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

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 $\nu$-directional Jacobians is tight in the Jakubowski S-topology with limit $J^{x,\nu}$, 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.