A Note on Transportation Cost Inequalities for Diffusions with Reflections

Authors

Soumik Pal Department of Mathematics, University of Washington, Seattle

Andrey Sarantsev Department of Mathematics and Statistics, University of Nevada, Reno

Electronic Communications in Probability (2019)

Abstract

We prove that reflected Brownian motion with normal reflections in a convex domain satisfies a dimension free Talagrand type transportation cost-information inequality. The result is generalized to other reflected diffusion processes with suitable drift and diffusion coefficients. We apply this to get such an inequality for interacting Brownian particles with rank-based drift and diffusion coefficients such as the infinite Atlas model. This is an improvement over earlier dimension-dependent results.

1. Introduction and the Main Results

Consider a metric space $(E, \rho)$. Fix $p \in [1, \infty)$. For any pair of Borel probability measures $\MP$ and $\MQ$ on $E$, define the Wasserstein distance of order $p$ as $$ W_p(\MP, \MQ) := \inf\limits_{\pi\in \Pi}\left[\iint\rho^p(x, y)\md\pi(x, y)\right]^{1/p}, $$ where the $\inf$ is taken over the set $\Pi$ of all couplings of $\MP$ and $\MQ$ (i.e., measures on $E\times E$ with marginal distributions $\MP$ and $\MQ$). Here and throughout, when we write $\mu(f)$, where $\mu$ is a probability measure and $f$ is a $\mu$-integrable function, we mean the expectation of $f$ with respect to $\mu$. The relative entropy $\CH(\MQ\mid\MP)$ of $\MQ$ with respect to $\MP$ is defined as $$ \CH(\MQ\mid\MP) := \MQ\left[\log\frac{\md\MQ}{\md\MP}\right] = \MP\left[\frac{\md\MQ}{\md\MP}\log\left(\frac{\md\MQ}{\md\MP}\right)\right], \quad \mbox{if}\; \MQ \ll \MP, $$ and $\CH(\MQ\mid\MP) = +\infty$ otherwise.

Definition

A Borel probability measure $\MP$ satisfies the transportation-cost information (TCI) inequality of order $p$ with constant $C > 0$ (we write: $\MP \in T_p(C)$) if for every Borel probability measure $\MQ$ on $E$ we have: \[\label{eq:TCI} W_p(\MP, \MQ) \le \sqrt{2C\CH(\MQ\mid\MP)}.\]

TCI is an example of the vast gallery of various inequalities linking transportation cost, relative entropy, and Fisher information. It is impossible to do justice to the enormous literature and its many uses. We refer the reader to an excellent survey by Gozlan and L\'eonard \cite{Gozlansurvey} and the recent book [NewBook] by Boucheron et al. Talagrand studied concentration for product spaces in \cite{Talagrand3, Talagrand5}. His idea is that a function of many variables which is Lipschitz in any variable but does not depend much on any single variable is close to a constant. A particularly useful application of TCI inequalities, such as above (for $p\ge 1$), is to prove Talagrand type Gaussian concentration. See the original article by Talagrand \cite{Talagrand5}, as well as Marton's derivation by using TCI inequality in \cite{Marton}. See also \cite{Bobkov1, C3, Otto} on relation between TCI and log-Sobolev inequalities.

In this paper, $\MP$ and $\MQ$ will represent laws of reflecting diffusion processes seen as probability measures on the set of continuous paths equipped with the uniform norm. Specifically we prove TCI inequalities for a certain class of interacting Brownian particle systems, called competing Brownian particles, with each particle moving as a Brownian motion on the real line with drift and diffusion coefficients dependent on the current rank of this particle relative to other particles. These systems were constructed in as a model for financial markets; see also . Our inequalities are dimension-free: that is, the constant $C$ is independent of the number of particles. This allows us to extend the inequality to infinite competing particle systems such as the infinite Atlas model . This is an improvement over the dimension-dependent inequalities in papers where applications of such inequalities can be found. See also on Poincaré inequalities for competing Brownian particles, and on large deviations for these particle systems.

The result for competing particles is a particular case of a general TCI inequality for normally reflected diffusion processes in convex domains. Reflected diffusions are defined as continuous-time stochastic processes in a certain domain $D \subseteq \BR^d$. As long as such process is in the interior, it behaves as a solution of a stochastic differential equation (SDE). As it hits the boundary, it is reflected back inside the domain. The simplest case is a reflected Brownian motion, which behaves as a Brownian motion inside the domain.

Dimension-free TCI inequalities are remarkable. Most known examples are in the case of product measures which utilize tensorization property of the entropy and the cost. Our examples are far from product measures since they involve motion of particles interacting with one another. Hence, dimension-free TCI inequalities in this context seem interesting. The proof, however, does not require much beyond existing machinery. Our novel contribution is essentially a single observation made in [eq:nonincreasing-1].

1.1. Notation

We denote by $a\cdot b = a_1b_1 + \ldots + a_Nb_N$ the dot product of vectors $a, b \in \BR^N$. The Euclidean norm of a vector $a$ is denoted by $\norm{a} := \left[a\cdot a\right]^{1/2}$. The matrix norm of a matrix $A$ is defined as $\norm{A} := \max_{\norm{x} = 1}\norm{Ax}$. We let $\BR_+ := [0, \infty)$. We denote the space $C\left([0, T], \BR^N\right)$ of continuous functions $[0, T] \to \BR^N$ with the sup-norm $\norm{x} := \sup_{0 \le t \le T}\norm{x(t)}$. We prove TCI inequality where the Wasserstein-$2$ transportation cost is measured in this norm.

1.2. TCI inequalities for competing Brownian particles

Fix an integer $N \ge 2$. For any vector $x = (x_1, \ldots, x_N) \in \BR^N$, there exists a unique ranking permutation: a one-to-one mapping $\mP_x : \{1, \ldots, N\} \to \{1, \ldots, N\}$, with the following properties:

$x_{\mP_x(i)} \le x_{\mP_x(j)}$ for $1 \le i < j \le N$; [1]
if $x_{\mP_x(i)} = x_{\mP_x(j)}$ for $1 \le i < j \le N$, then $\mP_x(i) < \mP_x(j)$. [2]

That is, $\mP_x$ arranges the coordinates of $x$ in increasing order, with ties broken by the increasing order of the index (or, name) of the coordinates that are tied.

Take a filtered probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t \ge 0}, \MP)$, with the filtration satisfying the usual conditions and supporting an $N$-dimensional Brownian motion $W = (W_1, \ldots, W_N)$. Fix constants $g_1, \ldots, g_N \in \BR$ and $\sigma_1, \ldots, \sigma_N > 0$.

Definition 1.1

Consider a continuous adapted process $X(t) = (X_1(t), \ldots, X_N(t)),\ t \ge 0$. Let $\mP_t = \mP_{X(t)}$. We say that $\mP_t^{-1}(i)$ is the rank of particle $i$ at time $t$, and $\mP_t(k)$ is the name of the $k$th ranked particle at time $t$. Then the following system of SDE: \[\label{eq:mainSDE} \md X_i(t) = \sum_{k=1}^N1(\mP_t(k) = i)\left(g_k\md t + \sigma_k\md W_i(t)\right),\] for $i = 1, \ldots, N$, defines a finite system of $N$ competing Brownian particles with drift coefficients $g_1, \ldots, g_N$ and diffusion coefficients $\sigma_1^2, \ldots, \sigma_N^2$. Let $Y_k(t) := X_{(k)}(t) := X_{\mP_t(k)}(t)$ be the position of the $k$th ranked particle, and let $Z_k(t) := Y_{k+1}(t) - Y_k(t)$ be the gap between the $k$th and $(k+1)$st ranked particles. The local time $L_{(k, k+1)} = (L_{(k, k+1)}(t), t \ge 0)$ of collision between $k$th and $k+1$st ranked particles is defined as the local time of the continuous semimartingale $Z_k$ at zero. The process $L(t) = \left(L_{(1, 2)}(t), \ldots, L_{(N-1, N)}(t)\right)$ is called the vector of local times.

From [Bass1987], this system exists in the weak sense and is unique in law. Strong existence and pathwise unqiueness are proved under the following assumptions [12, 14]: \[\label{eq:convex-diffusion} \sigma^2_n \ge \frac12\left(\sigma^2_{n-1} + \sigma^2_{n+1}\right),\, \mbox{if}\; 1 < n < N.\] Similar infinite systems can be defined for $N = \infty$; then we assume that the vector $X(t) = (X_i(t))_{i \ge 1}$ is rankable; that is, for every $t\ge 0$ there exists a unique permutation $\mP_{X(t)}$ of $\mathbb{N} := \{1, 2, \ldots\}$ which satisfies conditions ([1]) and ([2]). They were introduced in . See for weak existence and uniqueness in law under an assumption on initial conditions, \[\label{eq:initial-asmp} (4) \hspace{2cm} \lim\limits_{n \to \infty}X_n(0) \to \infty\quad \mbox{a.s., and}\quad \sum\limits_{n=1}^{\infty}e^{-\alpha X_n^2(0)} < \infty\quad \mbox{for all}\quad \alpha > 0,\] and the following assumptions on drift and diffusion coefficients: \[\label{eq:coeff-asmp} (5) \hspace{2cm} g_{n} = g_{n_0}\quad \mbox{and}\quad \sigma_n = \sigma_{n_0}\quad \mbox{for all}\quad n \ge n_0.\] See and for strong existence and pathwise uniqueness: We need [3] in addition to [4] and [5]. Two-sided infinite systems, indexed by $i \in \mathbb Z$, were introduced in [2]. The proofs and the results from this paper carry over to that set-up as well.

Theorem 1.1.

(a) For an $N \in \mathbb{N} \cup \{\infty\}$ assume that the drift and diffusion coefficients satisfy the following conditions: $g_1 \ge g_2 \ge \ldots$, and $\sigma_1 = \sigma_2 = \ldots = 1$. For the case of an infinite system, assume in addition [4] and [5]. Then for every finite $k \le N$, the distribution of $X = (X_1, \ldots, X_k)$ on $C([0, T], \BR^k)$ satisfies $T_2(C)$ with $C = T$.

(b) Assume weak existence and uniqueness in law. For an $N \in \mathbb{N}\cup\{\infty\}$, and a finite $k \le N$, the vector of ranked particles $Y = (Y_1, \ldots, Y_k)$ satisfies $T_2(C)$ on $C([0, T], \BR^k)$ with $C = T\sup\limits_{m \ge 1}\sigma_m^2$.

Theorem 1.1(a) follows from results from [LogConcave]. The more non-trivial Theorem 1.1(b) is based on Theorem 1.2 below, which is the main result of this paper. This is a general result that says that normally reflected Brownian motion in a convex domain satisfies a dimension-free TCI inequality as described below. It turns out that the vector of ranked particles $Y$ is a particular case of such normally reflected Brownian motion in a wedge $\{y = (y_1, \ldots, y_N)\mid y_1 \le \ldots \le y_N\}$.

Fix $d \ge 2$, the dimension. In this article a domain in $\BR^d$ is the closure of an open connected subset. We consider only convex domains. Following , we do not impose any additional smoothness conditions on such domain. For every $x \in \partial D$, we say that a unit vector $y \in \BR^d$ is an inward unit normal vector at point $x$, if \[\label{eq:inward-normal} z \in D\quad \mbox{implies}\quad (z - x)\cdot y \ge 0.\] The set of such inward unit normal vectors at $x$ is denoted as $\mathcal N(x)$. The most elementary example of this is a $C^1$ domain $D$; that is, with boundary $\pa D$ which can be locally (after a rotation) parametrized as a graph of a $C^1$ function. Then there exists a unique inward unit normal vector $\fn(x)$ at every point $x \in \pa D$, and $\mathcal N(x) = \{\fn(x)\}$.

A more complicated example is a convex piecewise smooth domain. Fix $m \ge 1$, the number of faces. Take $m$ domains $D_1, \ldots, D_m$ in $\BR^d$ which are $C^1$ and convex. Let $D = \bigcap_{i=1}^mD_i$. Assume $D \ne \cap_{j \ne i}D_j$ for every $i = 1, \ldots, m$; that is, each one of $m$ smooth domains is essential. Assume also that for each $i = 1, \ldots, m$, $F_i := \pa D\cap\pa D_i$ is a manifold of codimension $1$ and has nonempty relative interior. Then $D$ is called a convex piecewise smooth domain with $m$ faces $F_1, \ldots, F_m$, and $\pa D = \cup_{i=1}^mF_i$. For every $x \in F_i$, define the inward unit normal vector $\fn_i(x)$ to $\pa D_i$ at this point $x$, pointing inside $D_i$. For a point $x \in \pa D$ on the boundary, if $I(x) = \{i = 1, \ldots, m\mid x \in F_i\}$, then \[\mathcal N(x) = \Bigl\{\sum\limits_{i \in I}\alpha_i\fn_i(x)\mid \alpha_i \ge 0,\quad i \in I;\quad \sum\limits_{i \in I}\alpha_i^2 = 1\Bigr\}.\]

Definition 1.2.

For a vector field $g : \BR_+\times D \to \BR^d$ and a $z_0 \in D$, consider the following equation: \[\label{eq:SDE-main} (7) \quad Z(t) = z_0 + W(t) + \int_0^tg(s, Z(s))\,\md s + \int_0^t\fn(s)\,\md \ell(s),\ \ t \ge 0.\] Here, $Z : \BR_+ \to D$ is a continuous adapted process, $W$ is a $d$-dimensional Brownian motion with zero drift vector and constant, symmetric, positive definite $d\times d$ covariance matrix $A$, starting from the origin. For every $t \ge 0$, $\fn(t) \in \BR^d$ is a unit vector, and $\fn : \BR_+ \to \BR$ is a measurable function. The function $\ell : \BR_+ \to \BR$ is continuous, nondecreasing, and can increase only when $Z(t) \in \pa D$; for such $t$, $\fn(t) \in \mathcal N(Z(t))$. The solution $Z$ of (7) is called a reflected diffusion in $D$, with drift $g$ and (constant) diffusion matrix $A$, starting from $z_0$. If $g$ is a constant (does not depend on $x$ and $t$), then we call $Z$ a reflected Brownian motion (RBM) in $D$ with drift $g$ and diffusion matrix $A$.

Assumption 1.1.

For an integrable function $F : [0, T] \to \BR$, we have: \[\label{eq:monotone-F} \left(g(t, x) - g(t, y)\right)\cdot(x - y) \le \norm{x-y}^2F(t),\ t \in [0, T],\ x, y \in D,\]

Under Assumption 1.1, the equation (7) has a pathwise unique strong solution on time horizon $[0, T]$. This is proved similarly to [Theorem 4.1, Tanaka1979].

We start by proving that, under Assumption 1.1, the reflected diffusion satisfies a dimension-free $T_2(C)$. Our proof follows existing ideas in for non-reflected diffusions with one notable observation to handle the reflection. Consider an SDE in $\BR^d$ without reflection $\md X(t) = \md W(t) + g(t, X(t))\,\md t$, for some drift vector field $g$ defined on $[0, T]\times \mathbb{R}^d$, which satisfies contraction condition: \[\label{eq:contraction} (g(t, x) - g(t, y)) \cdot (x - y) \le 0,\quad \mbox{for all}\quad x, y \in \BR^d,\, t \in [0, T].\] It is shown in [LogConcave] that under condition the distribution of $X$ in the space $C([0, T], \BR^d)$ satisfies $T_2(C)$ with $C = T$. Our main observation in this article is that for a reflected diffusion in a convex domain $D$, the reflection term $\fn(t)\,\md\ell(t)$ plays the role of such drift.

Now comes the crucial observation: \[\label{eq:nonincreasing-1} \fn(t)\cdot(X(t) - X'(t))\,\md\ell(t) \le 0,\quad \text{for all}\; t \ge 0.\]