Towards sample path estimates for fast–slow stochastic partial differential equations

Gnann, Manuel V.; Kuehn, Christian; Pein, Anne

doi:10.1017/s095679251800061x

Cited by 6 publications

(4 citation statements)

References 28 publications

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“…The covariance operator Q is assumed to satisfy properties 1 ′ , 2 ′ , 3 ′ and 4 ′ from the previous section. The slow dependence on time is in accordance to the fact that in simulations of realistic applications, the previous parameter α in (4) is not constant but slowly changing; see also [31]. For fixed ϵ we will assume α(ϵt) < λ1 in 0 ≤ ϵt ≤ τ .…”

Section: Error and Moment Estimatesmentioning

confidence: 74%

Bifurcations and Early-Warning Signs for SPDEs with Spatial Heterogeneity

Bernuzzi

Kuehn

2023

Preprint

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Bistability is a key property of many systems arising in the nonlinear sciences. For example, it appears in many partial differential equations (PDEs). For scalar bistable reaction-diffusions PDEs, the bistable case even has taken on different names within communities such as Allee, Allen-Cahn, Chafee-Infante, Nagumo, Ginzburg-Landau, Φ4, Schlögl, Stommel, just to name a few structurally similar bistable model names. One key mechanism, how bistability arises under parameter variation is a pitchfork bifurcation. In particular, taking the pitchfork bifurcation normal form for reaction-diffusion PDEs is yet another variant within the family of PDEs mentioned above. More generally, the study of this PDE class considering steady states and stability, related to bifurcations due to a parameter is well-understood for the deterministic case. For the stochastic PDE (SPDE) case, the situation is less well-understood and has been studied recently. In this paper we generalize and unify several recent results for SPDE bifurcations. Our generalisation is motivated directly by applications as we introduce in the equation a spatially heterogeneous term and relax the assumptions on the covariance operator that defines the noise. For this spatially heterogeneous SPDE, we prove a finite-time Lyapunov exponent bifurcation result. Furthermore, we extend the theory of early warning signs in our context and we explain, the role of universal exponents between covariance operator warning signs and the lack of finite-time Lyapunov uniformity. Our results are accompanied and cross-validated by numerical simulations.

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Section: Error and Moment Estimatesmentioning

confidence: 74%

Bifurcations and Early-Warning Signs for SPDEs with Spatial Heterogeneity

Bernuzzi

Kuehn

2023

Preprint

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“…The second term is due to noise around the traveling wave, where once more the decay constant of (2.24) enters. From a theoretical viewpoint the multiscale estimate for the second moment can then be useful in Doob/Markov-type inequalities to control the probabilities of individual sample paths [1,6,37].…”

Section: Correction Of the Wave Velocitymentioning

confidence: 99%

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations

2022

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We investigate the stability of traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Krüger and Stannat (Nonlinear Anal. 162 (2017) 197-223) for scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications. CONTENTS

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“…One particular building block -initially developed by Berglund and Gentz -uses a sample paths viewpoint [2]. This approach has recently been extended to broader classes of spatial stochastic fast-slow systems [13] and it has found many successful applications; see e.g. [1,24,38,41].…”

Section: Introductionmentioning

confidence: 99%

Sample Paths Estimates for Stochastic Fast-Slow Systems Driven by Fractional Brownian Motion

2020

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We analyze the effect of additive fractional noise with Hurst parameter H > 1 2 on fastslow systems. Our strategy is based on sample paths estimates, similar to the approach by Berglund and Gentz in the Brownian motion case. Yet, the setting of fractional Brownian motion does not allow us to use the martingale methods from fast-slow systems with Brownian motion. We thoroughly investigate the case where the deterministic system permits a uniformly hyperbolic stable slow manifold. In this setting, we provide a neighborhood, tailored to the fast-slow structure of the system, that contains the process with high probability. We prove this assertion by providing exponential error estimates on the probability that the system leaves this neighborhood. We also illustrate our results in an example arising in climate modeling, where time-correlated noise processes have become of greater relevance recently.

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Towards sample path estimates for fast–slow stochastic partial differential equations

Cited by 6 publications

References 28 publications

Bifurcations and Early-Warning Signs for SPDEs with Spatial Heterogeneity

Bifurcations and Early-Warning Signs for SPDEs with Spatial Heterogeneity

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations

Sample Paths Estimates for Stochastic Fast-Slow Systems Driven by Fractional Brownian Motion

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