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

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• dc.contributor.author Dalang, Robert C.ca
• dc.contributor.author Khoshnevisan, Davarca
• dc.contributor.author Nualart, Eulàliaca
• dc.date.accessioned 2018-04-18T08:31:15Z
• dc.date.available 2018-04-18T08:31:15Z
• dc.date.issued 2009
• dc.description.abstract 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∈R+,x∈[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↦u(t,x) and x↦u(t,x) . Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes.en
• dc.description.sponsorship R.C. Dalang is supported in part by the Swiss National Foundation for Scientific Research. D. Khoshnevisan’s research is supported in part by a grant from the US National Science Foundation.
• dc.format.mimetype application/pdf
• dc.identifier.citation Dalang RD, Khoshnevisan D, Nualart E. Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise. Probab Theory Relat Fields. 2009;144(3-4):371–427. DOI: 10.1007/s00440-008-0150-1
• dc.identifier.doi http://dx.doi.org/10.1007/s00440-008-0150-1
• dc.identifier.issn 0178-8051
• dc.identifier.uri http://hdl.handle.net/10230/34390
• dc.language.iso eng
• dc.publisher Springerca
• dc.relation.ispartof Probability Theory and Related Fields. 2009;144(3-4):371–427.
• dc.rights © Springer The final publication is available at Springer via http://dx.doi.org/10.1007/s00440-008-0150-1
• dc.rights.accessRights info:eu-repo/semantics/openAccess
• dc.subject.keyword Hitting probabilitiesen
• dc.subject.keyword Stochastic heat equationen
• dc.subject.keyword Space-time white noiseen
• dc.subject.keyword Malliavin calculusen
• dc.title Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noiseca
• dc.type info:eu-repo/semantics/article
• dc.type.version info:eu-repo/semantics/acceptedVersion