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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A reference for this might be in the paper of Banuelos "Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators" . Between equations (1.3) and (1.4) he discussed the semigroup of Brownian motion conditioned to remain forever in D. |
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