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Scaling laws and warning signs for bifurcations of SPDEs

Part of: Stochastic analysis Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  18 September 2018

CHRISTIAN KUEHN  and
FRANCESCO ROMANO
CHRISTIAN KUEHN
Affiliation:
Faculty of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85747 Garching b. Munich, Germany email: ckuehn@ma.tum.de
FRANCESCO ROMANO
Affiliation:
Faculty of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85747 Garching b. Munich, Germany email: ckuehn@ma.tum.de Ludwig-Maximilians-Universität, Elite Graduate Course Theoretical and Mathematical Physics, Theresienstraße 37, 80333 Munich, Germany email: francesco1093@gmail.com
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Abstract

Critical transitions (or tipping points) are drastic sudden changes observed in many dynamical systems. Large classes of critical transitions are associated with systems, which drift slowly towards a bifurcation point. In the context of stochastic ordinary differential equations, there are results on growth of variance and autocorrelation before a transition, which can be used as possible warning signs in applications. A similar theory has recently been developed in the simplest setting for stochastic partial differential equations (SPDEs) for self-adjoint operators in the drift term. This setting leads to real discrete spectrum and growth of the covariance operator via a certain scaling law. In this paper, we develop this theory substantially further. We cover the cases of complex eigenvalues, degenerate eigenvalues as well as continuous spectrum. This provides a fairly comprehensive theory for most practical applications of warning signs for SPDE bifurcations.

Keywords

Critical transitiontipping pointwarning signscaling lawstochastic partial differential equation

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 5: Stochastic Analysis in Applications , October 2019 , pp. 853 - 868
Copyright
© Cambridge University Press 2018 

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