\operatorname{dist}_{g(\tau)}(x_{0},x),$
$\displaystyle L$ $\displaystyle:=\mathcal{L}(\gamma)$

and let $A=A(r,L,T,\alpha,\beta,\gamma)$ be the positive constant given by
Lemma 3.6. Then we have that, by Lemma 3.6, any minimizing sequence
$\\{\gamma_{i}\\}_{i=1}^{\infty}$ of the $\mathcal{L}$-energy with
$\mathcal{L}(\gamma_{i})\leq L$ from $(x_{0},0)$ to $(x,\tau)$ lies in the
space-time compact set

$\bigcup_{s\in[0,\tau]}\overline{B}_{g(s)}(x_{0},A)\times\\{s\\},$

and the existence of minimal $\mathcal{L}$-geodesic follows from the standard
theory of variation.

Finally, we need to show that $(M,g(\tau))_{\tau\in[0,T]}$ is
$\mathcal{L}$-complete, where $T\leq\overline{T}$ satisfies the condition in
the previous Lemma. For $0\leq\tau_{1}\leq\tau_{2}\leq T$, it is obvious that
$(M,g(\tau+\tau_{1}))_{\tau\in[0,\tau_{2}-\tau_{1}]}$ still satisfies (3.9)
and (3.10). Thus we may take $\tau_{1}=0$ and $\tau_{2}=\bar{\tau}\in[0,T]$.
Let $E$ be an arbitrary closed set in $M$. Since each time slice of the
backward Ricci flow is complete, we may, without loss of generality, assume
that $E=\overline{B}_{g(\bar{\tau})}(x_{0},r)$. Applying the standard distance
distortion estimate, while using the fact that $|\mathrm{Ric}|$ is uniformly
bounded in $\overline{B}_{g(\bar{\tau})(x_{0},r)}\times[0,\bar{\tau}]$, we
have

$r_{0}:=\sup_{\tau\in[0,\bar{\tau}]}\sup_{y\in\overline{B}_{g(\bar{\tau})(x_{0},r)}}\operatorname{dist}_{g(\tau)}(x_{0},y)<\infty.$

It clearly follows from the triangle inequality that, for any
$y_{0}\in\overline{B}_{g(\bar{\tau})(x_{0},r)}$,

(3.15)
$\displaystyle\mathrm{Ric}_{g(\tau)}(x)\geq-\alpha(T-\tau)^{-\gamma}\cdot\ln\left(\operatorname{dist}_{g(\tau)}(y_{0},x)+\beta+r_{0}\right)g(\tau)\quad\text{for
all}\quad(x,\tau)\in M\times[0,T].$

Now we may argue as in the previous lemma with (3.10) replaced by (3.15),
$x_{0}$ replaced by an arbitrary $x\in\overline{B}_{g(\bar{\tau})}(x_{0},r)$,
and $\beta$ replaced by $\beta+r_{0}$ (note that $\overline{T}(\alpha,\gamma)$
is independent of $\beta$). There exists
$A=A(2r,L(E,\bar{\tau}),T,\alpha,\beta+r_{0},\gamma)>0$, where
$L(E,\bar{\tau})$ is obtained by applying Lemma 3.3 to
$E=\overline{B}_{g(\bar{\tau})}(x_{0},r)$ and $\bar{\tau}\in[0,T]$, such that
for $x$, $y\in\overline{B}_{g(\bar{\tau})}(x_{0},r)$, any minimal
$\mathcal{L}$-geodesic connecting $(x,0)$ and $(y,\bar{\tau})$ lies in the
space-time compact set

$\bigcup_{\tau\in[0,\bar{\tau}]}\overline{B}_{g(\tau)}(x,A)\times\\{s\\}\subset\bigcup_{\tau\in[0,\bar{\tau}]}\overline{B}_{g(\tau)}(x_{0},A+r_{0})\times\\{s\\},$

which is clearly contained in the cross product of a spatial compact set $F$
and the time interval $[0,\bar{\tau}]$. ∎

##  4\. Perelman’s pseudolocality under the $\mathcal{L}$-complete assumption

In this section, we prove Theorem 1.4. Let $(M,g_{t})_{t\in[0,T]}$ be the
$\mathcal{L}$-complete Ricci flow in the statement of Theorem 1.4 satisfying

(4.1) $\displaystyle R(g_{0})\geq-1,$ (4.2)
$\displaystyle\left(\operatorname{Area}_{g_{0}}(\partial\Omega)\right)^{n}\geq(1-\delta)n^{n}\omega_{n}\left(\operatorname{Vol}_{g_{0}}(\Omega)\right)^{n-1}\
\text{ for all }\ \Omega\subset M.$

When formulating an argument of the reduced distance, we shall always consider
the backward Ricci flow $g(\tau):=g_{T-\tau}$ instead.

Let $\ell:M\times[0,T]\to\mathbb{R}$ be the reduced distance based at some
point on the $\tau=0$ (i.e. $t=T$) slice and $\mathcal{V}(\tau)$ the
corresponding reduced volume. First of all, we shall prove some preparatory
lemmas focusing on the “final” time-slice of the $\ell$-function

(4.3) $\displaystyle\ell:=\ell(\cdot,T):M^{n}\to\mathbb{R}.$

######  Lemma 4.1.

Under the $\mathcal{L}$-complete assumption and (4.1), we have

(4.4) $\displaystyle\int_{M^{n}}|\nabla\ell|^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu\leq\frac{2n}{T}+2+2T^{\frac{1}{2}}<\infty.$

Consequently, $(4\pi T)^{-n/4}e^{-\ell/2}$ is a $W^{1,2}$ function on $M^{n}$.

###### Proof.

This lemma is the same as [7, Lemma 6.6]. We shall include its proof for the
convenience of the reader. Rewriting (3.5) at $\tau=T$, we have

$\displaystyle\int_{M^{n}}\Big{(}-2\nabla\ell\cdot\nabla\phi-|\nabla\ell|^{2}\phi\Big{)}d\mu$
$\displaystyle\leq$
$\displaystyle-\int_{M^{n}}\left(R+\frac{\ell-n}{T}\right)\phi d\mu$
$\displaystyle\leq$
$\displaystyle\left(\frac{n}{T}+1+T^{\frac{1}{2}}\right)\int_{M^{n}}\phi
d\mu,$

where we have used the facts that $R_{g(\tau)}\geq-1$ and that
$\ell\geq-T^{\frac{3}{2}}$; to see that the latter is true, one needs only to
recall the definition (2.1)(2.2) of the reduced distance and the fact that
$R_{g(\tau)}\geq-1$ for all $\tau\in[0,T]$ .

Next, we take

(4.6) $\displaystyle\phi:=\varphi_{A}^{2}(4\pi T)^{-\frac{n}{2}}e^{-\ell},$

where

$\varphi_{A}(x)=\eta\left(\frac{d_{g(T)}(x,x_{0})}{A}\right),$

$x_{0}$ is the fixed base point, and $\eta:[0,\infty)\to[0,1]$ is a smooth and
decreasing function satisfying $\eta|_{[0,1]}\equiv 1$,
$\eta|_{[2,\infty)}\equiv 0$, and $-2\leq\eta^{\prime}(s)\leq 0$ for all
$s\in[0,\infty)$. Obviously, $\phi$ is a valid test function of (3.5), since
$\varphi_{A}$ is smooth and compactly supported and $\ell$ is locally
Lipschitz. Then, (4) becomes

$\displaystyle\int_{M^{n}}\Big{(}\varphi_{A}^{2}|\nabla\ell|^{2}-4\langle\nabla\ell,\nabla\varphi_{A}\rangle\varphi_{A}\Big{)}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu\leq\left(\frac{n}{T}+1+T^{\frac{1}{2}}\right)\int_{M^{n}}\varphi_{A}^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu.$

Hence we have

$\displaystyle\int_{M^{n}}\varphi_{A}^{2}|\nabla\ell|^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu$ $\displaystyle\leq$
$\displaystyle\left(\frac{n}{T}+1+T^{\frac{1}{2}}\right)\int_{M^{n}}\varphi_{A}^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu$
$\displaystyle+4\int_{M^{n}}\langle\nabla\ell,\nabla\varphi_{A}\rangle\varphi_{A}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu$ $\displaystyle\leq$
$\displaystyle\left(\frac{n}{T}+1+T^{\frac{1}{2}}\right)\int_{M^{n}}\varphi_{A}^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu$
$\displaystyle+\frac{1}{2}\int_{M^{n}}\varphi_{A}^{2}|\nabla\ell|^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu$
$\displaystyle+8\int_{M^{n}}|\nabla\varphi_{A}|^{2}_{g_{T}}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu,$

and subsequently

$\displaystyle\int_{M}\varphi_{A}^{2}|\nabla\ell|^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu$ $\displaystyle\leq$
$\displaystyle\left(\frac{64}{A^{2}}+\frac{2n}{T}+2+2T^{\frac{1}{2}}\right)\int_{M}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu$ $\displaystyle=$
$\displaystyle\left(\frac{64}{A^{2}}+\frac{2n}{T}+2+2T^{\frac{1}{2}}\right)\mathcal{V}(T)$
$\displaystyle\leq$
$\displaystyle\frac{64}{A^{2}}+\frac{2n}{T}+2+2T^{\frac{1}{2}},$

where we have applied the fact that $\mathcal{V}(T)\leq 1$ due to Theorem
3.2(1). Taking $A\to\infty$, we obtain (4.4). The boundedness of the
$L^{2}$-norm of $(4\pi T)^{-n/4}e^{-\ell/2}$ is but a consequence of the fact
that $\mathcal{V}(T)\leq 1$. ∎

######  Lemma 4.2.

Under the $\mathcal{L}$-complete assumption and (4.1)(4.2), we have

$\mathcal{V}(T)\geq e^{-T}(1-\delta).$

###### Proof.

Rewriting (4) using the same cut-off function as defined in (4.6) and the fact
that $R(g_{0})\geq-1$, we have

$\displaystyle\int_{M^{n}}\left(|\nabla\ell|^{2}+\frac{\ell-n}{T}\right)\varphi_{A}^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu$ $\displaystyle\leq$ $\displaystyle\
4\int_{M^{n}}\langle\nabla\ell,\nabla\varphi_{A}\rangle\varphi_{A}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu-\int_{M}R\varphi_{A}^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu$ $\displaystyle\leq$ $\displaystyle\
4\left(\int_{M^{n}}|\nabla\ell|^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu\right)^{\frac{1}{2}}\left(\int_{M^{n}}|\nabla\varphi_{A}|^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu\right)^{\frac{1}{2}}+\int_{M}\varphi_{A}^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu$ $\displaystyle\leq$ $\displaystyle\
4\left(\frac{2n}{T}+2+2T^{\frac{1}{2}}\right)^{\frac{1}{2}}\left(\frac{4}{A^{2}}\mathcal{V}(T)\right)^{\frac{1}{2}}+\int_{M}\varphi_{A}^{2}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu,$

where we have applied Lemma 4.1. Taking $A\to\infty$, we have

(4.7) $\int_{M^{n}}\left(T|\nabla\ell|^{2}+\ell-n\right)(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu\leq T\int_{M}(4\pi
T)^{-\frac{n}{2}}e^{-\ell}d\mu.$

Rescaling $g(\tau)$ as

$\tilde{g}(\tau)=(2T)^{-1}g(2T\tau).$

and denote by $\tilde{\ell}$ the reduced length with respect to
$\tilde{g}(\tau)$. Then (4.7) becomes

(4.8)
$\int_{M^{n}}\left(\tfrac{1}{2}|\nabla\tilde{\ell}_{1/2}|^{2}+\tilde{\ell}_{1/2}-n\right)(2\pi)^{-\frac{n}{2}}e^{-\tilde{\ell}_{1/2}}d\tilde{\mu}\leq
T\int_{M}(2\pi)^{-\frac{n}{2}}e^{-\tilde{\ell}_{1/2}}d\tilde{\mu}.$

Letting

$\varphi(x):=(2\pi)^{-\frac{n}{4}}e^{-\frac{1}{2}\tilde{\ell}_{1/2}(x)}\in
W^{1,2}(M),$

we can rewrite (4.8) as

$\displaystyle\int_{M}\left(2|\nabla\varphi|^{2}-\varphi^{2}\log\varphi^{2}\right)d\tilde{\mu}$
$\displaystyle\leq$ $\displaystyle\
\left(n+\frac{n}{2}\log(2\pi)\right)\int_{M}\varphi^{2}d\tilde{\mu}+T\int_{M}\varphi^{2}d\tilde{\mu}$
$\displaystyle\leq$ $\displaystyle\
\int_{M}\left(2|\nabla\varphi|^{2}-\varphi^{2}\log\varphi^{2}\right)d\tilde{\mu}+\left(\log\left(\int_{M}\varphi^{2}d\tilde{\mu}\right)-\log(1-\delta)+T\right)\int_{M}\varphi^{2}d\tilde{\mu},$

where in the second inequality we have applied Theorem 2.2. Consequently, we
have

$\int_{M}\varphi^{2}d\tilde{\mu}\geq e^{-T}(1-\delta).$

∎

Now we proceed to prove Theorem 1.4 by contradiction. Assume that there exist
$\alpha$ and sequences of positive numbers $\delta_{i}\rightarrow
0^{+},\varepsilon_{i}\rightarrow 0^{+}$, and smooth $\mathcal{L}$-complete
pointed Ricci flows
$\left(\mathcal{M}_{i}^{n},g_{i,t}\right)_{t\in\left[0,\varepsilon_{i}^{2}\right]},i\in\mathbb{N},$
satisfying (4.1) (4.2). Furthermore, there exist “bad” points in space-time
$(x_{i},t_{i})\in\mathcal{M}_{i}\times\left(0,\varepsilon_{i}^{2}\right]$
satisfying

(4.9)
$\displaystyle\left|\mathrm{Rm}_{g_{i}}\right|\left(x_{i},t_{i}\right)>\frac{\alpha}{t_{i}}$

for all $i\in\mathbb{N}$.

We can apply a point-picking method as in Perelman’s proof of pseudolocality
theorem. The details can be found in [10]. Arguing in the same way as [10,
Lemma 21.12] and [10, Chapter 22 $\S 1$], we may adjust
$\left(x_{i},t_{i}\right)$ such that they satisfy not only (4.9), but also

(4.10) $\displaystyle\left|\mathrm{Rm}_{g_{i}}\right|(x,t)\leq
4Q_{i}\qquad\text{ for all }\qquad(x,t)\in
B_{g_{i,t_{i}}}\left(x_{i},A_{i}Q_{i}^{-1/2}\right)\times\left[t_{i}-\tfrac{\alpha}{2}Q_{i}^{-1},t_{i}\right]$

where
$Q_{i}:=\left|\mathrm{Rm}_{g_{i}}\right|(x_{i},t_{i})>\frac{\alpha}{t_{i}}$
and $A_{i}\to\infty$.

######  Lemma 4.3.

For each $i$, there exists a time

$\tilde{t}_{i}\in\left[t_{i}-\frac{\alpha}{2}Q_{i}^{-1},t_{i}\right)$

such that

$\mathcal{V}_{i}(t_{i}-\tilde{t}_{i})\leq 1-\beta_{1}$

where $\mathcal{V}_{i}$ is the reduced volume of $g_{i}$ based at
$\left(x_{i},t_{i}\right)$ and $\beta_{1}\in(0,1)$ is a constant independent
of $i$.

###### Proof.

We apply the following rescaling and time-shifting to the Ricci flows

(4.11) $g_{i,t}\longrightarrow\hat{g}_{i,t}\doteqdot
Q_{i}g_{i,t_{i}+Q_{i}^{-1}t},$

and proceed to show that for each $i\in\mathbb{N}$, there is a
$\hat{t}_{i}\in[-\frac{\alpha}{2},0)$ with

$\mathcal{\hat{V}}_{i}(-\hat{t}_{i})\leq 1-\beta_{1},$

where $\mathcal{\hat{V}}_{i}$ is the reduced volume of $\hat{g}_{i}$ based at
$(x_{i},0)$. (4.10) now becomes

(4.12) $\displaystyle\left|\mathrm{Rm}_{\hat{g}_{i}}\right|(x,t)\leq
4\qquad\text{ for all }\qquad(x,t)\in
B_{\hat{g}_{i,0}}\left(x_{i},A_{i}\right)\times\left[-\tfrac{\alpha}{2},0\right].$

We argue by contradiction. Assume that, by passing to a subsequence,

(4.13)
$\displaystyle\lim_{i\to\infty}\inf_{\tau\in(0,\frac{\alpha}{2}]}\mathcal{\hat{V}}_{i}(\tau)=1.$

Claim. There are positive constants $c(\alpha)$ and $C(\alpha)$ with the
following property. Fix any large $i\in\mathbb{N}$ and any
$\bar{\tau}\in(0,\frac{\alpha}{4}]$. Let $\gamma_{V}:[0,\bar{\tau}]\to M$ be a
minimal $\mathcal{L}$-geodesic with respect to
$\hat{g}_{i}(\tau)=\hat{g}_{i,-\tau}$ starting from $(x_{i},0)$ with
$\lim_{\tau\to 0}\sqrt{\tau}\gamma^{\prime}_{V}(\tau)=V\in T_{x_{i}}M$. For
any $A\in[10C(\alpha),A_{i}]$, if $|V|_{\hat{g}_{i}(0)}\leq c(\alpha)A$, then
$\gamma_{V}\big{|}_{[0,\bar{\tau}]}\subset
B_{\hat{g}_{i}(0)}(x_{i},\tfrac{1}{2}A)$.

###### Proof of the Claim.

Writing $X(\tau)=\gamma_{V}^{\prime}(\tau)$, we may apply the
$\mathcal{L}$-geodesic equation to obtain (c.f. [16, (26.4)])

$\displaystyle\frac{d}{d\tau}(\tau|X|^{2}_{\hat{g}_{i}(\tau)})=-2\tau\mathrm{\mathrm{Ric}}_{\hat{g}_{i}}(X,X)+\tau\langle
X,\nabla R_{\hat{g}_{i}}\rangle_{\hat{g}_{i}(\tau)}.$

By (4.12) and Shi’s estimates, we have

$\displaystyle\left|\mathrm{Rm}_{\hat{g}_{i}}\right|\leq 4,\qquad\left|\nabla
R_{\hat{g}_{i}}\right|\leq C(n,\alpha),\qquad\text{for all}\qquad(x,\tau)\in
B_{\hat{g}_{i}(0)}(x_{i}\tfrac{1}{2}A_{i})\times[0,\tfrac{\alpha}{4}].$

Then, as long as $\gamma_{V}\big{|}_{[0,\tau]}\subset
B_{\hat{g}_{i}(0)}(x_{i},\tfrac{1}{2}A_{i})$, we may estimate

$\displaystyle\frac{d}{d\tau}\left(\tau|X|^{2}_{\hat{g}_{i}(\tau)}\right)\leq
8\tau|X|^{2}_{\hat{g}_{i}(\tau)}+C\tau|X|_{\hat{g}_{i}(\tau)}\leq
9\tau|X|^{2}_{\hat{g}_{i}(\tau)}+C$

and by integrating

(4.14) $\displaystyle\tau|X|^{2}_{\hat{g}_{i}(\tau)}\leq
e^{4\alpha}|V|^{2}_{\hat{g}_{i}(0)}+C.$

Let us fix an $A\leq A_{i}$. Assume that
$\tau^{\prime}\in(0,\tfrac{\alpha}{4})$ is the first time when $\gamma_{V}$
has reached the boundary of $B_{\hat{g}_{i}(0)}(x_{i},\frac{1}{2}A)$. Thus,
$\gamma_{V}\big{|}_{[0,\tau^{\prime}]}\subset
B_{\hat{g}_{i}(0)}(x_{i},\frac{1}{2}A)$,
$\operatorname{dist}_{\hat{g}_{i}(0)}(\gamma_{V}(0),\gamma_{V}(\tau^{\prime}))=\frac{1}{2}A$,
and

$\displaystyle|X(\tau)|_{\hat{g}_{i}(\tau)}^{2}\geq
e^{-\alpha}|X(\tau)|^{2}_{\hat{g}_{i}(0)}\qquad\text{ for all
}\qquad\tau\in[0,\tau^{\prime}].$

Then,

$\displaystyle\tfrac{1}{2}A$
$\displaystyle=\operatorname{dist}_{\hat{g}_{i}(0)}(\gamma_{V}(0),\gamma_{V}(\tau^{\prime}))\leq\int_{0}^{\tau^{\prime}}|X(\tau)|_{\hat{g}_{i}(0)}d\tau\leq
e^{\alpha/2}\int_{0}^{\tau^{\prime}}|X(\tau)|_{\hat{g}_{i}(\tau)}d\tau$
$\displaystyle\leq
e^{\alpha/2}\left(\int_{0}^{\tau^{\prime}}\sqrt{\tau}|X(\tau)|^{2}_{\hat{g}_{i}(\tau)}d\tau\right)^{\frac{1}{2}}\left(\int_{0}^{\tau^{\prime}}\tau^{-\frac{1}{2}}d\tau\right)^{\frac{1}{2}}$
$\displaystyle\leq
C(\alpha)\left(\int_{0}^{\tau^{\prime}}\tau^{-\frac{1}{2}}\left(e^{4\alpha}|V|^{2}_{\hat{g}_{i}(0)}+C\right)d\tau\right)^{\frac{1}{2}}$
$\displaystyle\leq C(\alpha)(|V|_{\hat{g}_{i}(0)}+1),$

where we have applied (4.14). If $A\geq 10C(\alpha)$, then the above
computation yields a contradiction if we assume
$|V|_{\hat{g}_{i}(0)}\leq\frac{1}{4C(\alpha)}A$. This proves the claim. ∎

By contradictory assumption (4.13) and the proof of Perelman’s no local
collapsing theorem (c.f. [16, Theorem 26.2]), it is easy to see that

$\displaystyle\inf_{i}\operatorname{inj}_{g_{i,0}}(x_{i})>0.$

Therefore, we may extract a subsequence from
$\big{\\{}\big{(}M_{i},\hat{g}_{i,t},x_{i}\big{)}_{t\in[-\frac{\alpha}{2},0]}\big{\\}}_{i=1}^{\infty}$
which converges to a smooth Ricci flow with bounded curvature

$\displaystyle(M_{\infty},\hat{g}_{\infty,t},x_{\infty})_{t\in(-\frac{\alpha}{2},0]}.$

Since
$|\mathrm{Rm}_{\hat{g}_{\infty}}|(x_{\infty},0)=\lim_{i\to\infty}|\mathrm{Rm}_{\hat{g}_{i}}|(x_{i},0)=\lim_{i\to\infty}Q_{i}^{-1}|\mathrm{Rm}_{g_{i}}|(x_{i},t_{i})=1$,
$\hat{g}_{\infty}$ is not flat.

On the other hand, letting $\mathcal{L}exp_{x_{i}}^{\tau}$ be the
$\mathcal{L}$-exponential map of $\hat{g}_{i}$ based at $(x_{i},0)$. The claim
above shows that

$\displaystyle M\setminus
B_{\hat{g}_{i}(0)}(x_{i},\tfrac{1}{2}A)\subset\mathcal{L}exp_{x_{i}}^{\tau}\Big{(}\mathbb{R}^{n}\setminus\\{|V|\leq
cA\\}\Big{)},\ \ \ \text{ for all }\ \tau\in[0,\tfrac{\alpha}{4}]\ \text{ and
}\ A\in[10C(\alpha),A_{i}].$

By the monotonicity of the $\mathcal{L}$-Jacobian (3.6) and the fact that

$\lim_{\tau\to
0}(4\pi\tau)^{-\frac{n}{2}}e^{-l(\gamma_{V}(\tau),\tau)}\mathcal{L}J_{V}(\tau)=(4\pi)^{-\frac{n}{2}}\exp\left(-\tfrac{|V|^{2}}{4}\right),$

we have that, for each $\tau\in(0,\frac{\alpha}{4}]$,

$\displaystyle\int_{M\setminus
B_{\hat{g}_{i}(0)}(x_{i},\frac{1}{2}A)}(4\pi\tau)e^{-\ell_{\hat{g}_{i}}}(\cdot,\tau)d\hat{g}_{i}(\tau)\leq\int_{|V|\geq
cA}(4\pi)^{-\frac{n}{2}}\exp\left(-\tfrac{|V|^{2}}{4}\right)dV,\ \ \ \text{
for all }\ A\in[10C(\alpha),A_{i}].$

It is then clear from (4.13) that

$\mathcal{V}_{\infty}(\tau)=\lim_{i\to\infty}\mathcal{V}_{i}(\tau)=1\qquad\text{
for all }\qquad\tau\in(0,\tfrac{\alpha}{4}],$

where $\mathcal{V}_{\infty}$ is the reduced volume of $\hat{g}_{\infty}$ based
at $(x_{\infty},0)$. Thus, $\hat{g}_{\infty}$ is the flat Euclidean space;
this is a contradiction. ∎

Now we can give the proof of Theorem 1.4.

###### Proof of Theorem 1.4.

Applying Lemma 4.2 to the reduced volumes $\mathcal{V}_{i}$ of the Ricci flows
$(M_{i},g_{i})$ based at $(x_{i},t_{i})$, since
$t_{i}\leq\varepsilon_{i}^{2}$, we have that

$\mathcal{V}_{i}(t_{i})\geq e^{-\varepsilon_{i}^{2}}(1-\delta_{i}).$

By the monotonicity of the reduced volume, we also have

$\mathcal{V}_{i}(\tau)\geq
e^{-\varepsilon_{i}^{2}}(1-\delta_{i})\qquad\text{for all
}\qquad\tau\in(0,t_{i}].$

This is clearly a contradiction to Lemma 4.3 when $i$ is large.

∎

Theorem 1.5 just follows from Theorem 1.4 and Theorem 3.1 in [5].

##  5\. uniqueness on Eulidean space

###### Proof of Theorem 1.6.

By Theorem 1.3, the Ricci flow in question is $\mathcal{L}$-complete (in the
second case, we also assume $T\leq\overline{T}(\alpha,\gamma)$ for the
moment). Then Theorem 1.5 implies that the Ricci flow is one with bounded
curvature. The conclusion then follows from [6].

We remark here that in case (2), once we established the strong uniqueness
assuming $T\leq\overline{T}(\alpha,\gamma)$, then the same result holds also
for any $T>0$ by splitting $[0,T]$ into smaller intervals, each with length no
greater than $\overline{T}(\alpha,\gamma)$. ∎

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# Interaction and Decision Making-aware Motion Planning using Branch Model
Predictive Control††thanks: This work was partially supported by the
Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the
Knut and Alice Wallenberg Foundation.

Rui Oliveira  Division of Decision and Control Systems
KTH Royal Institute of Technology
Stockholm, Sweden
<EMAIL_ADDRESS>Siddharth H. Nair  Model Predictive Control Laboratory
UC Berkeley
Berkeley, USA
<EMAIL_ADDRESS>Bo Wahlberg  Division of Decision and Control
Systems
KTH Royal Institute of Technology
Stockholm, Sweden
<EMAIL_ADDRESS>
###### Abstract

Motion planning for autonomous vehicles sharing the road with human drivers
remains challenging. The difficulty arises from three challenging aspects:
human drivers are 1) multi-modal, 2) interacting with the autonomous vehicle,
and 3) actively making decisions based on the current state of the traffic
scene. We propose a motion planning framework based on Branch Model Predictive
Control to deal with these challenges. The multi-modality is addressed by
considering multiple future outcomes associated with different decisions taken
by the human driver. The interactive nature of humans is considered by
modeling them as reactive agents impacted by the actions of the autonomous
vehicle. Finally, we consider a model developed in human neuroscience studies
as a possible way of encoding the decision making process of human drivers. We
present simulation results in various scenarios, showing the advantages of the
proposed method and its ability to plan assertive maneuvers that convey intent
to humans.

##  I Introduction

Autonomous vehicles (AVs) must drive in the presence of other traffic
participants, such as human-driven vehicles (HVs) and pedestrians. To this
date, sharing the road with human traffic participants is one of the biggest
challenges hindering AVs from being deployed at a large scale.

Current state-of-the-art sensor and perception technology already provides an
accurate understanding of the traffic scene’s current state. However, the
irregularity of human behavior makes the prediction task, i.e., knowing how
the traffic scene evolves, reliable only for a few seconds into the future.

Besides being hard to predict, the traffic scene evolution is also directly
impacted by the decisions taken by the AV. Human drivers will react
differently depending on other surrounding vehicles’ maneuvers.

Most planning approaches assume that other traffic participant predictions are
fixed, resulting in the autonomous vehicle performing maneuvers to avoid said
predictions. The result is overly conservative driving from the autonomous
vehicle. To consider that the AV’s decisions impact the traffic scene
evolution and avoid conservativeness, one must solve the joint prediction and
planning problem.

This work presents a novel motion planning approach to tackle the joint
prediction and planning problem, making the following contributions:

  * •

Proposal of a framework for handling interaction-heavy scenarios, considering
the aspects of _multi-modality_ , _interaction_ , and _decision making_ of
human drivers;

  * •

Approximating human drivers’ decision making process through models developed
in neuroscience studies, allowing the autonomous vehicle to take proactive and
assertive maneuvers that convey intent to human drivers;

  * •

Performance evaluation and comparison against relevant works in the area,
showing an increase in average performance without sacrificing safety.

$v=v_{r}$Initial state ($t=0$)$v=0$Outcome 1 ($t=4$)$v=v_{r}$Outcome 2 ($t=4$)

(a) Multi-modality

$v=v_{r}$$v=v_{r}$Initial state ($t=0$)$v=2$$v=2$Outcome 1
($t=4$)$v=v_{r}$$v=v_{r}$Outcome 2 ($t=4$)

(b) Interaction

$v=v_{r}$$v=v_{r}$Initial state ($t=0$)$v=0$$v=v_{r}$Outcome 1
($t=4$)$v=v_{r}$$v=0$Outcome 2 ($t=4$)

(c) Decision making

Figure 1: Challenges associated with an autonomous vehicle (yellow car)
driving in the presence of human drivers (red car).

###  I-A Related work

We start by introducing three challenging aspects of human drivers and
existing works relating to them.

####  I-A1 Multi-modality

Consider a human driver arriving at an intersection (_initial state_ of
LABEL:\baselabelfig:multi-modal). A defensive driver slows down and stops to
check for oncoming vehicles safely (_outcome 1_). On the other hand, an
aggressive driver speeds through the intersection to achieve a shorter
traveling time (_outcome 2_). A planner must consider different outcomes and
plan a motion that guarantees safety for all.

Model Predictive Control (MPC) approaches for tackling uncertainties stemming
from the multi-modality are introduced in [1, 2]. However, they are limited to
uncertainties stemming from the unknown existence of static obstacles or
intents of pedestrians.

The works [3, 4] use Scenario MPC to consider the multi-modality arising from
the uncertainty over different maneuvers types of other drivers (such as keep
or change lanes). [5] combines Scenario MPC and Stochastic MPC to improve
other vehicles’ motion predictions.

Contingency MPC is an approach that tracks a desired nominal plan while
maintaining a contingency plan to deal with possible emergencies [6, 7].
Multi-modality is tackled by considering both the nominal and contingency
outcomes.

####  I-A2 Interaction

Driving is a highly interactive task, where drivers adapt their actions in
response to those of other drivers. Consider the HV (red car) and AV (yellow
car) approaching the intersection shown in _initial state_ of
LABEL:\baselabelfig:interaction. If the AV turns right, the HV will slow down
to avoid a collision (_outcome 1_). On the other hand, if the AV proceeds
through the intersection, the HV keeps its speed (_outcome 2_). Interactions
occur in most traffic situations, and considering them is crucial to reducing
AVs’ conservativeness.

The problem of driving an autonomous vehicle through an intersection is
tackled with Stochastic MPC in [8]. The approach considers the interaction
aspects of driving by modeling human drivers as closed-loop predictions
tracking a constant headway to the AV in front of them.

The work in [9] introduces a formulation of interaction with HVs as an
underactuated dynamical system. This formulation allows the AV to perform
complex interaction behaviors, such as accelerating or slowing down, to show
intent to humans.

####  I-A3 Decision making

Humans often make decisions while driving, as shown in the initial state of
LABEL:\baselabelfig:decision-making), where the HV (red car) has to decide if
it will cross the intersection. The decision is affected by the perceived
intended behavior of the AV (yellow car). If the AV approaches at high speed
(_outcome 1_), the human decides to stop at the intersection. On the other
hand, if the AV slows down (_outcome 2_), the human drives through the
intersection. This example shows how the traffic scene, namely the AV’s
position and velocity, affects the HV’s decision [10]. This example
significantly differs from the interaction aspects shown in
LABEL:\baselabelfig:interaction, as no imminent collision forces the HV to
stop. Instead, the AV shows an intent to drive through or stop at the
intersection, leading the HV to react accordingly.

Branch MPC [11] is used to tackle the multi-modality arising from other human
driver’s decision making. The human is modeled using a finite set of policies
that build a scenario tree. The planned solution is a feedback policy in the
form of a trajectory tree accounting for all possible scenarios. The HV
policies are propagated independently of other agents, which can lead to the
freezing robot problem [12]. Realizing this drawback, the authors of [13]
consider closed-loop models to propagate the control policies of other
vehicles. However, both works lack a driver decision making model based on
human behavior research.

####  I-A4 Summary

LABEL:\baselabeltable:approaches_comparison outlines the previous works
regarding their ability to tackle the mentioned challenges. Only recently has
[13] considered all challenges. We build upon [11] by considering the reactive
behavior of humans and addressing the _interaction_ challenge. Moreover, we
model human _decision making_ using neuroscience studies, adding a sound
sociological model lacking in [13].

TABLE I: Comparison of different planning approaches according to the challenges identified in Section I-A.  Approach | Multi-modal | Interaction | Decision making
---|---|---|---
[1],[2],[4],[7] | ✓ | $\times$ | $\times$
[11] | ✓ | $\times$ | ✓
[9] | $\times$ | ✓ | ✓
[8] | ✓ | ✓ | $\times$
[13], our approach | ✓ | ✓ | ✓

##  II Modeling

###  II-A Scenarios

We consider two scenarios that force the AV to interact with a HV. In the
first scenario, the AV merges onto a road, as shown in
LABEL:\baselabelfig:y_merging_scenario. The second scenario considers a non-
signalized intersection, i.e., an intersection without traffic lights
LABEL:\baselabelfig:intersection_scenario. Both scenarios lack priority rules,
and thus no vehicle has the right of way over the other, requiring interaction
and unspoken negotiation between them.

(a) Merging scenario. The AV (yellow) needs to merge into a lane where an HV
(red) is also merging into.

(b) Intersection scenario. The AV (yellow) and the HV (red) need to ensure
that they do not cross simultaneously.

Figure 2: The scenarios considered in this work.

We assume that a vehicle $i$, $i\in\\{H,A\\}$, human-driven ($H$) or
autonomous ($A$), moves along the road centerline along a path length $s^{i}$.
In both scenarios in LABEL:\baselabelfig:y_merging_scenario and
LABEL:\baselabelfig:intersection_scenario, the vehicles are on separate lanes
until the merging point $s_{\text{conflict}}$. After $s_{\text{conflict}}$ the
vehicles are on the same lane (merging), or share a conflict region
(intersection), and must keep a safe distance to avoid a collision. We assume
that as the HV approaches the conflict point, it eventually makes a decision
at $s_{\text{br}}$, possibly changing its behavior. $s_{\text{br}}$ is a point
located before $s_{\text{conflict}}$, and it is chosen so as to represent a
point on the road where the HV becomes aware of the AV, and takes a decision
according to its preferences and driving style, [14], and even the intention
shown by the AV [10].

###  II-B Vehicle models

We assume the AV to drive along the lane center and only plan for its
longitudinal motion. This assumption is justified as there is little to no
advantage in considering the possibility of steering the vehicle laterally in
these scenarios. Both vehicles follow the model

$x=\begin{bmatrix}s&v\end{bmatrix}^{\intercal},\quad\dot{x}=\begin{bmatrix}v&u\end{bmatrix}^{\intercal},$
(1)

where $s$ is the current position along the centerline path, $v$ is the
vehicle longitudinal velocity, and $u$ is the acceleration, corresponding to
the control input.

The AV input $u^{A}$ is determined by the planning framework presented in
LABEL:\baselabelsec:mpc_formulation. The HV input $u^{H}$ follows a certain
driving policy $\pi^{\text{before}}$, up to $s_{\text{br}}$, and afterwards
$\pi^{\text{after}}$:

$u^{H}=\begin{cases}\pi^{\text{before}}(s^{H},v^{H})&\text{if
}s^{H}<s_{\text{br}}\\\ \pi^{\text{after}}(s^{H},v^{H},s^{A},v^{A})&\text{if
}s^{H}\geq s_{\text{br}}\end{cases}$ (2)

Note that $\pi^{\text{after}}$ is a function of $s^{A}$ and $v^{A}$, due to
the interaction behavior between the HV and AV.

###  II-C Human driving policies

We assume two types of policies for the HV, _velocity tracking_ when there is
no vehicle ahead, and _vehicle following_ when there is a vehicle ahead.

####  II-C1 Velocity tracking

When there is no leading vehicle ahead, the HV will track a desired reference
speed $v_{\text{ref}}$. We then have
$u^{H}=\pi^{v_{\text{ref}},\text{track}}$, where

$\pi^{v_{\text{ref}},\text{track}}\left(v^{H},v_{\text{ref}}\right)=K_{v}\left(v_{\text{ref}}-v^{H}\right),$
(3)

and $K_{v}>0$ is a constant gain. Policy $\pi^{v_{\text{ref}},\text{track}}$
takes as inputs the current vehicle velocity $v^{H}$ and the desired reference
velocity $v_{\text{ref}}$, and outputs an acceleration command proportional to
their difference.

####  II-C2 Vehicle following

If there is another vehicle ahead, the HV adapts its speed to avoid a rear-end
collision. In this case, $u^{H}=\pi^{\text{va}}$, where va stands for _vehicle
ahead_ , and:

$\pi^{\text{va}}=\begin{cases}\pi^{v_{\text{ref}},\text{track}}\left(v^{H},v_{\text{ref}}\right)&\text{if
}v^{H}\geq v_{\text{ref}}\\\
K_{v}(v^{A}-v^{H})+K_{d}\left(d-d_{\text{ref}}\right)&\text{if
}v^{H}<v_{\text{ref}}\end{cases}$ (4)

where $d=s^{A}-s^{H}$ corresponds to the distance from the human-driven
vehicle to the vehicle ahead, and $K_{d}>0$ is a constant gain.

When $v^{H}\geq v_{\text{ref}}$, the vehicle tracks its desired velocity
$v_{\text{ref}}$, resulting in braking. When $v^{H}<v_{\text{ref}}$, the
vehicle speeds up while taking into account the vehicle ahead. The term
$K_{v}(v^{A}-v^{H})$ performs velocity tracking, and
$K_{d}\left(d-d_{\text{ref}}\right)$ keeps a safe distance $d_{\text{ref}}$ to
the vehicle in front. Policy $\pi^{\text{va}}$ is inspired by the Intelligent
Driver Model [15].

###  II-D Human decision making

In the considered scenarios we assume the human has two different behaviors,
one before arriving at $s_{\text{br}}$, and another after it, as given by
LABEL:\baselabeleq:policies_before_after_branching. At $s_{\text{br}}$ the HV
starts interacting with the AV and changes its behavior. The new behavior
depends on the type of driver profile and on the traffic scene, and it is not
known in advance. In the merge scenario we consider behaviors corresponding to
three types of driver:

  * •

$\pi^{\text{va,fast}}$ \- egoistic driver who speeds up
($v_{\text{ref}}=v_{\text{ref}}^{\text{fast}}$),

  * •

$\pi^{\text{va,keep}}$ \- neutral driver who keeps speed
($v_{\text{ref}}=v_{\text{ref}}^{\text{keep}}$),

  * •

$\pi^{\text{va,slow}}$ \- altruistic driver who slows down
($v_{\text{ref}}=v_{\text{ref}}^{\text{slow}}$).

In the intersection scenario we consider two types of driver:

  * •

$\pi^{\text{cross}}$ \- driver keeps speed
($v_{\text{ref}}=v_{\text{ref}}^{\text{keep}}$),

  * •

$\pi^{\text{stop}}$ \- driver slows down to a stop.

At point $s_{\text{br}}$ the human decides on one of the policies to follow.
We make use of research in the field of neuroscience, namely we consider the
work in [10], where the authors study the decision making process of human
drivers approaching an intersection. The authors propose that drivers’
decision making depends on the degree of safety of the two co-existing
possibilities of either crossing or stopping at the intersection. The degree
of safety can be quantified by the critical time to cross $CT_{\text{cross}}$,
and the critical time to stop $CT_{\text{stop}}$. The probability of the human
choosing to cross, i.e., applying policy $\pi^{\text{cross}}$, is defined as
[10]:

$P_{\pi^{\text{cross}}}=\frac{w}{1+e^{-aCT_{\text{cross}}}}+\frac{1-w}{1+e^{-bCT_{\text{stop}}}}.$
(5)

Where the parameters $a$, $b$, and $w$ are found by fitting the model
LABEL:\baselabeleq:crossing_frequency to experimental data. The experimental
data is obtained from thirty experienced drivers whose decision making is
studied in a driving simulator.

LABEL:\baselabeleq:crossing_frequency provides a decision making model that
can be used to estimate the probabilities of the human driver taking different
decisions. This allows the AV to understand the likelihood of different future
scene outcomes stemming from different decision taken by the human driver.
Moreover, the dependency of LABEL:\baselabeleq:crossing_frequency on both the
HV and AV states allows the planner to reason about how different maneuvers
affect the likelihood of different scenarios. This is fundamental in order for
the AV to achieve assertive behavior. The following section proposes a
framework that can take advantage of model
LABEL:\baselabeleq:crossing_frequency, providing the planner with a better
knowledge of HV behavior and allowing the AV to drive assertively.

##  III Motion Planning Framework

###  III-A Tree formulation - human-driven vehicle

In order to take into account the possible future decisions of the human, and
the resulting states of its vehicle, we use a tree structure [11]. Figure 3
shows a tree with $J+1$ branches (branches $[2,\ldots,J-1]$ not visible). Each
branch $j$ in the tree has an associated human driver policy $\pi^{j}$. Within
a branch $j$, the HV states are propagated assuming they follow the associated
policy $\pi^{j}$. The evolution of states corresponds to a discretized model
of LABEL:\baselabeleq:vehicle_model so that

$x_{t+1}^{H,\pi^{j}}=f^{H,\pi^{j}}\left(x_{t}^{H,\pi^{j}},x_{t}^{A,\pi^{j}}\right),$
(6)

where the vehicle input $u_{t}^{H,\pi^{j}}$ is determined according the active
policy $\pi^{j}$, and can depend on the AV state, as in
LABEL:\baselabeleq:policy_velocity_following. For the transition states
between branches

$x_{t_{\text{br}}+1}^{H,\pi^{j}}=f^{H,\pi^{j}}\left(x_{t_{\text{br}}}^{H,\pi^{0}},x_{t_{\text{br}}}^{A,\pi^{0}}\right),$
(7)

where $u_{t_{\text{br}}}^{H,\pi^{j}}$ already follows the policy $\pi^{j}$.

The tree in Fig. 3 is composed of a root branch and $J$ child branches,
corresponding to the set of branches $\mathcal{J}=\\{0,1,2,\ldots,J\\}$. The
root branch splits into different branches at time $t_{\text{br}}$,
corresponding to the state at which the HV crosses the path length
$s_{\text{br}}$. At this path length $s_{\text{br}}$, the human changes to a
policy that interacts with the AV. Since the human can take multiple policies,
$x_{t_{\text{br}}}^{H,\pi^{0}}$ propagates into $J$ different states
$x_{t_{br}+1}^{H,\pi^{j}}$, corresponding to the different branches with
policies $\pi^{j}$. Each branch has an associated probability $P_{\pi^{j}}$.
For the first branch $P_{\pi^{0}}=1$, and for the remaining branches
$\sum_{j=1}^{J}P_{\pi^{j}}=1$.

$x_{t}^{H,\pi^{0}}$$x_{t+1}^{H,\pi^{0}}$$x_{t_{\text{br}}}^{H,\pi^{0}}$$\pi_{0},P_{\pi_{0}}$$x_{t_{\text{br}}+1}^{H,\pi^{1}}$$x_{t_{\text{br}}+2}^{H,\pi^{1}}$$x_{t+N}^{H,\pi^{1}}$$\pi_{1},P_{\pi_{1}}$$x_{t_{\text{br}}+1}^{H,\pi^{J}}$$x_{t_{\text{br}}+2}^{H,\pi^{J}}$$x_{t+N}^{H,\pi^{J}}$$\pi_{J},P_{\pi_{J}}$$\vdots$

Figure 3: Diagram of the scenario tree used to model the HV possible future
states.

###  III-B Tree formulation - autonomous vehicle

The AV state evolution follows a similar tree structure as the HV, illustrated
in Fig. 4, and corresponding to a discretized model of
LABEL:\baselabeleq:vehicle_model so that

$x_{t+1}^{A,\pi^{j}}=f^{A,\pi^{j}}\left(x_{t}^{A,\pi^{j}},u_{t}^{A,\pi^{j}}\right).$
(8)

The vehicle input $u_{i}^{A,\pi^{j}}$ is determined by the solution to the
optimal control problem introduced later in
LABEL:\baselabelsubsec:MPC_formulation. Similarly, for the branching states,
we have:

$x_{t_{\text{br}}+1}^{A,\pi^{j}}=f^{A,\pi^{j}}\left(x_{t_{\text{br}}}^{A,\pi^{0}},u_{i}^{A,\pi^{j}}\right).$
(9)

$x_{t}^{A,\pi^{0}}$$x_{t+1}^{A,\pi^{0}}$$x_{t_{\text{br}}}^{A,\pi^{0}}$$P_{\pi^{0}}$$x_{t_{\text{br}}+1}^{A,\pi^{1}}$$x_{t_{\text{br}}+\Delta
t_{\text{obs}}}^{A,\pi^{1}}$$x_{t+N}^{A,\pi^{1}}$$P_{\pi^{1}}$$x_{t_{\text{br}}+1}^{A,\pi^{J}}$$x_{t_{\text{br}}+\Delta
t_{\text{obs}}}^{A,\pi^{J}}$$x_{t+N}^{A,\pi^{J}}$$P_{\pi^{J}}$$\vdots$

Figure 4: Diagram of the scenario tree used to model the AV possible future
states. The arcs between states in branch $1$ and branch $J$ correspond to
autonomous vehicle states that are forced to be equal, according to
LABEL:\baselabeleq:observation_constraints.

In a practical setting, it is not possible to immediately estimate the policy
decision made by the human-driven vehicle, as prediction systems have a delay
until accurately estimating the new human behavior. Therefore, we force the
autonomous vehicle states to be equal between the different branches
$[1,\ldots,J]$ for the first $\Delta t_{\text{obs}}$ seconds of the branch:

$x_{t_{i}}^{A,\pi^{j}}=x_{t_{i}}^{A,\pi^{j^{\prime}}},\\\ \forall
t_{i}\in[t_{\text{br}}+1,\ldots,t_{\text{br}}+\Delta
t_{\text{obs}}],\\{\forall j,j^{\prime}\in\mathcal{J}|j\neq j^{\prime}\\}.$
(10)

LABEL:\baselabeleq:observation_constraints forces the solutions to not assume
immediate knowledge of the HV policy, and instead delay by $\Delta
t_{\text{obs}}$ the adaption to the new behavior. $\Delta t_{\text{obs}}$ is
tuned based on considerations of expected time to perceive a new vehicle
behavior and feasibility of the planning problem.

###  III-C MPC formulation

For each branch $j\in\mathcal{J}$, consider the vectors

$\displaystyle\bm{x}^{H}_{j}$
$\displaystyle=[x_{t_{i}^{j}}^{H,\pi^{j}},x_{t_{i}^{j}+1}^{H,\pi^{j}},\ldots,x_{t_{f}^{j}}^{H,\pi^{j}}],$
$\displaystyle\bm{x}^{A}_{j}$
$\displaystyle=[x_{t_{i}^{j}}^{A,\pi^{j}},x_{t_{i}^{j}+1}^{A,\pi^{j}},\ldots,x_{t_{f}^{j}}^{A,\pi^{j}}],$
$\displaystyle\bm{u}^{A}_{j}$
$\displaystyle=[u_{t_{i}^{j}}^{A,\pi^{j}},u_{t_{i}^{j}+1}^{A,\pi^{j}},\ldots,u_{t_{f}^{j}}^{A,\pi^{j}}],$

where $t_{i}^{j}$ and $t_{f}^{j}$ are initial and final times associated with
the first and last states in branch $j$. $\bm{x}^{H}_{j}$ corresponds to the
HV states, and $\bm{x}^{A}_{j}$, $\bm{u}^{A}_{j}$ to the AV states, and
inputs, respectively.

The MPC is solved at the beginning of every planning cycle at time $t$, its
formulation is as follows:

$\displaystyle\underset{\\{\bm{u}^{A}_{j}\\}_{j\in\mathcal{J}}}{\text{min}}{\hskip
2.5pt}$
$\displaystyle\sum_{j\in\mathcal{J}}~{}P_{\pi^{j}}J(\bm{x}^{H}_{j},\bm{x}^{A}_{j},\bm{u}^{A}_{j})$
(11a) s.t.
$\displaystyle\mathrm{\lx@cref{creftype~refnum}{\baselabeleq:discretized_human_model}},\forall
i=t_{i}^{j},t_{i}^{j}+1,\ldots,t_{f}^{j},\forall j\in\mathcal{J}$ (11b)
$\displaystyle\mathrm{\lx@cref{creftype~refnum}{\baselabeleq:discretized_human_model_branching_states}},\forall
j\in\mathcal{J}\setminus\\{0\\},$ (11c)
$\displaystyle\mathrm{\lx@cref{creftype~refnum}{\baselabeleq:discretized_autonomous_model}},\forall
i=t_{i}^{j},t_{i}^{j}+1,\ldots,t_{f}^{j},\forall j\in\mathcal{J},$ (11d)
$\displaystyle\mathrm{\lx@cref{creftype~refnum}{\baselabeleq:discretized_autonomous_model_branching_states}},\forall
j\in\mathcal{J}\setminus\\{0\\},$ (11e)
$\displaystyle\mathrm{\lx@cref{creftype~refnum}{\baselabeleq:observation_constraints}},\forall
j\in\mathcal{J}\setminus\\{0\\},$ (11f)
$\displaystyle\begin{cases}P_{\pi^{0}}=1&\text{if }t\leq t_{\text{br}}\\\
P_{\pi^{0}}=0&\text{otherwise}\end{cases},$ (11g)
$\displaystyle\begin{cases}\mathrm{\lx@cref{creftype~refnum}{\baselabeleq:crossing_frequency}},\;j\in\mathcal{J}\setminus\\{0\\}&\text{if
}t\leq t_{\text{br}}+\Delta t_{\text{obs}}\\\
P_{\pi^{j^{\prime}}}\leavevmode\resizebox{3.33333pt}{0.0pt}{=}1,P_{\pi^{j}}\leavevmode\resizebox{3.33333pt}{0.0pt}{=}0,\;j\in\mathcal{J}\setminus\\{{j^{\prime}}\\}&\text{otherwise}\end{cases}$
(11h)
$\displaystyle\begin{cases}x_{t}^{H,\pi^{0}}\leavevmode\resizebox{3.33333pt}{0.0pt}{=}x_{\text{init}}^{H},x_{t}^{A,\pi^{0}}\leavevmode\resizebox{3.33333pt}{0.0pt}{=}x_{\text{init}}^{A}&\text{if
}t\leq t_{\text{br}}\\\
x_{t}^{H,\pi^{j}}\leavevmode\resizebox{3.33333pt}{0.0pt}{=}x_{\text{init}}^{H},x_{t}^{A,\pi^{j}}\leavevmode\resizebox{3.33333pt}{0.0pt}{=}x_{\text{init}}^{A},j\in\mathcal{J}\setminus\\{0\\}&\text{otherwise}\end{cases}$
(11i)

LABEL:\baselabeleq:constraint_1 and LABEL:\baselabeleq:constraint_2 encode the
state evolution constraints within the branches for the HV and AV. Similarly,
LABEL:\baselabeleq:constraint_3 and LABEL:\baselabeleq:constraint_4 encode the
state evolution constraints between branches. LABEL:\baselabeleq:constraint_5
sets the observation constraints forcing the AV states to be equal across
branches $j\in\mathcal{J}\setminus\\{0\\}$, from time $t_{\text{br}}$ until
time $t_{\text{br}}+\Delta t_{\text{obs}}$.
LABEL:\baselabeleq:cnt_prob_root_node defines the probability of the root
branch: in case the current planning cycle happens before the HV has taken a
decision, i.e., $t\leq t_{\text{br}}$, the branch exists and is certain,
$P_{\pi^{0}}=1$. Otherwise, the branch does not exist anymore, and it is
disregarded in the optimization objective, $P_{\pi^{0}}=0$.
LABEL:\baselabeleq:cnt_prob_leaf_nodes defines the probabilities of the leaf
branches: in case the current planning cycle happens before the AV can
identify the true mode of the human, i.e., $t\leq t_{\text{br}}+\Delta
t_{\text{obs}}$, its probability is given by the behavioral decision making
model in [10]. Otherwise, the probability of the branch corresponding to the
true mode is set to 1, and the probability of all other branches is 0, and
they are ignored from LABEL:\baselabeleq:optimization_problem. Finally,
LABEL:\baselabeleq:cnt_initial_states makes the initial HV and AV states of
the root branch correspond to the current measured states
$x_{\text{init}}^{H},x_{\text{init}}^{A}$ of the vehicles if $t\leq
t_{\text{br}}$. Otherwise it makes the initial state of all possible branches
equal to $x_{\text{init}}^{H},x_{\text{init}}^{A}$.

Solving optimization problem LABEL:\baselabeleq:optimization_problem, we find
the optimal AV input vectors ${\bm{u}^{A}_{j}}^{\star}$ for each branch of the
tree. At each planning cycle, the optimization problem
LABEL:\baselabeleq:optimization_problem is solved, considering the latest
sensor measurements $x_{\text{init}}^{H},x_{\text{init}}^{A}$. The vehicle
control input corresponds to the first element of the root branch input vector
${\bm{u}^{A}_{0}}^{\star}$ if $t\leq t_{\text{br}}$. Otherwise, it corresponds
to the first element of an active child branch.

##  IV Results

The following results are obtained in a custom Python simulation environment.
The MPC problem presented in LABEL:\baselabeleq:optimization_problem is
formulated using CasADi [16], and the corresponding nonlinear optimization
problem is solved using Ipopt [17].

###  IV-A Uncertain Traffic Light

We present the results of a scenario where a traffic light on the road cannot
be adequately perceived by the AV sensors, illustrated in
LABEL:\baselabelfig:uncertain_obstacle_scenario. The AV knows about the
existence of the traffic light, due to information contained in maps or since
the perception systems detect the existence of the traffic light pole.
However, it does not know the state of the traffic light due to poor
visibility conditions. We assume that at $s_{\text{br}}$, the AV accurately
detects the state of the traffic light.

The tree structure for this scenario corresponds to a root node lasting until
the AV crosses $s_{\text{br}}$, with two child nodes, one that continues
driving if the traffic light is green, $\pi^{\text{green}}$, and one coming to
a stop if the traffic light is red, $\pi^{\text{red}}$.

Figure 5: Scenario with an uncertain traffic light.

We study the performances of a Robust, Prescient, and Contingency MPC and
compare them to our proposed Branch MPC. The different formulations are as
follows:

  * •

Robust MPC - assumes the traffic light is red until $s_{\text{br}}$, afterward
it detects the actual traffic light state;

  * •

Prescient MPC - assumes a perfect perception system that knows the state of
the traffic light even before $s_{\text{br}}$;

  * •

Contingency MPC - optimizing the driving behavior as if the traffic light was
green, but keeping a contingency plan if the light is red;

  * •

Branch MPC - our approach, considering both traffic light possibilities and
assuming $P_{\pi^{\text{green}}}=P_{\pi^{\text{red}}}=0.5$.

We run closed-loop simulations for different probabilities of the traffic
light being red. The average performance of the different approaches is shown
in LABEL:\baselabelfig:uncertain_obstacles_comparison_average.

Figure 6: Average closed-loop costs of Robust, Prescient, Contingency, and
Branch MPC for different probabilities of the scenario’s traffic light being
red.

The Robust MPC has an expected cost that is never better than the Prescient or
Branch MPCs. Since Robust MPC always assumes that the obstacle exists, it
achieves an acceptable performance when the probability of the obstacle is
high enough. However, in most cases, it has the worst performance. On the
other hand, the Prescient MPC consistently achieves the best performance as it
plans its maneuver according to the actual (unknown) state of the traffic
light.

The Contingency MPC performance is comparable to the Branch and Prescient MPCs
for small obstacle probabilities. However, as the obstacle probability
increases, its performance degrades. The Contingency MPC is an optimistic
planner, assuming a green light until proven otherwise. This results in abrupt
braking maneuvers when the light is red, leading to poor performance.

Our proposed method achieves, on average, a performance that is always better
than Robust MPC while being equally as safe. For lower probabilities of red
light, the Branch MPC performance is comparable to that of Contingency MPC.
For higher probabilities, Branch MPC outperforms Contingency MPC due to not
being optimistic about the traffic light state.

LABEL:\baselabelfig:uncertain_obstacle_single_planning shows a single planning
cycle for the different MPC approaches starting at identical initial vehicle
states. In this initial state, the AV is aware of the traffic light but does
not know its state yet. Robust MPC plans a single velocity profile assuming a
red light and bringing the vehicle to a stop. Contingency MPC plans two
velocity profiles to deal with the possibility of green and red lights. The
velocity profile associated with a red light is not penalized in the cost
function, resulting in an abrupt and uncomfortable contingency maneuver.
Branch MPC reduces its driving speed, at the cost of increasing travel time,
to deal with the possibility of a red light comfortably. Therefore, Branch MPC
chooses a tradeoff between the performances in both future possible scenarios.

Figure 7: Single planning cycle for the autonomous vehicle approaching an
uncertain obstacle under different planning frameworks: Branch, Robust, and
Contingency MPCs.

###  IV-B Multi-modality

We consider the merging scenario in LABEL:\baselabelfig:y_merging_scenario,
where the HV can have three possible future velocity tracking policies
$\pi^{\text{va,fast}}$, $\pi^{\text{va,keep}}$, or $\pi^{\text{va,slow}}$. The
tree structure consists of a root node where the human driver keeps a constant
speed policy $\pi_{0}$ until the branching point, and three child nodes
corresponding to the HV taking the three possible policies. Although chosen
manually in this study, the tracking policies could be given by a prediction
module providing a multi-modal set of outcomes, as in [18, 19].

LABEL:\baselabelfig:merge_scenario_single_plan_tree (bottom) shows the planned
velocity profiles of a planning cycle occurring when the HV and AV are heading
toward the merging point. The HV is predicted to have the possibility to take
either of the three different velocity profiles. The AV plans a set of
velocity profiles, going behind the HV in case it decides to keep its speed or
accelerate or going ahead in case the HV slows down. The AV plan is better
visualized by looking at the planned path lengths in
LABEL:\baselabelfig:merge_scenario_single_plan_tree (top). The path lengths
(subtracted by the predicted path length of the HV with policy
$\pi^{\text{va,keep}}$) show the distinctive decisions made by the planner to
either go behind or ahead of different possible policies of the HV.

Figure 8: Planned autonomous vehicle trajectory (solid line) and predicted
human-driven vehicle trajectory (dashed line) for the Branch MPC and Robust
MPC cases. Top: Path lengths (centered around the path executed by
$\pi^{\text{va,keep}}$). The rounded markers correspond to the instant when
the vehicles cross the conflict point. Bottom: Velocity profiles.

LABEL:\baselabelfig:merge_scenario_single_plan_tree also shows the velocity
planned by a Robust MPC approach. The planned maneuver is conservative, opting
to go behind all possible realizations of future human policies. Considering
the multi-modality and planning feedback policies dependent on the mode that
the human eventually decides upon allows the planned profiles to squeeze in
between different possible policies, reducing conservativeness compared to a
robust approach.

###  IV-C Non-interaction vs. interaction-aware human models

We now consider an AV that accelerates from a standstill and merges onto a
road with an oncoming vehicle driving at high speed. The AV can decide to go
ahead of the other vehicle or wait for it to pass and go behind it.

Figure 9 shows a single planning cycle, where the AV is at a standstill, and
an oncoming vehicle drives faster than the AV’s desired cruising velocity. In
the non-interacting vehicle model case, the AV predicts the HV to keep its
speed constant and decides to wait for it to pass and, afterward, go behind
it. In the case of packed traffic with several oncoming vehicles, the AV could
be stuck indefinitely at the junction [12]. However, when considering an
interaction-aware model, the AV decides to go ahead of the oncoming vehicle,
as it predicts that the HV will slow down and adapt its speed. We remark that
the planner assumes a limit to the HV’s braking capabilities to guarantee that
the AV does not act discourteously to other traffic participants [20].

Figure 9: Predicted and planned velocity profiles when considering non-
interacting and interaction-aware HV models.

###  IV-D Intersection scenario

We consider the non-signalized intersection scenario in
LABEL:\baselabelfig:intersection_scenario. Since there are no priority rules,
the vehicles must negotiate who goes first. The tree structure consists of a
root node and two child nodes corresponding to the HV keeping its speed,
$\pi^{\text{cross}}$, or stopping, $\pi^{\text{stop}}$.

LABEL:\baselabelfig:intersection_fixed_probabilities shows the AV planned
velocities when assuming that both HV policies have an equal and fixed
probability $P_{\pi^{\text{cross}}}=P_{\pi^{\text{stop}}}=0.5$. The AV decides
to slow down to deal with both possible outcomes of the human decision.

(a) Fixed probabilities.

(b) Adapting probabilities.

Figure 10: Intersection scenario for different configurations of human-vehicle
policy probabilities.

We note that this intersection scenario resembles the one considered in [10],
and therefore we use the human decision making model
LABEL:\baselabeleq:crossing_frequency to determine probabilities
$P_{\pi^{\text{cross}}}$ and $P_{\pi^{\text{stop}}}$.
LABEL:\baselabelfig:intersection_varying_probabilities shows the results for
this scenario when considering that the human policy probabilities follow
LABEL:\baselabeleq:crossing_frequency. In this case, the planned velocity
profile increases at around $t=5$ s to assertively indicate to the human
driver that the AV intends to cross. This results in a higher predicted
probability of the human deciding to slow down and give way to the autonomous
vehicle, $P_{\pi^{\text{stop}}}=0.856$. With a lower probability of the human
keeping its speed, $P_{\pi^{\text{cross}}}=0.144$, the AV plans a more abrupt
maneuver in the unlikely event of this outcome occurring.

These results show that considering the human decision making model
LABEL:\baselabeleq:crossing_frequency allows the planner to make assertive
maneuvers and improve driving performance. The AV drives in a way that shows
intent to other HVs, intending to optimize its expected driving outcomes. This
behavior comes out naturally due to minimizing the objective cost of
LABEL:\baselabeleq:optimization_objective, and therefore, does not require
manually tuned driving strategies for different traffic situations.

##  V Conclusions

We presented a motion planning framework tackling the challenges of autonomous
vehicles in the presence of human drivers, namely, multi-modality,
interaction, and decision making. A Branch MPC problem is formulated by
combining scenario trees and research from the neuroscience field to model the
human driver’s decision making process. We show that Branch MPC and
interaction-aware models achieve better average performance than alternative
formulations in the literature. Furthermore, we show that using a human
decision making model leads the planner to find proactive maneuvers that
convey intent to the human driver. The proposed framework plans assertive
maneuvers that influence human drivers to make decisions favorable to the
autonomous vehicle.

In future work, it is interesting to analyze the impact of the problem
modeling assumptions and to consider behavioral decision making for more
general driving scenarios. Finally, practical implementation is needed to
validate the suitability of the proposed approach to model and solve the joint
prediction and planning problem accurately.

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