Large deviations of empirical measures of diffusions in weighted topologies

Ferré, Grégoire; Stoltz, Gabriel

doi:10.1214/20-ejp514

Cited by 10 publications

(9 citation statements)

References 111 publications

(269 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is however present in the initial approach by Donsker and Varadhan. To our knowledge, such approach is only explicitly described in [23], in a few derived papers as in [21], and in [17]. As in our approach, the latter extend by this way the alternative approach for the convergence to QSDs that they have just obtained.…”

Section: Introductionmentioning

confidence: 81%

Large deviations for surviving trajectories of general Markov processes

Velleret

2020

Preprint

View full text Add to dashboard Cite

The purpose of this paper is to ensure the conditions of Gärtner-Ellis Theorem for evaluations of the empirical measure. We show that up-to-date conditions for ensuring the convergence to a quasi-stationary distribution can be applied efficiently. By this mean, we are able to prove Large Deviation results even with a conditioning that the process is not extinct at the end of the evaluation. The domain on which these Large Deviation results apply is implicitly given by the range of penalization for which one can prove the above-mentioned results of quasi-stationarity. We propose a way to relate the range of controlled deviations to the range of admissible penalization. Central Limit Theorems are deduced from these results of Large Deviations. As an application, we consider the empirical measure of position and jumps of a continuous-time process on a unbounded domain of R d . This model is inspired by the adaptation of a population to a changing environment. Jumps makes it possible for the process to face a deterministic dynamics leading to high extinction areas.

show abstract

Section: Introductionmentioning

confidence: 81%

Large deviations for surviving trajectories of general Markov processes

Velleret

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…For time homogeneous models q t,t`τ " q τ condition ( 20) is clearly met as soon as q τ is a positive continuous density on the compact sets V 2 ǫ . These absolutely continuous integral operators arise in a natural way in discrete time settings [63,67] and continuous time elliptic diffusion-type absorption models [6,98,99,168]. In connection to this, two-sided estimates for stable-like processes are provided in [167,177].…”

Section: Regularity Conditionsmentioning

confidence: 99%

On the Stability of Positive Semigroups

Moral¹,

Horton²,

Jasra³

2021

Preprint

View full text Add to dashboard Cite

The stability and contraction properties of positive integral semigroups on Polish spaces are investigated. We provide a novel analysis based on an extension of V -norm, Dobrushin-type, contraction techniques on functionally weighted Banach spaces for Markov operators. These are applied to a general class of positive and possibly time-inhomogeneous bounded integral semigroups and their normalised versions. Under mild regularity conditions, the Lipschitz-type contraction analysis presented in this article simplifies and extends several exponential estimates developed in the literature. The spectral-type theorems that we develop can also be seen as an extension of Perron-Frobenius and Krein-Rutman theorems for positive operators to time-varying positive semigroups. Incidentally, in the context of time-homogeneous models, the regularity conditions discussed in the present article appears to be a necessary and sufficient condition for the existence of leading eigenvalues. We review and illustrate in detail the impact of these results in the context of positive semigroups arising in transport theory, physics, mathematical biology and advanced signal processing.

show abstract

“…Similarly, the global Lipschitz condition is used in Section 4 to obtain standard error estimates, but could be replaced by a local Lipschitz condition with an appropriate control. Finally, the upper boundedness of f could also be weakened under a Lyapunov condition, see the technique used in [13].…”

Section: The Function F Belongs To C ∞ (R D ) and Is Upper Boundedmentioning

confidence: 99%

“…We can prove by standard Itô calculus on ψε together with the logarithmic transform (8) that gε as defined by (13) is the solution to (11), see [17, Chapter 2, Section 3]. As a natural consequence of ( 13)-( 14), we have…”

Section: The Optimal Controlmentioning

confidence: 99%

Stochastic viscosity approximations of Hamilton-Jacobi equations and variance reduction

Ferré¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We consider the computation of free energy-like quantities for diffusions in high dimension, when resorting to Monte Carlo simulation is necessary. Such stochastic computations typically suffer from high variance, in particular in a low noise regime, because the expectation is dominated by rare trajectories for which the observable reaches large values. Although importance sampling, or tilting of trajectories, is now a standard technique for reducing the variance of such estimators, quantitative criteria for proving that a given control reduces variance are scarce, and often do not apply to practical situations. The goal of this work is to provide a quantitative criterion for assessing whether a given bias reduces variance, and at which order. We rely for this on a recently introduced notion of stochastic solution for Hamilton-Jacobi-Bellman equations. Based on this tool, we introduce the notion of k-stochastic viscosity approximation of a HJB equation. We next prove that such approximate solutions are associated with estimators having a relative variance of order k − 1 at log-scale. Finally, in order to show that our definition is relevant, we provide examples of stochastic viscosity approximations of order one and two, with a numerical illustration confirming our theoretical findings.

show abstract

Large deviations of empirical measures of diffusions in weighted topologies

Cited by 10 publications

References 111 publications

Large deviations for surviving trajectories of general Markov processes

Large deviations for surviving trajectories of general Markov processes

On the Stability of Positive Semigroups

Stochastic viscosity approximations of Hamilton-Jacobi equations and variance reduction

Contact Info

Product

Resources

About