Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise

• 成果类型: Article
• 署名作者: Dalang, Robert C.; Khoshnevisan, Davar; Nualart, Eulalia
• 署名单位: Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Utah System of Higher Education; University of Utah; Universite Paris 13
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-008-0150-1
• 发表日期: 2009
• 页码: 371-427

摘要

We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish upper and lower bounds on the one-point density of the solution u(t, x), and upper bounds of Gaussian-type on the two-point density of (u(s, y), u(t, x)). In particular, this estimate quantifies how this density degenerates as (s, y) -> (t, x). From these results, we deduce upper and lower bounds on hitting probabilities of the process {u(t, x)}(t is an element of R+,x is an element of[0,1]), in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d >= 7 and are not polar when d <= 5. We also show that the Hausdorff dimension of the range of the process is 6 when d > 6, and give analogous results for the processes t bar right arrow u(t, x) and x bar right arrow u(t, x). Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes.