Hitting properties of a random string
概率论
2011-02-18 v3 偏微分方程分析
摘要
We consider Funaki's model of a random string taking values in R^d. It is specified by the following stochastic PDE, du = u_{xx} + W, where W=W(x,t) is two-parameter white noise, also taking values in R^d. We study hitting properties, double points, and recurrence. The main difficulty is that the process has the Markov property in time, but not in space. We find: (1) The string hits points if d<6. (2) For fixed t, there are points x,y such that u(t,x)=u(t,y) iff d < 4. (3) There exist points t,x,y such that u(t,x)=u(t,y) iff d < 8. (4) There exist points s,t,x,y such that u(t,x)=u(s,y) iff d < 12. (5) The string is recurrent iff d < 7.
引用
@article{arxiv.math/0112315,
title = {Hitting properties of a random string},
author = {Carl Mueller and Roger Tribe},
journal= {arXiv preprint arXiv:math/0112315},
year = {2011}
}