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?

## 4 Answers

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.

It is hard to say what is a "typical" diffusion process without knowing further details, so let us assume that it is at least governed by a transition kernel, and in particular its solution to be positive a.e. at any initial data if its initial data is positive a.e.: typically, diffusion equations on smooth domains with Dirichlet or Neumann boundary conditions satisfy these properties.

Then, it is known that the semigroup is irreducible (i.e., the motion is not eventually confined in any subdomain) if and only if a strong parabolic maximum principle holds, i.e., the solution is immediately strictly positive a.e. whenever the initial data is merely non-negative and not identically zero. For this reason, such a semigroup can not possibly be confined in any *non-trivial* subdomain for some time but eventually spread everywhere. Does this answer your question?

See p.83 in Engländer, János; Kyprianou, Andreas E. Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32 (2004), no. 1A, 78–99.

It is the h-transformed diffusion appearing there.

A related result can be found in `A Fleming-Viot particle representation of the Dirichlet Laplacian' by Krzysztof Burdzy, Robert Holyst and Peter March.

anydiffusion process. It won't work for an OU process, it won't work for a non isotropic diffusion process, etc. $\endgroup$