This work addresses some asymptotic behavior of solutions to the stochastic convective Brinkman-Forchheimer (SCBF) equations perturbed by multiplicative Gaussian noise in bounded domains. Using a weak convergence approach of Budhiraja and Dupuis, we establish the Laplace principle for the strong solution to the SCBF equations in a suitable Polish space. Then, the Wentzell-Freidlin large deviation principle is derived using the well known results of Varadhan and Bryc. The large deviations for short time are also considered in this work. Furthermore, we study the exponential estimates on certain exit times associated with the solution trajectory of the SCBF equations. Using contraction principle, we study these exponential estimates of exit times from the frame of reference of Freidlin-Wentzell type large deviations principle. This work also improves several LDP results available in the literature for the tamed Navier-Stokes equations as well as Navier-Stokes equations with damping in bounded domains.
Since T. Lyons invented rough path theory, one of its most successful applications is a new proof of Freidlin-Wentzell's large deviation principle for diffusion processes. In this paper we extend this method to the case of pinned diffusion processes under a mild ellipticity assumption. Besides rough path theory, our main tool is quasi-sure analysis, which is a kind of potential theory in Malliavin calculus.
Freidlin and Wentzell characterized the logarithmic asymptotics of the exit time from a basin of attraction for a finite dimensional diffusion with small noise. After that, several authors studied the same properties for exit problems associated to specific infinite dimensional system. In this paper, we present a general method, based a control theoretic approach, to establish exit time and exit place results for a large class of stochastic equations in Banach spaces.
We show two Freidlin-Wentzell type Large Deviations Principles (LDP) in path space topologies (uniform and Hölder) for the solution process of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) using techniques which directly address the presence of the law in the coefficients and altogether avoiding decoupling arguments or limits of particle systems. We provide existence and uniqueness results along with several properties for a class of MV-SDEs having random coefficients and drifts of super-linear growth. As an application of our results, we establish a Functional Strassen type result…
This work is concerned with Freidlin-Wentzell type large deviation principle for a family of multi-scale quasilinear and semilinear stochastic partial differential equations. Employing the weak convergence method and Khasminskii's time discretization approach, the Laplace principle (equivalently, large deviation principle) for a general class of multi-scale SPDEs is derived. In particular, we succeed in dropping the compactness assumption of embedding in the Gelfand triple in order to deal with the case of bounded and unbounded domains in applications. Our main results are applicable to variou…
The Large Deviations Principle (LDP) is verified for a homogeneous diffusion process with respect to a Brownian motion $B_t$, $$ X^\eps_t=x_0+\int_0^tb(X^\eps_s)ds+ \eps\int_0^tσ(X^\eps_s)dB_s, $$ where $b(x)$ and $σ(x)$ are are locally Lipschitz functions with super linear growth. We assume that the drift is directed towards the origin and the growth rates of the drift and diffusion terms are properly balanced. Nonsingularity of $a=σσ^*(x)$ is not required.
We prove a Freidlin-Wentzell large deviation principle for general stochastic evolution equations with small perturbation multiplicative noises. In particular, our general result can be used to deal with a large class of quasi linear stochastic partial differential equations, such as stochastic porous medium equations and stochastic reaction diffusion equations with polynomial growth zero order term and $p$-Laplacian second order term.
The convective Brinkman-Forchheimer (CBF) equations characterize the motion of incompressible fluid flows in a saturated porous medium. The small noise asymptotic for the two-time-scale stochastic convective Brinkman-Forchheimer (SCBF) equations in two and three dimensional bounded domains is carried out in this work. More precisely, we establish a Wentzell-Freidlin type large deviation principle for stochastic partial differential equations with slow and fast time-scales, where the slow component is the SCBF equations in two and three dimensions perturbed by small multiplicative Gaussian nois…
Exponential mixing and Freidlin--Wentzell large deviation principle for Markov cocycles
School of Mathematics, Sichuan University（四川大学数学学院）；School of Mathematical Sciences, University of Electronic Science and Technology of China（电子科技大学数学科学学院）；Department of Mathematics, New Mexico Institute of Mining and Technology（新墨西哥矿业与技术大学数学系）
We establish the Freidlin--Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation with multiplicative noise in one spatial dimension. That is, we introduce a small parameter $ \sqrt{\varepsilon} $ to the noise, and establish an LDP for the trajectory of the solution. Such a Freidlin--Wentzell LDP gives the short-time, one-point LDP for the KPZ equation in terms of a variational problem. Analyzing this variational problem under the narrow wedge initial data, we prove a quadratic law for the near-center tail and a $ \frac52 $ law for the deep lower tail. These power laws confi…
This is an expository note on large deviations, Hamilton-Jacobi-Bellman (HJB) equations, and the role of the Freidlin-Wentzell quasipotential in Chemical Reaction Networks (CRNs). The note was motivated by observations which identified Lyapunov functions for deterministic descriptions of CRNs by taking appropriate scaling limits of invariant distributions for the corresponding stochastic dynamics. We explain how this is a special case of a classical theory due to Freidlin and Wentzell. We also show that this Lyapunov function is a solution to the HJB partial differential equation if and only i…
We prove a large deviation principle of Freidlin-Wentzell's type for the multivalued stochastic differential equations with monotone drifts, which in particular contains a class of SDEs with reflection in a convex domain.
In this paper, we establish the Freidlin-Wentzell's large deviations for quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone. The proof is based on the weak convergence approach.
In this paper, we establish the Freidlin-Wentzell type large deviation principles for porous medium-type equations perturbed by small multiplicative noise. The porous medium operator $Δ(|u|^{m-1}u)$ is allowed. Our proof is based on weak convergence approach.
In this paper, we prove a large deviation principle of Freidlin-Wentzell's type for the multivalued stochastic differential equations. As an application, we derive a functional iterated logarithm law for the solutions of multivalued stochastic differential equations.
A Freidlin-Wentzell type large deviation principle is established for stochastic partial differential equations with slow and fast time-scales, where the slow component is a one-dimensional stochastic Burgers equation with small noise and the fast component is a stochastic reaction-diffusion equation. Our approach is via the weak convergence criterion developed in [3].
For a heat equation with memory driven by a Lévy-type noise we establish the existence of a unique solution. The main part of the article focuses on the Freidlin-Wentzell large deviation principle of the solutions of heat equation with memory driven by a Lévy-type noise. For this purpose, we exploit the recently introduced weak convergence approach.
This paper provides a large deviation principle for Non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao and Liu \cite{GL}, this extends the corresponding results collected in Freidlin and Wentzell \cite{FreidlinWentzell}. However, we use a different line of argument, adapting the PDE method of Fleming \cite{Fleming} and Evans and Ishii \cite{EvansIshii} to the path-dependent case, by using backward stochastic differential techniques. Similar to the Markovian case, we obtain a characterization of the action function as the unique boun…
We consider a generalization of classical results of Freidlin and Wentzell to the case of time dependent dissipative drifts. We show the convergence of diffusions with multiplicative noise in the zero limit of a diffusivity parameter to the related dynamical systems. The solution to the associated transport equation is obtained as an application.
We prove a Freidlin-Wentzell result for stochastic differential equations in infinite-dimensional Hilbert spaces perturbed by a cylindrical Wiener process. We do not assume the drift to be Lipschitz continuous, but only continuous with at most linear growth. Our result applies, in particular, to a large class of nonlinear fractional diffusion equations perturbed by a space-time white noise.
This paper establishes a Freidlin-Wentzell large deviation principle for stochastic differential equations(SDEs) under locally weak monotonicity conditions and Lyapunov conditions. We illustrate the main result of the paper by showing that it can be applied to SDEs with non-Lipschitzian coefficients, which can not be covered in the existing literature. These include the interesting biological models like stochastic Duffing-van der Pol oscillator model, stochastic SIR model, etc.
We present two examples of a large deviations principle where the rate function is not strictly convex. This is motivated by a model used in mathematical finance (the Heston model), and adds a new item to the zoology of non strictly convex large deviations. For one of these examples, we show that the rate function of the Cramer-type of large deviations coincides with that of the Freidlin-Wentzell when contraction principles are applied.
We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and 2D magnetic Bénard problem and also some shell models of turbulence. We first prove the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by weak convergence method.
In this paper, we established the Freidlin-Wentzell type large deviation principles for first-order scalar conservation laws perturbed by small multiplicative noise. Due to the lack of the viscous terms in the stochastic equations, the kinetic solution to the Cauchy problem for these first-order conservation laws is studied. Then, based on the well-posedness of the kinetic solutions, we show that the large deviations holds by utilising the weak convergence approach.
The primary goal of this paper is to prove a near-martingale optional stopping theorem and establish solvability and large deviations for a class of anticipating linear stochastic differential equations. We prove the existence and uniqueness of solutions using two approaches: (1) Ayed-Kuo differential formula using an ansatz, and (2) a novel braiding technique by interpreting the integral in the Skorokhod sense. We establish a Freidlin-Wentzell type large deviations result for solution of such equations.
In this paper we study a large deviation principle of Freidlin-Wentzell type for pinned hypoelliptic diffusion measures associated with a natural sub-Laplacian on a compact sub-Riemannian manifold. To prove this large deviation principle, we use rough path theory and manifold-valued Malliavin calculus.
We develop a path integral framework for determining most probable paths in a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin-Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider an $n-$dimensional system with a switching manifold in the drift that forms an $(n-1)-$dimensional hyperplane and investigate noise-induced transitions between metastable states on either side of the switching manifold. To do this, we mollify the dr…
We prove large deviations for $g(t)$-Brownian motion in a complete, evolving Riemannian manifold $M$ with respect to a collection $\{g(t)\}_{t\in [0,1]}$ of Riemannian metrics, smoothly depending on $t$. We show how the large deviations are obtained from the large deviations of the (time-dependent) horizontal lift of $g(t)$-Brownian motion to the frame bundle $FM$ over $M$. The latter is proved by embedding the frame bundle into some Euclidean space and applying Freidlin-Wentzell theory for diffusions with time-dependent coefficients, where the coefficients are jointly Lipschitz in space and t…
We study Donsker-Watanabe's delta functions associated with strongly hypoelliptic diffusion processes indexed by a small parameter. They are finite Borel measures on the Wiener space and admit a rough path lift. Our main result is a large deviation principle of Schilder type for the lifted measures on the geometric rough path space as the scale parameter tends to zero. As a corollary, we obtain a large deviation principle conjectured by Takanobu and Watanabe, which is a generalization of a large deviation principle of Freidlin-Wentzell type for pinned diffusion processes.
We formulate large deviations principle (LDP) for diffusion pair $(X^ε,ξ^ε)=(X_t^ε,ξ_t^ε)$, 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 $ξ_t^ε$. In some sense we obtain a combination of Freidlin-Wentzell's and Donsker-Varadhan's results. Our approach relies the concept of the exponential tightness and Puhalskii's theorem.
Freidlin-Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well-posed matrix Riccati equations involving the minimize…
We study the large deviations principle (LDP) for stationary solutions of a class of stochastic differential equations (SDE) in infinite time intervals by the weak convergence approach, and then establish the LDP for the invariant measures of the SDE by the contraction principle. We further point out the equivalence of the rate function of the LDP for invariant measures induced by the LDP for stationary solutions and the rate function defined by quasi-potential. This fact gives another view of the quasi-potential introduced by Freidlin and Wentzell.
The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principl…
We consider delay differential equations (DDE) that are on the verge of an instability, i.e. the characteristic equation for the linearized equation has one root as zero and all other roots have negative real parts. In presence of small mean-zero noise, we study the large deviations from the corresponding deterministic system. Using spectral theory for DDE it is easy to see that, the projection on to the one dimensional space corresponding to the zero root is exponentially equivalent with the original process. For the one-dimensional process we make the observation that the results of Freidlin…
We prove the Freidlin-Wentzell type large deviations principle for the family of stationary measures of stochastic nonlinear wave (NLW) equation with white noise. We do not assume that the limiting equation possesses a unique equilibrium and do not impose roughness on the noise. This allows to provide the first such result in the PDE setting.
We study the exit-time from a domain of a self-interacting diffusion, where the Brownian motion is replaced by $σB_t$ for a constant $σ$. The first part of this work consists in showing that the rate of convergence (of the occupation measure of the self-interacting process toward some explicit Gibbs measure) previously obtained in \cite{kk-ejp} for a convex confinment potential $V$ and a convex interaction potential can be bounded uniformly with respect to $σ$. Then, we prove an Arrhenius-type law for the first exit-time from a domain (satisfying classical hypotheses of Freidlin-Wentzell theor…
Large deviations principles characterize the exponential decay rates of the probabilities of rare events. Cerrai and Rockner [13] proved that systems of stochastic reaction-diffusion equations satisfy a large deviations principle that is uniform over bounded sets of initial data. This paper proves uniform large deviations results for a system of stochastic reaction--diffusion equations in a more general setting than Cerrai and Rockner. Furthermore, this paper identifies two common situations where the large deviations principle is uniform over unbounded sets of initial data, enabling the chara…
This paper is devoted to investigating Freidlin-Wentzell's large deviation principle for one (spatial) dimensional nonlinear stochastic wave equation $\frac{\partial^2 u^{\e}(t,x)}{\partial t^2}=\frac{\partial^2 u^{\e}(t,x)}{\partial x^2}+\sqrt{\e}σ(t, x, u^{\e}(t,x))\dot{W}(t,x)$, where $\dot{W}$ is white in time and fractional in space with Hurst parameter $H\in(\frac 14,\frac 12)$. The variational framework and the modified weak convergence criterion proposed by Matoussi et al. \cite{MSZ} are adopted here.
We establish large deviation principle (LDP) for the family of vector-valued random processes $(X^ε,Y^ε),ε\to 0$ defined as $$ X^ε_t=\frac{1}{ε^κ}\int_0^t H(ξ^ε_s,Y^ε_s)ds, dY^ε_t=F(ξ^ε_t,Y^ε_t)dt+ Dε^{1/2-κ}G(ξ^ε_t,Y^ε_t)dW_t,$$ where $W_t$ is Wiener process and $ξ^ε_t$ is fast ergodic diffusion. We show that, under $κ<{1/2}$ or less and Veretennikov-Khasminskii type condition for fast diffusion, the LDP holds with rate function of Freidlin-Wentzell's type.
This paper is devoted to investigating the Freidlin-Wentzell's large deviation principle for a class of McKean-Vlasov quasilinear SPDEs perturbed by small multiplicative noise. We adopt the variational framework and the modified weak convergence criteria to prove the Laplace principle for McKean-Vlasov type SPDEs, which is equivalent to the large deviation principle. Moreover, we do not assume any compactness condition of embedding in the Gelfand triple to handle both the cases of bounded and unbounded domains in applications. The main results can be applied to various McKean-Vlasov type SPDEs…
We study controlled differential equations driven by a rough path (in the sense of T. Lyons) with an additional, possibly unbounded drift term. We show that the equation induces a solution flow if the drift grows at most linearly. Furthermore, we show that the semiflow exists assuming only appropriate one-sided growth conditions. We provide bounds for both the flow and the semiflow. Applied to stochastic analysis, our results imply "strong completeness" and the existence of a stochastic (semi)flow for a large class of stochastic differential equations. If the driving process is Gaussian, we ca…
We establish Freidlin-Wentzell results for a nonlinear ordinary differential equation starting close to the stable state $0$, say, subject to a perturbation by a stochastic integral which is driven by an $\varepsilon$-small and $(1/\varepsilon)$-accelerated Lévy process with exponentially light jumps. For this purpose we derive a large deviations principle for the stochastically perturbed system using the weak convergence approach developed by Budhiraja, Dupuis, Maroulas and collaborators in recent years. In the sequel we solve the associated asymptotic first escape problem from the bounded ne…
We prove Freidlin-Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth-death processes, Galton-Watson trees, epidemic SI models, and prey-predator models. The proofs are carried out using a general analytic approach based on the well-posedness of a class of associated Hamilton-Jacobi equations. The notable feature for these Hamilton-Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton-Jacobi equations corresponding…
We utilize the weak convergence method to establish the Freidlin--Wentzell large deviations principle (LDP) for stochastic delay differential equations (SDDEs) with super-linearly growing coefficients, which covers a large class of cases with non-globally Lipschitz coefficients. The key ingredient in our proof is the uniform moment estimate of the controlled equation, where we handle the super-linear growth of the coefficients by an iterative argument. Our results allow both the drift and diffusion coefficients of the considered equations to super-linearly grow not only with respect to the del…
The aim of this paper is threefold. Firstly, we prove the existence and the uniqueness of a global strong (in both the probabilistic and the PDE senses) $\mathrm{H}^{1}_2$-valued solution to the 2D stochastic Navier-Stokes equations (SNSEs) driven by a multiplicative Lévy noise under the natural Lipschitz on balls and linear growth assumptions on the jump coefficient. Secondly, we prove a Girsanov-type theorem for Poisson random measures and apply this result to a study of the well-posedness of the corresponding stochastic controlled problem for these SNSEs. Thirdly, we apply these results to …
We provide a large deviations analysis of deadlock phenomena occurring in distributed systems sharing common resources. In our model transition probabilities of resource allocation and deallocation are time and space dependent. The process is driven by an ergodic Markov chain and is reflected on the boundary of the d-dimensional cube. In the large resource limit, we prove Freidlin-Wentzell estimates, we study the asymptotic of the deadlock time and we show that the quasi-potential is a viscosity solution of a Hamilton-Jacobi equation with a Neumann boundary condition. We give a complete analys…
