Hitting Properties of a Random String

By C. Mueller, R. Tribe. 2001

82 citations recorded by the source corpus.

We consider Funaki's model of a random string taking values in $\mathbf{R}^d$. It is specified by the following stochastic PDE, \[ \frac{\partial u(x)}{\partial t}=\frac{\partial^2 u(x)}{\partial x^2} +\dot{W}. \] where $\dot{W}=\dot{W}(x,t)$ is two-parameter white noise, also taking values in $\mathbf{R}^d$. We find the dimensions in which the string hits points, and in which it has double points of various types. We also study the question of recurrence and transience.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ae13d51056efb8d553f64f4d27f2439b53d1b01f

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In Mathematical Physics

Citations in the corpus

Hitting probabilities for systems of non-linear stochastic heat equations with additive noise (2007, background): “If d = 6, we observe that Mueller and Tribe Mueller and Tribe (2002, Corollary 4) show that the law of their stationary pinned string Mueller and Tribe (2002, (2.1)) is mutually equivalent, on compact subsets of ]0 , T [×]0 , 1[, to the law of the solution of (1.1) (see Mueller and Tribe (2002, Coro”
Hitting properties of parabolic s.p.d.e.’s with reflection (2004, background): “By Lemmas 1 and 2 of [10], the laws of Z and U over [0, T ] × R are mutually absolutely continuous.”
Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise (2007, methodology): “Hitting probabilities for systems of linear heat equations have been obtained in Mueller and Tribe [ MT03 ].”