Front Propagation in Stochastic Neural Fields: A Rigorous Mathematical Framework

Krüger, Jennifer; Stannat, Wilhelm

doi:10.1137/13095094x

Cited by 28 publications

(34 citation statements)

References 19 publications

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“…Furthermore, travelling waves have been studied, particularly in the bistable case, in quite some detail for stochastic neural fields, see e.g. [21,22,83,91,99,112]. An important topic directly related to the bistable setting is the case a = 1/2 for f = f 3 , so that the PDE has a standing wave.…”

Section: Discussionmentioning

confidence: 99%

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Kuehn

2019

Jahresber. Dtsch. Math. Ver.

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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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Section: Discussionmentioning

confidence: 99%

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Kuehn

2019

Jahresber. Dtsch. Math. Ver.

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“…In many examples, using trace class noise implies L 2 -valued Wiener processes and thus a decay condition both of solutions and of the noise at infinity. This leads to L 2 -valued solutions, as for example by Brzezniak and Li [7] or by Krüger and Stannat [21], where an integral equation is considered. In the next paragraph we will comment on the fact that a decay at infinity rules out the effect we want to study here using modulation equations.…”

Section: Spdes In Weighted Spaces On Unbounded Domainsmentioning

confidence: 96%

Modulation Equation and SPDEs on Unbounded Domains

Bianchi

Blömker

Schneider

2019

Commun. Math. Phys.

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We consider the approximation via modulation equations for nonlinear SPDEs on unbounded domains with additive space time white noise. Close to a bifurcation an infinite band of eigenvalues changes stability, and we study the impact of small space-time white noise on this bifurcation.As a first example we study the stochastic Swift-Hohenberg equation on the whole real line. Here due to the weak regularity of solutions the standard methods for modulation equations fail, and we need to develop new tools to treat the approximation.As an additional result we sketch the proof for local existence and uniqueness of solutions for the stochastic Swift-Hohenberg and the complex Ginzburg Landau equations on the whole real line in weighted spaces that allow for unboundedness at infinity of solutions, which is natural for translation invariant noise like space-time white noise. Moreover we use energy estimates to show that solutions of the Ginzburg-Landau equation are Hölder continuous and have moments in those functions spaces. This gives just enough regularity to proceed with the error estimates of the approximation result.

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“…In [23], the idea of minimizing α → u t − ϕ α 2 is used as we do to keep track of the position of the stochastic front. However, rather than describing the dynamics of the minima of α → u t − ϕ α 2 explicitly, a gradient-descent adaptation procedure is proposed, whereby (β t ) t≥0 in (5.1) is defined via an ODE to converge dynamically towards the nearest local minimum with a certain speed.…”

Section: Now By Proposition 42 We See That Formallymentioning

confidence: 99%

“…This goes further than the work of [4] and [23], since our description is exact rather than a first order ε-expansion or an approximation. We can also see that the solution of the SDE exists exactly up until the point at which the local minimum may become a saddle point.…”

Section: Introductionmentioning

confidence: 98%

A General Framework for Stochastic Traveling Waves and Patterns, with Application to Neural Field Equations

Inglis¹,

MacLaurin²

2016

SIAM J. Appl. Dyn. Syst.

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In this paper we present a general framework in which to rigorously study the effect of spatio-temporal noise on traveling waves and stationary patterns. In particular the framework can incorporate versions of the stochastic neural field equation that may exhibit traveling fronts, pulses or stationary patterns. To do this, we first formulate a local SDE that describes the position of the stochastic wave up until a discontinuity time, at which point the position of the wave may jump. We then study the local stability of this stochastic front, obtaining a result that recovers a well-known deterministic result in the smallnoise limit. We finish with a study of the long-time behavior of the stochastic wave.

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Front Propagation in Stochastic Neural Fields: A Rigorous Mathematical Framework

Cited by 28 publications

References 19 publications

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Modulation Equation and SPDEs on Unbounded Domains

A General Framework for Stochastic Traveling Waves and Patterns, with Application to Neural Field Equations

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