Asymptotic evaluation of certain markov process expectations for large time, II

Donsker, M. D.; Varadhan, S. R. S.

doi:10.1002/cpa.3160280206

Cited by 612 publications

(471 citation statements)

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“…This type of large deviations results have been first obtained by Donsker and Varadhan [5] and later extended by many others (see the books [8,4,3] and the references therein). The LDP is well understood in the case of finite-dimensional diffusions and Markov processes with compact phase space, provided that the randomness is sufficiently non-degenerate and ensures mixing in the total variation norm.…”

Section: Xvii-3supporting

confidence: 72%

Large deviations results for the stochastic Navier–Stokes equations

Nersesyan¹

2017

Séminaire Laurent Schwartz — EDP Et Applications

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In this note, we discuss the problem of large deviations for the stochastic 2D Navier-Stokes equations. We show that the occupation measures of the trajectories of the system satisfy a large deviations principle, provided that the noise acts on all Fourier modes. In the case when the noise is more degenerate and acts on all the determining modes, we obtain an LDP of local type. The proofs use the methods introduced in [13,20] based on a Kifer-type sufficient condition for LDP and a multiplicative ergodic theorem. AMS subject classifications: 35Q30, 60B12, 60F10Keywords: Navier-Stokes system, white-in-time noise, large deviations principle, occupation measures, coupling IntroductionIn this note, we review the results of the paper [21], where the large deviations principle (LDP) is studied for the stochastic 2D Navier-Stokes (NS) equations driven by a white-in-time noiseThis system describes the motion of an incompressible fluid in a bounded domain D ⊂ R 2 with a smooth boundary ∂D, where ν > 0 is the kinematic viscosity, u = (u 1 (t, x), u 2 (t, x)) and p = p(t, x) are unknown velocity field and pressure of the fluid, f is the external (random) force, and u,Before stating our results, let us make some comments about this system and recall some previous results on its ergodicity. The system is considered in the usual space of divergence-free vector fields where L = −Π∆ is the Stokes operator, B(u) = Π( u, ∇ u), and Π is the orthogonal projection onto H in L 2 (Leray projector).Structure of the noise. We assume that f is of the formwhere the functions h and η are, respectively, the deterministic and random components of the external force with h ∈ H andHere {b j } is a sequence in R + such that B 0 = ∞ j=1 b 2 j < ∞, {β j } is a sequence of independent standard Brownian motions defined on a filtered probability space (Ω, F, {F t }, P) satisfying the usual conditions (see Definition 2.25 in [14]), and {e j } is an orthonormal basis in H consisting of the eigenfunctions of L with eigenvalues {α j }.Under these conditions, problem (0.3), (0.2) admits a unique solution and defines a Markov family (u t , P u ) parametrised by the initial condition u = u 0 ∈ H. The corresponding Markov semigroups are given bywhere P t (u, Γ) = P u {u t ∈ Γ} is the transition function. Recall that a measure µ ∈ P(H) is stationary if P * t µ = µ for any t > 0. The existence of a stationary measure is a relatively simple question, it is proved by using the classical Bogolyubov-Krylov argument. The uniqueness and the mixing are much more difficult problems which have been extensively studied in recent years. We refer the reader to the papers [7,16,6,17,2,11,22] and the book [18] for a detailed discussion of this topic. In particular, as it is stated in the following theorem, if η is sufficiently non-degenerate, then the family (u t , P u ) admits a unique stationary measure, which is exponentially mixing (see Theorem 3.5.2 in [18]).1 To simplify the notation, we shall assume that ν = 1. 2 P(H) is the set of probability Borel measures...

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Section: Xvii-3supporting

confidence: 72%

Large deviations results for the stochastic Navier–Stokes equations

Nersesyan¹

2017

Séminaire Laurent Schwartz — EDP Et Applications

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“…Note at first LDP for family µ ε (dz) = ν ε ([0, 1], dz) (on the space of probability measures supplied by Levy-Prohorov's metric) proved by Donsker and Varadhan [6], [7], [8], [9] for a wide class of Markov processes ξ ε t = ξ t/ε . Corresponding rate function obeys an invariant form: for any probabilistic measure µ on R…”

Section: ∆ ∈ B(r + ) γ ∈ B(r) (13)mentioning

confidence: 99%

Large deviations for two scaled diffusions

Liptser¹

1996

Probab Theory Relat Fields

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Abstract. We formulate large deviations principle (LDP) for diffusion pair (X ε , ξ ε ) = (X ε t , ξ ε t ), where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, the LDP is established for (X ε , ν ε ) with ν ε (dt, dz) being an occupation type measure corresponding to ξ ε t . In some sense we obtain a combination of FreidlinWentzell's and Donsker-Varadhan's results. Our approach relies the concept of the exponential tightness and Puhalskii's theorem. IntroductionLet ε be a small positive parameter, (X ε , ξ ε ) = (X ε t , ξ ε t ) t≥0 be a diffusion pair defined on some stochastic basis (Ω, F , F = (F t ) t≥0 , P) by Itô's equations w.r.t. independent Wiener processes W t and V t :is an ergodic process in the following sense. Let p(z) be the unique invariant density of ξ ε ,andX t is a solution of an ordinary differential equationẊ t =Ā(X t ) withĀ(x) = R A(x, z)p(z)dz subject to the same initial point x 0 . Then for any bounded continuous function h(t, z) and T > 0where r T is the uniform metric on [0, T ]. The above-mentioned ergodic property is a motivation to examine LDP for pair (X ε , ξ ε ), or more exactly for pair (X ε , ν ε ), where ν ε = ν ε (dt, dz) is an occupation measure on R + ×R, B(R + )⊗B(R) (B(R + ), B(R) are the Borel σ-algebras on R + and R respectively) corresponding to ξ ε :A choice of ν ε as the occupation measure is natural since the first ergodic property in (1.2) is nothing but1991 Mathematics Subject Classification. 60F10.

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“…The rate function for the S n -valued process {φ(x l 1 )} was first derived by Donsker and Varadhan in a series of papers [12,13,14], using a characterization of the spectral radius of a positive matrix [11]. In [10], an alternate derivation of these results is given based on the Gärtner-Ellis theorem [18,17].…”

Section: Rate Function For Singleton Frequencies Of a Markov Chainmentioning

confidence: 99%

“…This is the approach adopted in [19,Chapter IV]. In some earlier work such as [12,13,14], the rate function for singleton frequencies of a Markov processes (Theorem 4) is derived first, and this is used to derive the rate function for doublet frequencies (Theorem 5). This is also the approach followed in [10].…”

Section: The Rate Function For Singleton Frequenciesmentioning

confidence: 99%

An Elementary Derivation of the Large Deviation Rate Function for Finite State Markov Chains

Vidyasagar¹

2013

Asian Journal of Control

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Large deviation theory is a branch of probability theory that is devoted to a study of the "rate" at which empirical estimates of various quantities converge to their true values. The object of study in this paper is the rate at which estimates of the doublet frequencies of a Markov chain over a finite alphabet converge to their true values. In case the Markov process is actually an i.i.d. process, the rate function turns out to be the relative entropy (or Kullback-Leibler divergence) between the true and the estimated probability vectors. This result is a special case of a very general result known as Sanov's theorem and dates back to 1957. Moreover, since the introduction of the "method of types" by Csiszár and his co-workers during the 1980s, the proof of this version of Sanov's theorem has been "elementary," using some combinatorial arguments. However, when the i.i.d. process is replaced by a Markov process, the available proofs are far more complex. The main objective of this paper is therefore to present a first-principles derivation of the LDP for finite state Markov chains, using only simple combinatorial arguments (e.g. the method of types), thus gathering in one place various arguments and estimates that are scattered over the literature.

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Asymptotic evaluation of certain markov process expectations for large time, II

Cited by 612 publications

References 3 publications

Large deviations results for the stochastic Navier–Stokes equations

Large deviations results for the stochastic Navier–Stokes equations

Large deviations for two scaled diffusions

An Elementary Derivation of the Large Deviation Rate Function for Finite State Markov Chains

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