Exit from a basin of attraction for stochastic weakly damped nonlinear Schrödinger equations

Gautier, Éric

doi:10.1214/07-aop344

Cited by 19 publications

(8 citation statements)

References 23 publications

(59 reference statements)

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“…Because the set {x} ∪ n {x n } is compact, Theorem 5. 19 guarantees that there exists a subsequence such that y n → y. By the definitions of G (ρ) j , we know that y ∈ K j .…”

Section: Ldp For Exit Shape Under Extra Assumptionsmentioning

confidence: 99%

Metastability and exit problems for systems of stochastic reaction-diffusion equations

Salins¹,

Spiliopoulos²

2019

Preprint

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In this paper we develop a metastability theory for a class of stochastic reaction-diffusion equations exposed to small multiplicative noise. We consider the case where the unperturbed reaction-diffusion equation features multiple asymptotically stable equilibria. When the system is exposed to small stochastic perturbations, it is likely to stay near one equilibrium for a long period of time, but will eventually transition to the neighborhood of another equilibrium. We are interested in studying the exit time from the full domain of attraction (in a function space) surrounding an equilibrium and therefore do not assume that the domain of attraction features uniform attraction to the equilibrium. This means that the boundary of the domain of attraction is allowed to contain saddles and limit cycles. Our method of proof is purely infinite dimensional, i.e., we do not go through finite dimensional approximations. In addition, we address the multiplicative noise case and we do not impose gradient type of assumptions on the nonlinearity. We prove large deviations logarithmic asymptotics for the exit time and for the exit shape, also characterizing the most probable set of shapes of solutions at the time of exit from the domain of attraction.

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“…Because the set {x} ∪ n {x n } is compact, Theorem 5. 19 guarantees that there exists a subsequence such that y n → y. By the definitions of G (ρ) j , we know that y ∈ K j .…”

Section: Ldp For Exit Shape Under Extra Assumptionsmentioning

confidence: 99%

Metastability and exit problems for systems of stochastic reaction-diffusion equations

Salins¹,

Spiliopoulos²

2019

Preprint

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“…As random disturbances may affect the qualitative behaviors drastically and result in new properties for this quantum model, stochastic Schrödinger equations have attracted attentions recently (e.g., [26,27,28,29,30]).…”

Section: Introductionmentioning

confidence: 99%

Effective Approximation for a Nonlocal Stochastic Schrödinger Equation with Oscillating Potential

Li¹,

Yang²,

Duan³

2019

Preprint

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We consider the homogenization of a nonlocal stochastic Schrödinger equation with a rapidly oscillating, periodically time-dependent potential. With help of a two-scale convergence technique, we establish a homogenization principle for this nonlocal stochastic partial differential equation. We explicitly derive the homogenized model. In particular, this homogenization principle holds when the nonlocal operator is the fractional Laplacian.

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“…By using large deviations methods, [23] gets an estimate for {µ ε } in any sufficiently small neighborhood of equivalent set. There are also some relative works in SPDEs such as [7] [19] [21] [20] [12] [9] [11] [6].…”

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confidence: 99%

The concentration of zero-noise limits of invariant measures for stochastic dynamical systems

Zhang¹,

Gu²,

Li³

2022

Preprint

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In this paper, we study the concentration phenomena of invariant measures for stochastic differential equations with locally Lipschitz continuous coefficients and more than one ergodic state in R d . Under some dissipative conditions, by using Lyapunov-like functions and large deviations methods, we estimate the invariant measures in neighborhoods of stable sets, neighborhoods of unstable sets and their complement, respectively. Our result illustrates that the invariant measures concertrate on the intersection of stable sets where a functional W (K i ) are minimized and the Birkhoff center of the corresponding deterministic systems as noise tends down to zero. Furthermore, we show the large deviations principle of the invariant measures. At the end of this paper, we provide some explicit examples and their numerical simulations.

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Exit from a basin of attraction for stochastic weakly damped nonlinear Schrödinger equations

Cited by 19 publications

References 23 publications

Metastability and exit problems for systems of stochastic reaction-diffusion equations

Metastability and exit problems for systems of stochastic reaction-diffusion equations

Effective Approximation for a Nonlocal Stochastic Schrödinger Equation with Oscillating Potential

The concentration of zero-noise limits of invariant measures for stochastic dynamical systems

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