We know that the solution of the heat equation $\partial\_tu=\frac 12\Delta u$ with Dirichlet boundary condition $u\rvert\_{\partial\Omega}=g$ is $u(t,x)=\mathbb{E}[g(B\_\tau)\mid B\_t=x]$, with $\tau$ the hitting time of the boundary by standard Brownian motion $B\_t$.


I am looking for a reference that provides a similar stochastic representation of heat equation on a domain with smooth boundary with **Neumann boundary condition**, using reflected Brownian motion. The equation I just described is the following: $$\partial\_t u = \frac 12\Delta u,\quad\frac{\partial u}{\partial\mathbf{n}}(t,x)=g(x),\quad\forall x\in\partial\Omega,t>0.$$ I suspect this should be in some standard textbooks, but I can't find it in some common stochastic calulus books….



Reference for representation of heat equation with Neumann boundary condition on smooth domain using reflected Brownian motion
References are given here for multiple boundaries:


[\*Full proof\* references for Markov generators with various boundary conditions](https://mathoverflow.net/questions/306978/full-proof-references-for-markov-generators-with-various-boundary-conditions/307054#307054)


Here too: [Continuity of green functions](https://mathoverflow.net/questions/333632/continuity-of-green-functions/333640#333640) and
[Martingales associated with heat equation](https://mathoverflow.net/questions/385030/martingales-associated-with-heat-equation)


For more references see the thesis ["Path integral methods using Feynman-Kac formula and reflecting. Brownian motions for Neumann and Robin Problems."](https://math.charlotte.edu/sites/math.charlotte.edu/files/fields/preprint_archive/paper/2016_09.pdf) , where they also reference the work by E.Hsu
“Reflecting Brownian motion, boundary local time and the Neumann problem”


