Ergodic Properties of Multidimensional Brownian Motion with Rebirth

Ergodic Properties of Multidimensional Brownian Motion with Rebirth

In a bounded open region of the d dimensional space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. The evolution is invariant with respect to a density equal, modulo a constant, to the Green function of the Dirichlet Laplacian centered at the point of return. We calculate the resolvent in closed form, study its spectral properties and...

Path Collapse for Multidimensional Brownian Motion with Rebirth†

In a bounded open region of the d dimensional Euclidean space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. It was shown that in dimension one coupled paths starting at different points but driven by the same Brownian motion either collapse with probability one or never meet. In higher dimensions, for convex or polyhedral regions the...

Ergodic Property of the Brownian Motion Process.

This work was carried out while the author was a National Science Foundation predoctoral fellow. The author expresses his appreciation to D. C. Spencer for suggesting this problem and for his help and advice in the preparation of this paper. 1 M. P. Gaffney, "The Heat Equation Method of Milgram and Rosenbloom for Open Riemannian Manifolds," Ann. Math. (to appear). 2 D. C. Spencer, "The Heat Equ...

Penalisations of multidimensional Brownian motion, VI

Analytic Properties of Brownian Motion

This paper largely deals with some analytic properties of Brownian motion and those of its discrete counterpart, random walks. I prove that the random walker returns to the origin infinitely with probability one if and only if his coin is not biased. I then then show how to construct a continuous Brownian motion over the reals. While demonstrating this existence is in itself non-trivial, I show...