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2020 Optimal lower bounds on hitting probabilities for stochastic heat equations in spatial dimension $k \geq 1$
Robert C. Dalang, Fei Pu
Electron. J. Probab. 25: 1-31 (2020). DOI: 10.1214/20-EJP438

Abstract

We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix $\gamma _{Z}$ of $Z := (u(s, y), u(t, x) - u(s, y))$, where $u$ is the solution to a system of $d$ non-linear stochastic heat equations in spatial dimension $k \geq 1$. We also obtain the optimal exponents for the $L^{p}$-modulus of continuity of the increments of the solution and of its Malliavin derivatives. These lead to optimal lower bounds on hitting probabilities of the process $\{u(t, x): (t, x) \in [0, \infty [ \times \mathbb{R} ^{k}\}$ in the non-Gaussian case in terms of Newtonian capacity, and improve a result in Dalang, Khoshnevisan and Nualart [Stoch PDE: Anal Comp 1 (2013) 94–151].

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Robert C. Dalang. Fei Pu. "Optimal lower bounds on hitting probabilities for stochastic heat equations in spatial dimension $k \geq 1$." Electron. J. Probab. 25 1 - 31, 2020. https://doi.org/10.1214/20-EJP438

Information

Received: 30 November 2018; Accepted: 28 February 2020; Published: 2020
First available in Project Euclid: 31 March 2020

zbMATH: 1441.60044
MathSciNet: MR4089790
Digital Object Identifier: 10.1214/20-EJP438

Subjects:
Primary: 60G60, 60H07, 60H15, 60J45

JOURNAL ARTICLE
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