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

2025/10/02 by Madani, Abdelatif Benchérif
#35CXX #60HXX #FOS: Mathematics #G.3 #Probability (math.PR)

paper · doi:10.48550/arxiv.2510.01898

Abstract

We give a probabilistic representation for the gradient of a 2nd order linear parabolic PDE ∂_tu(t,x)=(1/2)a^ij∂_iju(t,x)+bⁱ∂_iu(t,x) with Cauchy initial condition u(0,x)=f(x) and Neumann boundary condition in a (closed) convex bounded smooth domain D in ℝ^d, d≥ 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 𝔼(∇ f(X^x(t))⋅ J^x,e_i(t)) for the gradient ∂_iu(t,x), where x∈ D, t≥ 0, e_i the canonical basis of 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.

Citations

• Pathwise differentiability of reflected diffusions in convex polyhedral domains
• New characterizations of the S topology on the Skorokhod space
• Reflected brownian motion: selection, approximation and linearization
• Differentiability of stochastic flow of reflected Brownian motions
• Multiplicative functional for reflected Brownian motion via deterministic ODE
• Multiplicative functional for reflected Brownian motion via deterministic ODE
• Bismut–Elworthy's formula and random walk representation for SDEs with reflection
• Strong Approximations for Time-Dependent Queues

Related

• Pointwise explicit estimates for derivatives of solutions to linear parabolic PDEs with Neumann boundary conditions
• Probabilistic representation of weak solutions to a parabolic boundary value problem on a non-smooth domain
• A probabilistic approach to large time behaviour of viscosity solutions of parabolic equations with Neumann boundary conditions
• Probabilistic representation of the gradient of a killed diffusion semigroup: The half-space case
• Gradient Holder regularity for parabolic normalized p(x,t)-Laplace equation