Carl Mueller, Roger Tribe
Published 2001-12-31, updated 2002-01-03Version 3
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.
Journal: Electronic J. Probab., 7, paper 10, 1-29, (2002)
The Isochronal Phase of Stochastic PDE and Integral Equations: Metastability and Other Properties
Existence and Uniqueness of Stochastic PDEs associated with the Forward Equations: An Approach using Alternate Norms
On the rate of escape or approach to the origin of a random string
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