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, say Xnx, of the associated reflecting diffusion Xx and proceed with weak ν-directional derivatives, leading to a Jacobian matrix Jnx(t) whose columns are noted Jnx,ν=ei(t) (ei being the canonical basis of Rd) for insights to begin with. The weakly coupled SDE (Xn⋅,Jn⋅) is then shown to have a suitable realization which is a unique point-wise solution for all integers. This family is tight in the Jakubowski S-topology with limit (Xx,Jx), Jx is the solution of a certain linear SDE. For a generator in Hörmander form the quantities E(∇f(Xx(t))⋅Jx,ei(t)) (where t≥0, x∈D and f the differentiable initial condition) are the gradient (∂iu(t,x)) when a certain system of parabolic PDEs with the absolute boundary condition is uniquely solvable. Some more extensions and applications are discussed in the closing Section.