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Jahresbericht der Deutschen Mathematiker-Vereinigung
Erschienen in: Jahresbericht der Deutschen Mathematiker-Vereinigung 2/2020

29.08.2019 | Survey Article

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

verfasst von: Christian Kuehn

Erschienen in: Jahresbericht der Deutschen Mathematiker-Vereinigung | Ausgabe 2/2020

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Abstract

In this review, we provide a concise summary of several important mathematical results for stochastic travelling waves generated by monostable and bistable reaction-diffusion stochastic partial differential equations (SPDEs). In particular, this survey is intended for readers new to the topic but who have some knowledge in any sub-field of differential equations. The aim is to bridge different backgrounds and to identify the most important common principles and techniques currently applied to the analysis of stochastic travelling wave problems. Monostable and bistable reaction terms are found in prototypical dissipative travelling wave problems, which have already guided the deterministic theory. Hence, we expect that these terms are also crucial in the stochastic setting to understand effects and to develop techniques. The survey also provides an outlook, suggests some open problems, and points out connections to results in physics as well as to other active research directions in SPDEs.
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Fußnoten
1
To quote an opinion from 1991 by Glimm [67]: “The major problems for partial differential equations are either nonlinear or stochastic or both.” Pattern formation for SPDEs evidently involves the study of nonlinear stochastic PDEs.
 
2
There are several slightly different definitions of the essential spectrum of an operator, so one should check carefully, which definition each author uses.
 
3
I would like to thank Eulalia Nualart for pointing out the alternative attribution to Furutsu [63] to me, which I had not been previously aware of.
 
4
The terminology of ‘theorem’ is potentially not ideal. Although the formal relation seems evident from a calculation, most justifications in the literature of the Furutsu-Novikov Theorem seem to be extremely concise in terms of their explanation and formalization.
 
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Metadaten
Titel
Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations
verfasst von
Christian Kuehn
Publikationsdatum
29.08.2019
Verlag
Springer Berlin Heidelberg
Erschienen in
Jahresbericht der Deutschen Mathematiker-Vereinigung / Ausgabe 2/2020
Print ISSN: 0012-0456
Elektronische ISSN: 1869-7135
DOI
https://doi.org/10.1365/s13291-019-00206-9