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 \({\{u(t,x)\}_{t \in \mathbb{R}_+, x\in [0,1]}}\), 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.
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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.
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Dalang, R.C., Khoshnevisan, D. & Nualart, E. Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise. Probab. Theory Relat. Fields 144, 371–427 (2009). https://doi.org/10.1007/s00440-008-0150-1
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DOI: https://doi.org/10.1007/s00440-008-0150-1
Keywords
- Hitting probabilities
- Stochastic heat equation
- Space-time white noise
- Malliavin calculus
Mathematics Subject Classification (2000)
- Primary: 60H15
- 60J45
- Secondary: 60H07
- 60G60