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

Dalang, Robert ; Khoshnevisan, Davar ; Nualart, Eulalia

In: Probability Theory and Related Fields, 2009, vol. 144, no. 3-4, p. 371-427

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    Title
    • Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise
    Author
    • Dalang, Robert. Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Station 8, 1015, Lausanne, Switzerland
    • Khoshnevisan, Davar. Department of Mathematics, The University of Utah, 155 S.1400 E, 84112-0090, Salt Lake City, UT, USA
    • Nualart, Eulalia. Institut Galilée, Université Paris 13, 93430, Villetaneuse, France
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Probability Theory and Related Fields, 2009, vol. 144, no. 3-4, p. 371-427. Springer-Verlag
    Other Version
    OAI-PMH Identifier
    • oai:doc.rero.ch:319066
    Summary
    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 $${\{u(t,x)\}_{t \in \mathbb{R}_+, x\in [0,1]}}$$ , 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 $${t \mapsto u(t,x)}$$ and $${x \mapsto u(t,x)}$$ . Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes