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Optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations

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  • Dalang, Robert C.
  • Pu, Fei

Abstract

We consider a system of d non-linear stochastic fractional heat equations in spatial dimension 1 driven by multiplicative d-dimensional space–time white noise. We establish a sharp Gaussian-type upper bound on the two-point probability density function of (u(s,y),u(t,x)). From this result, we deduce optimal lower bounds on hitting probabilities of the process {u(t,x):(t,x)∈[0,∞[×R} in the non-Gaussian case, in terms of Newtonian capacity, which is as sharp as that in the Gaussian case. This also improves the result in Dalang et al. (2009) for systems of classical stochastic heat equations. We also establish upper bounds on hitting probabilities of the solution in terms of Hausdorff measure.

Suggested Citation

  • Dalang, Robert C. & Pu, Fei, 2021. "Optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 359-393.
    • Handle: RePEc:eee:spapps:v:131:y:2021:i:c:p:359-393
    DOI: 10.1016/j.spa.2020.07.015
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    References listed on IDEAS

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    1. Sanz-Solé, Marta & Viles, Noèlia, 2018. "Systems of stochastic Poisson equations: Hitting probabilities," Stochastic Processes and their Applications, Elsevier, vol. 128(6), pages 1857-1888.
    2. Debbi, Latifa & Dozzi, Marco, 2005. "On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1764-1781, November.
    3. Wu, Dongsheng, 2011. "On the solution process for a stochastic fractional partial differential equation driven by space-time white noise," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1161-1172, August.
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