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May 2020 The almost-sure asymptotic behavior of the solution to the stochastic heat equation with Lévy noise
Carsten Chong, Péter Kevei
Ann. Probab. 48(3): 1466-1494 (May 2020). DOI: 10.1214/19-AOP1401


We examine the almost-sure asymptotics of the solution to the stochastic heat equation driven by a Lévy space-time white noise. When a spatial point is fixed and time tends to infinity, we show that the solution develops unusually high peaks over short time intervals, even in the case of additive noise, which leads to a breakdown of an intuitively expected strong law of large numbers. More precisely, if we normalize the solution by an increasing nonnegative function, we either obtain convergence to $0$, or the limit superior and/or inferior will be infinite. A detailed analysis of the jumps further reveals that the strong law of large numbers can be recovered on discrete sequences of time points increasing to infinity. This leads to a necessary and sufficient condition that depends on the Lévy measure of the noise and the growth and concentration properties of the sequence at the same time. Finally, we show that our results generalize to the stochastic heat equation with a multiplicative nonlinearity that is bounded away from zero and infinity.


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Carsten Chong. Péter Kevei. "The almost-sure asymptotic behavior of the solution to the stochastic heat equation with Lévy noise." Ann. Probab. 48 (3) 1466 - 1494, May 2020.


Received: 1 November 2018; Revised: 1 June 2019; Published: May 2020
First available in Project Euclid: 17 June 2020

zbMATH: 07226367
MathSciNet: MR4112721
Digital Object Identifier: 10.1214/19-AOP1401

Primary: 35B40 , 60F15 , 60G17 , 60G55 , 60H15

Keywords: Additive intermittency , almost-sure asymptotics , Integral test , Lévy noise , Poisson noise , Stochastic heat equation , Stochastic pde , Strong law of large numbers

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 3 • May 2020
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