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

We give a probabilistic representation for the gradient of a 2nd order linear parabolic PDE ${\mathrm{\partial }}_{t}u(t,x)=(1\mathrm{/}2)a^{ij}{\mathrm{\partial }}_{ij}u(t,x)+b^i{\mathrm{\partial }}_{i}u(t,x)$ with Cauchy initial condition $u(0,x)=f(x)$ and Neumann boundary condition in a (closed) convex bounded smooth domain $D$ in ${\mathbb{R}}^d$, $d\ge 1$. 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 $\mathbb{E}(\mathrm{\nabla }f(X^x(t))\cdot J^{x,e_i}(t))$ for the gradient ${\mathrm{\partial }}_{i}u(t,x)$, where $x\in D$, $t\ge 0$, $e_i$ 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.