Semiclassical Analysis for Diffusions and Stochastic Processes

Kolokoltsov, Vassili N.

doi:10.1007/bfb0112488

Cited by 100 publications

(97 citation statements)

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“…The methods of those works as well as the one of Eckmann and Hairer [6] are inspired by those of hypoellipicity for Hörmander type operators. We will not repeat here all the motivations of [15], coming also from the works [4,17] and others, including recent developments in pseudospectral theory.…”

Section: Introductionmentioning

confidence: 99%

Tunnel Effect for Kramers–Fokker–Planck Type Operators

Hérau

Hitrik

Sjöstrand

2008

Ann. Henri Poincaré

133

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We consider operators of Kramers-Fokker-Planck type in the semi-classical limit such that the exponent of the associated Maxwellian is a Morse function with two local minima and a saddle point. Under suitable additional assumptions we establish the complete asymptotics of the exponentially small splitting between the first two eigenvalues. RésuméOn considère des opérateurs du type de Kramers-Fokker-Planck dans la limite semi-classique tels que l'exposant du maxwellien associé soit une fonction de Morse avec deux minima et un point selle. Sous des hypothèses supplémentaires convenables onétablit un développement asymptotique complet de l'écart exponentiellement petit entre les deux premières valeurs propres.1

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Section: Introductionmentioning

confidence: 99%

Tunnel Effect for Kramers–Fokker–Planck Type Operators

Hérau

Hitrik

Sjöstrand

2008

Ann. Henri Poincaré

133

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“…Similarly one may obtain a formal "dual" Feynman-Kac formula (in the sense of this Fourier transform representation) under the complex measure condition on coefficients given by Itô [29]; see Kolokoltsov [31], and Chen et al [12]. In particular this approach makes Itô's complex measure condition completely natural from a probabilistic point of view.…”

Section: Multiplicative Stochastic Cascadementioning

confidence: 97%

“…The simplest way to check the commutativity required for the second equation in the display (39) is by Fourier transforms along the lines indicated earlier for (31).…”

Section: Vorticity Pressure Incompressibility and Background Radiationmentioning

confidence: 99%

Some Recent Developments

Albeverio¹,

Høegh-Krohn²,

Mazzucchi³

2008

Lecture Notes in Mathematics

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This is largely an attempt to provide probabilists some orientation to an important class of non-linear partial differential equations in applied mathematics, the incompressible Navier-Stokes equations. Particular focus is given to the probabilistic framework introduced by LeJan and Sznitman [Probab. Theory Related Fields 109 (1997) , vol. 140, 2004, in press]. In particular this is an effort to provide some foundational facts about these equations and an overview of some recent results with an indication of some new directions for probabilistic consideration.

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“…[48] for its application to nonlinear Schrodinger equation and [55] for stochastic Ito's equations describing the evolution (5) with differential generators) a general well posedness result (Theorem 3.1) allowing to reduce the study of nonlinear problems (5)- (7) to some regularity properties of the corresponding linear problems. This result can be applied to a variety of models with pseudo-differential generators (beyond standard diffusions), where the corresponding regularity results for linear generators are available, for instance in case of decomposable generators (see [37]), regular enough degenerate diffusions like interacting curvilinear Ornstein-Uhlenbeck processes or stochastic geodesic flows on the cotangent bundles to Riemannian manifolds (see [1], [33]), or stable jump diffusions. As the latter are of special interest among Lévy type processes, due to their applications in a wide variety of models from plasma physics to finances (see e.g.…”

Section: Content Of the Paper And Bibliographical Commentsmentioning

confidence: 99%

Nonlinear Markov Semigroups and Interacting Lévy Type Processes

Kolokoltsov

2006

J Stat Phys

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Semigroups of positivity preserving linear operators on measures of a measurable space X describe the evolutions of probability distributions of Markov processes on X. Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions B(X) on X describe the evolutions of averages over the trajectories of these Markov processes. In this paper we introduce and study the general class of semigroups of non-linear positivity preserving transformations on measures that is non-linear Markov or Feller semigroups. An explicit structure of generators of such groups is given in case when X is the Euclidean space R d (or more generally, a manifold) showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for Markov models of interacting particles. Well posedness results for these equations are given together with applications to interacting particles: dynamic law of large numbers and central limit theorem, the latter being new already for the standard coagulation-fragmentation models.

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Semiclassical Analysis for Diffusions and Stochastic Processes

Cited by 100 publications

References 0 publications

Tunnel Effect for Kramers–Fokker–Planck Type Operators

Tunnel Effect for Kramers–Fokker–Planck Type Operators

Some Recent Developments

Nonlinear Markov Semigroups and Interacting Lévy Type Processes

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