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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….

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  • $\begingroup$ It would be helpful if you spelled out the boundary-value problem you are interested in in full detail. In your claim about Dirichlet case, $\mathbb{E}_xg(B_\tau)$ does not depend on $t$ at all, and it is in fact the solution to the Laplace equation, not heat equation. The solution to heat equation should also depend on the initial data. It's hard to guess what do you mean by this example. $\endgroup$
    – Kostya_I
    Commented Dec 1, 2022 at 13:49
  • $\begingroup$ @Kostya_I: Thanks for your advice! Does it look clear to you right now? $\endgroup$
    – MikeG
    Commented Dec 1, 2022 at 18:52
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    $\begingroup$ Answer would involve $g(B_t)$ integrated against the accumulated intersection (local) time with the boundary up to time $t$. Take expectation of that, then you add $\Bbb E[u(0,B_t)]$ to get the final answer. Not sure where to find a reference though. $\endgroup$
    – shalop
    Commented Dec 4, 2022 at 1:44
  • $\begingroup$ @Kostya_I I guess if $g$ is a continuous function on the boundary $([0,t]\times \partial\Omega )\cup (\{0\}\times \Omega)$ then the function $u(t,x) = \Bbb E_x [g(t-\tau,B_\tau)]$ would likely solve the heat equation with boundary data $g$ where $\tau = t\wedge \inf\{s\ge 0: B_s \in \partial\Omega\}$. I guess this was what was meant. $\endgroup$
    – shalop
    Commented Dec 4, 2022 at 5:44

1 Answer 1

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References are given here for multiple boundaries:

*Full proof* references for Markov generators with various boundary conditions

Here too: Continuity of green functions and 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." , where they also reference the work by E.Hsu “Reflecting Brownian motion, boundary local time and the Neumann problem”

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