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A probabilistic representation for the gradient in a linear parabolic PDE with Neumann boundary condition

Abdelatif Benchérif Madani

TL;DR

This work develops a probabilistic representation for the gradient of the solution to a linear parabolic PDE with Neumann boundary in a smooth bounded domain by penalizing the reflecting diffusion and differentiating the resulting flow pathwise. By proving tightness of the penalized Jacobian processes in the Jakubowski $S$-topology and identifying limit points via a linear SDE with boundary interactions, the authors obtain a gradient formula: $\\partial_\\nu u(t,x) = \\mathbb{E}[\\nabla f(X^{x}(t)) \\cdot J^{x,\\nu}(t)]$, tying the gradient to the Jacobian along the diffusion flow. A differential-geometric viewpoint is employed, where the gradient is treated as a 1-form and the Weingarten map (shape operator) naturally appears in boundary contributions, leading to an absolute boundary condition framework. The paper also discusses extensions to nonconvex domains via localization, and to curved manifolds, highlighting the potential of this probabilistic approach for gradient computation in PDEs with complex boundaries.

Abstract

We give a probabilistic representation for the gradient of a 2nd order linear parabolic PDE $\partial_{t}u(t,x)=(1/2)a^{ij}\partial_{ij}u(t,x)+b^{i}\partial_{i}u(t,x)$ with Cauchy initial condition $u(0,x)=f(x)$ and Neumann boundary condition in a (closed) convex bounded smooth domain $D$ in $\mathbb{R}^{d}$, $d\geq 1$. The idea is to start from a penalized version of the associated reflecting diffusion $X^{x}$, proceed with a pathwise derivative, show that the resulting family of $ν$-directional Jacobians is tight in the Jakubowski S-topology with limit $J^{x,ν}$, solution of a certain linear SDE, and set $\mathbb{E}\left(\nabla f(X^{x}(t))\cdot J^{x,e_{i}}(t)\right)$ for the gradient $\partial_{i}u(t,x)$, where $x\in D$, $t\geq 0$, $e_{i}$ the canonical basis of $\mathbb{R}^{d}$ and $f$, the initial condition of the semigroup of $X^{x}$, is differentiable. Some more extensions and applications are discussed in the concluding remarks.

A probabilistic representation for the gradient in a linear parabolic PDE with Neumann boundary condition

TL;DR

This work develops a probabilistic representation for the gradient of the solution to a linear parabolic PDE with Neumann boundary in a smooth bounded domain by penalizing the reflecting diffusion and differentiating the resulting flow pathwise. By proving tightness of the penalized Jacobian processes in the Jakubowski -topology and identifying limit points via a linear SDE with boundary interactions, the authors obtain a gradient formula: , tying the gradient to the Jacobian along the diffusion flow. A differential-geometric viewpoint is employed, where the gradient is treated as a 1-form and the Weingarten map (shape operator) naturally appears in boundary contributions, leading to an absolute boundary condition framework. The paper also discusses extensions to nonconvex domains via localization, and to curved manifolds, highlighting the potential of this probabilistic approach for gradient computation in PDEs with complex boundaries.

Abstract

We give a probabilistic representation for the gradient of a 2nd order linear parabolic PDE with Cauchy initial condition and Neumann boundary condition in a (closed) convex bounded smooth domain in , . The idea is to start from a penalized version of the associated reflecting diffusion , proceed with a pathwise derivative, show that the resulting family of -directional Jacobians is tight in the Jakubowski S-topology with limit , solution of a certain linear SDE, and set for the gradient , where , , the canonical basis of and , the initial condition of the semigroup of , is differentiable. Some more extensions and applications are discussed in the concluding remarks.

Paper Structure

This paper contains 9 sections, 7 theorems, 61 equations.

Table of Contents

  1. Introduction and statement of the result
  2. Notation
  3. Main result
  4. Proof of theorem \ref{['ThmMain']}
  5. Tightness
  6. The limit points
  7. A probabilistic formula for the gradient
  8. The Weingarten map
  9. Concluding remarks

Key Result

Theorem 1.1

Given our assumptions on the coefficients and $D$, let $L$ be the generator $L=L^{X}+L^{J}$, where with domain $\mathcal{D}$ the set of functions $F(x,\nu)\in \mathcal{C}^{2}_{b}(D\times\mathbb{R}^{d})$ subject to the following boundary conditions: for all $\nu\in\mathbb{R}^{d}$ we have and Let $x\in D$ and $\nu$ be a (unit) direction in $\mathbb{R}^{d}$. The family of Markov processes $(X^{x}_

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 4 more