On Euclidean space, (d-dimensional) Brownian motion is intimately related to the heat kernel, where if $f$ is sufficiently nice initial data on $\mathbb{R}^d$ and $W_x(t)$ is a Brownian motion started at position $x$, then the solution to the heat equation with initial data $f$ is given at time $t$ by $E[f(W_x(t))]$, with the expectation ranging over all Brownian motions starting at $x$ and running for $t$ units of time. While reading a paper, I saw that this relation holds as well for bounded domains in Euclidean space, but was not familiar with this result, or even with how Brownian motion was defined on proper subdomains of Euclidean space. Could anyone provide me with a reference and/or explanation for how Brownian motion is defined on these bounded subdomains (or better yet, on manifolds with or without boundary), and how it is related to heat flow? Thanks.
2026-04-29 14:24:50.1777472690
Brownian motion and the heat equation on domains with boundary and/or manifolds
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