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September 2021 Small ball probabilities and a support theorem for the stochastic heat equation
Siva Athreya, Mathew Joseph, Carl Mueller
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Ann. Probab. 49(5): 2548-2572 (September 2021). DOI: 10.1214/21-AOP1515

Abstract

We consider the following stochastic partial differential equation on t0,x[0,J],J1, where we consider [0,J] to be the circle with end points identified,

tu(t,x)=1 2x2u(t,x)+g(t,x,u)+σ(t,x,u)W˙(t,x),

W˙(t,x) is 2-parameter d-dimensional vector valued white noise and σ is function from R+×R×Rd to space of symmetric d×d matrices which is Lipschitz in u. We assume that σ is uniformly elliptic and that g is uniformly bounded. Assuming that u(0,x)0, we prove small ball probabilities for the solution u. We also prove a support theorem for solutions, when u(0,x) is not necessarily zero.

Funding Statement

The first author was supported in part by MATRICS and CPDA grants. The second author was supported in part by a CPDA grant. The third author was supported in part by a Simons grant.

Acknowledgments

A significant part of this work was done during a Research in Pairs visit by the authors to Centre International De Recontres Mathématiques (CIRM) in July 2019. We thank the centre for the wonderful environment and hospitality.

Citation

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Siva Athreya. Mathew Joseph. Carl Mueller. "Small ball probabilities and a support theorem for the stochastic heat equation." Ann. Probab. 49 (5) 2548 - 2572, September 2021. https://doi.org/10.1214/21-AOP1515

Information

Received: 1 June 2020; Revised: 1 February 2021; Published: September 2021
First available in Project Euclid: 24 September 2021

Digital Object Identifier: 10.1214/21-AOP1515

Subjects:
Primary: 60H15
Secondary: 60G17 , 60G60

Keywords: heat equation , small ball , Stochastic partial differential equations , Support , White noise

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 5 • September 2021
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