Fractal properties of the random string processes

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dc.creatorWu, Dongsheng
dc.creatorXiao, Yimin
dc.date2006-12-22
dc.date2006-12-27
dc.date.accessioned2026-07-07T07:37:15Z
dc.date.available2026-07-07T07:37:15Z
dc.descriptionLet $\{u_t(x),t\ge 0, x\in {\mathbb{R}}\}$ be a random string taking values in ${\mathbb{R}}^d$, specified by the following stochastic partial differential equation [Funaki (1983)]: \[\frac{\partial u_t(x)}{\partial t}=\frac{{\partial}^2u_t(x)}{\partial x^2}+\dot{W},\] where $\dot{W}(x,t)$ is an ${\mathbb{R}}^d$-valued space-time white noise. Mueller and Tribe (2002) have proved necessary and sufficient conditions for the ${\mathbb{R}}^d$-valued process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$ to hit points and to have double points. In this paper, we continue their research by determining the Hausdorff and packing dimensions of the level sets and the sets of double times of the random string process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$. We also consider the Hausdorff and packing dimensions of the range and graph of the string.
dc.descriptionPublished at http://dx.doi.org/10.1214/074921706000000806 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)
dc.identifierhttps://arxiv.org/abs/math/0612700
dc.identifierhttp://arxiv.org/abs/math/0612700
dc.identifierIMS Lecture Notes Monograph Series 2006, Vol. 51, 128-147
dc.identifierdoi:10.1214/074921706000000806
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/120710
dc.subjectProbability
dc.subject60H15, 60G15, 60G17 (Primary) 28A80 (Secondary)
dc.titleFractal properties of the random string processes
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