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Large deviations for sticky-reflecting Brownian motion with boundary diffusion

Jean-Baptiste Casteras, Leonard Monsaingeon, Luca Nenna

TL;DR

The paper derives a Schilder-type large deviation principle for sticky-reflected Brownian motion with boundary diffusion in short time, uncovering a phase transition at a = 1 that modifies the intrinsic distance and induces a nonstandard optimal transport geometry. It provides an explicit transition kernel in half-spaces, a static LDP with a piecewise rate function c(x,y) that combines interior and boundary contributions, and a dynamical LDP for path measures, thereby connecting stochastic diffusion with entropic optimal transport and Schrödinger-type problems. The analysis shows that for a ≤ 1 the cost reduces to the Euclidean metric, while for a > 1 the optimal paths may travel along the boundary, giving rise to angle-dependent geodesics and a cone-based cost structure. These results justify and illuminate a boundary-influenced transport framework and set the stage for gradient-flow formulations in non-smooth domains.

Abstract

We study a Schilder-type large deviation principle for sticky-reflected Brownian motion with boundary diffusion, both at the static and sample path level in the short-time limit. A sharp transition for the rate function occurs, depending on whether the tangential boundary diffusion is faster or slower than in the interior of the domain. The resulting intrinsic distance naturally gives rise to a novel optimal transport model, where motion and kinetic energy are treated differently in the interior and along the boundary.

Large deviations for sticky-reflecting Brownian motion with boundary diffusion

TL;DR

The paper derives a Schilder-type large deviation principle for sticky-reflected Brownian motion with boundary diffusion in short time, uncovering a phase transition at a = 1 that modifies the intrinsic distance and induces a nonstandard optimal transport geometry. It provides an explicit transition kernel in half-spaces, a static LDP with a piecewise rate function c(x,y) that combines interior and boundary contributions, and a dynamical LDP for path measures, thereby connecting stochastic diffusion with entropic optimal transport and Schrödinger-type problems. The analysis shows that for a ≤ 1 the cost reduces to the Euclidean metric, while for a > 1 the optimal paths may travel along the boundary, giving rise to angle-dependent geodesics and a cone-based cost structure. These results justify and illuminate a boundary-influenced transport framework and set the stage for gradient-flow formulations in non-smooth domains.

Abstract

We study a Schilder-type large deviation principle for sticky-reflected Brownian motion with boundary diffusion, both at the static and sample path level in the short-time limit. A sharp transition for the rate function occurs, depending on whether the tangential boundary diffusion is faster or slower than in the interior of the domain. The resulting intrinsic distance naturally gives rise to a novel optimal transport model, where motion and kinetic energy are treated differently in the interior and along the boundary.
Paper Structure (7 sections, 8 theorems, 144 equations, 3 figures)

This paper contains 7 sections, 8 theorems, 144 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Sticky-reflected Brownian Motion with boundary diffusion
  3. Heuristics and technical obstructions
  4. Transition kernel in half-spaces
  5. Static Large Deviation Principle
  6. Study of the cost function
  7. Sample path Large Deviation Principle

Key Result

Theorem 5.1

Let $a>0$ and write for convenience $A=a-1$. For any fixed $x=(x_1,x')\in\bar{D}=\mathbb{R}^+\times\mathbb{R}$ the sequence $\left\{ \rho^\varepsilon_x\right\}_{\varepsilon>0}\in \mathcal{P}(\bar{D})$ from eq:def_rhoeps_2d satisfies the LDP with good rate function $y\mapsto c(x,y)$ given by where for $A=a-1>0$ the "cone" $\mathfrak C$ is Moreover $(x,y)\mapsto c(x,y)$ is symmetric and continuou

Figures (3)

  • Figure 1: The cone $\mathfrak C(x)$.
  • Figure 2: The alternatives in Lemma \ref{['lem:I2_epxlicit']} and Lemma \ref{['lem:min_I1_I2']} (for $a>1$).
  • Figure 3: Geodesics for all possible configurations of $(x,y)\in {\bar{D}}^2$

Theorems & Definitions (16)

  • Theorem 5.1
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • proof : Proof of Theorem \ref{['theo:LDP_static']}
  • Theorem 6.1
  • Corollary 6.2
  • proof
  • proof : Proof of Theorem \ref{['theo:c_static_dynamic-geodesics']}
  • ...and 6 more