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Scaling laws and warning signs for bifurcations of SPDEs

Published online by Cambridge University Press:  18 September 2018

Faculty of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85747 Garching b. Munich, Germany email:
Faculty of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85747 Garching b. Munich, Germany email: Ludwig-Maximilians-Universität, Elite Graduate Course Theoretical and Mathematical Physics, Theresienstraße 37, 80333 Munich, Germany email:


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.

© Cambridge University Press 2018 

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