Random Perturbations of Dynamical Systems

Freidlin, Mark; Wentzell, Alexander D.

doi:10.1007/978-1-4684-0176-9

Cited by 2,024 publications

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“…• In the case of diffusions with constant noise power σ(x) = σ, the low-noise limit σ → 0 yields for the driven process a deterministic differential equation for the (unique) fluctuation path characterizing the conditioning A T = a. This can be used to recover known results from the Freidlin-Wentzell theory of fluctuation paths and instantons for noise-perturbed SDEs [51]. • The conditions leading to the equivalence of X t |A T = a and Y t prevent many processes from being treated within our theory.…”

Section: Canonical-microcanonical Observable Equivalencementioning

confidence: 94%

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Nonequilibrium Markov Processes Conditioned on Large Deviations

Chétrite

Touchette

2014

Ann. Henri Poincaré

327

762

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International audienceWe consider the problem of conditioning a Markov process on a rare event and of representing this conditioned process by a conditioning-free process, called the effective or driven process. The basic assumption is that the rare event used in the conditioning is a large-deviation-type event, characterized by a convex rate function. Under this assumption, we construct the driven process via a generalization of Doob's h-transform, used in the context of bridge processes, and show that this process is equivalent to the conditioned process in the long-time limit. The notion of equivalence that we consider is based on the logarithmic equivalence of path measures, and implies that the two processes have the same typical states. In constructing the driven process, we also prove equivalence with the so-called exponential tilting of the Markov process, often used with importance sampling to simulate rare events and giving rise, from the point of view of statistical mechanics, to a nonequilibrium version of the canonical ensemble. Other links between our results and the topics of bridge processes, quasi-stationary distributions, stochastic control, and conditional limit theorems are mentioned

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Section: Canonical-microcanonical Observable Equivalencementioning

confidence: 94%

“…These paths are described mathematically by the Freidlin-Wentzell theory of large deviations [51] and are fundamental for characterizing many noise-activated (escape-type) processes arising in chemical reactions, biological processes, magnetic systems, and glassy systems [52][53][54].…”

Section: Fluctuation Paths and Fluctuation Dynamicsmentioning

confidence: 99%

Nonequilibrium Markov Processes Conditioned on Large Deviations

Chétrite

Touchette

2014

Ann. Henri Poincaré

327

762

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“…Some local results about these sequences are known on bounded domain of R n (see Friedmann [3] [4] [5], Friedlin Ventcell' [2]) : when b has only one attracting point, u ǫ converges uniformly on every compact set to a constant as ǫ converges to 0, but when the point is repulsive the sequence converges in the distribution sense to a Dirac distribution centered at this point.…”

Section: Introductionmentioning

confidence: 99%

Singular perturbations and first order PDE on manifolds

Holcman

Kupka

2001

Comptes Rendus de l'Académie des Sciences - Series I - Mathemat

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In this note we present some results concerning the concentration of sequences of first eigenfunctions on the limit sets of a Morse-Smale dynamical system on a compact Riemanniann manifold. More precisely a renormalized sequence of eigenfunctions converges to a measure µ concentrated on the hyperbolic sets of the field. The set of all possible measure turns out to be a sum of a finite Dirac distributions localized at the critical point of the field and absolutely continuous measure with respect to the Lebesgue measure on each limit cycles : the coefficients which appear in the limit measure can be characterized using the concentration theory.In the second part, certain aspects of some first order PDE on manifolds are studied. We study the limit of a sequence solutions of a second order PDE, when a parameter of viscosity tends to zero. Under some explicit assumptions on some vector fields, bounded and differentiable solutions are obtained. We exibit the role played by the limit sets of the dynamical systems and provide in some cases an explicit representation formula.

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“…(i) The left hand side of (8.2) is called a quasi potential. It is known that, if the classical dynamics which is the minimizer of I is a gradient flow for a certain potential F and if all stable equilibrium points are global minimal points of F, the quasi potential actually coincides with the potential F itself, see Theorem 3.1, p. 118 of [5] in a finite dimensional setting.…”

Section: Discussionmentioning

confidence: 99%

Large deviations for the Ginzburg–Landau ∇φ interface model

Funaki

Nishikawa

2001

Probab Theory Relat Fields

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Hydrodynamic large scale limit for the Ginzburg-Landau ∇φ interface model was established in [6]. As its next stage this paper studies the corresponding large deviation problem. The dynamic rate functional is given byis the surface tension for mean tilt u ∈ R d . Our main tool is H −1 -method exploited by Landim and Yau [9]. The relationship to the rate functional obtained under the static situation by Deuschel et al.[3] is also discussed.

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Random Perturbations of Dynamical Systems

Cited by 2,024 publications

References 0 publications

Nonequilibrium Markov Processes Conditioned on Large Deviations

Nonequilibrium Markov Processes Conditioned on Large Deviations

Singular perturbations and first order PDE on manifolds

Large deviations for the Ginzburg–Landau ∇φ interface model

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