Open Access
2020 Local bounds for stochastic reaction diffusion equations
Augustin Moinat, Hendrik Weber
Electron. J. Probab. 25: 1-26 (2020). DOI: 10.1214/19-EJP397


We prove a priori bounds for solutions of stochastic reaction diffusion equations with super-linear damping in the reaction term. These bounds provide a control on the supremum of solutions on any compact space-time set which only depends on the specific realisation of the noise on a slightly larger set and which holds uniformly over all possible space-time boundary values. This constitutes a space-time version of the so-called “coming down from infinity” property. Bounds of this type are very useful to control the large scale behaviour of solutions effectively and can be used, for example, to construct solutions on the full space even if the driving noise term has no decay at infinity. Our method shows the interplay of the large scale behaviour, dictated by the non-linearity, and the small scale oscillations, dictated by the rough driving noise. As a by-product we show that there is a close relation between the regularity of the driving noise term and the integrability of solutions.


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Augustin Moinat. Hendrik Weber. "Local bounds for stochastic reaction diffusion equations." Electron. J. Probab. 25 1 - 26, 2020.


Received: 17 December 2018; Accepted: 15 November 2019; Published: 2020
First available in Project Euclid: 5 February 2020

zbMATH: 1445.60047
MathSciNet: MR4073678
Digital Object Identifier: 10.1214/19-EJP397

Primary: 35B45 , 35K57 , 60H15

Keywords: a priori bounds , maximum principle , non-linear stochastic PDE , reaction-diffusion equation

Vol.25 • 2020
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