Ergodic Property of the Brownian Motion Process

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...

Nonincrease Everywhere of the Brownian Motion Process

The (linear, separable) Brownian motion process has been studied more than any other stochastic process. It has many applications and, at least since Bachelier, probabilists have been at(tract,ed by its delicate and curious properties. It furnished, in the hands of N. Jyiener, the first instance of a satisfactorily defined nondiscrete stochastic process n-ith continuous time parameter, and it i...

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...

The Strong Markov Property of the Brownian Motion

The compact support property for the Λ - Fleming - Viot process with underlying Brownian motion ∗

Using the lookdown construction of Donnelly and Kurtz we prove that, at any fixed positive time, the Λ-Fleming-Viot process with underlying Brownian motion has a compact support provided that the corresponding Λ-coalescent comes down from infinity not too slowly. We also find both upper bound and lower bound on the Hausdorff dimension for the support.