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 ∂ₜu(t,x)=(1/2)aⁱʲ∂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 Rᵈ, d≥ 1. The idea is to start from a penalized version of the associated reflecting diffusion Xˣ, proceed with a pathwise derivative, show that the resulting family of $ν$-directional Jacobians is tight in the Jakubowski S-topology with limit Jx,ν, solution of a certain linear SDE, and set E(∇ f(Xˣ(t))· J^x,eᵢ(t)) for the gradient ∂ᵢu(t,x), where x∈ D, t≥ 0, eᵢ the canonical basis of Rᵈ and f, the initial condition of the semigroup of Xˣ, is differentiable. Some more extensions and applications are discussed in the concluding remarks.