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.
Categories: math.PR