## Brownian motion inside a domain

Consider a regular domain $D \subset \mathbb{R}^d$ and a Brownian motion $B_t$ conditioned to stay inside $D$ for time $t \in [0,T]$. In the limit $T \to \infty$ the conditioned Brownian motion behaves like an homogenous diffusion in $D$ whose invariant distribution is the first eigenfunction of the Laplacian in $D$ (which gives another proof that the first eigenfunction is positive).This should also work for more general diffusion processes. Where can I find a reference for this (and similar) results?

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 Stating the obvious here, but it's not going to work for just any diffusion process. It won't work for an OU process, it won't work for a non isotropic diffusion process, etc. – Arthur B Oct 11 at 16:06