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

By Robert C. Dalang, Davar Khoshnevisan, E. Nualart. 2007

72 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a5d074547129b4c5e1e65d6d4ab8c8dc26afd1b9

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Citations in the corpus

Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields (2009, background): “It includes not only the well-known fractional Brownian motion and the Brownian sheet, but also such anisotropic random fields as fractional Brownian sheets ([1]), solutions to stochastic heat equation driven by space-time white noise ([5], [ 6 ], [16]) and many more.”
Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions (2013, background): “…Statistics, 2015 MOMENTS AND GROWTH INDICES FOR THE NONLINEAR STOCHASTIC HEAT EQUATION WITH ROUGH INITIAL CONDITIONS1 BY LE CHEN AND ROBERT C. DALANG École Polytechnique Fédérale de Lausanne We study the nonlinear stochastic heat equation in the spatial domain R, driven by space–time white noise.”