Large deviations for stochastic flows and their applications

Large deviations for stochastic flow solutions to SDEs containing a small parameter are studied. The obtained results are applied to establish a C_p,r,-large deviation principle for stochastic flows and for solutions to anticipating SDEs. The recent results of Millet-Nualart-Sans and Yoshida are improved and refined.     

Large deviations and stochastic homogenization

Summary A general theorem is stated providing large deviations estimates for a family of measures on a topological vector space. Applications are given in the second part, where large deviations problems arising in stochastic homogenization are discussed. Another application is given in similar problems connected with Donsker's invariance principle.     

Large deviations for stochastic nonlinear beam equations

We establish a large deviation principle for the solutions of stochastic partial differential equations for nonlinear vibration of elastic panels (also called stochastic nonlinear beam equations).     

Large deviations for stochastic differential delay equations

In this work, a Freidlin–Wentzell type large deviation principle is established for stochastic differential delay equations. The result in Mohammed and Zhang (2006) [6] is improved.     

Large deviations and approximations for slow-fast stochastic reaction-diffusion equations

A large deviation principle is derived for a class of stochastic reaction-diffusion partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This result also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.     

Large deviations principles for stochastic scalar conservation laws

Large deviations principles for a family of scalar 1 + 1 dimensional conservative stochastic PDEs (viscous conservation laws) are investigated, in the limit of jointly vanishing noise and viscosity. A first large deviations principle is obtained in a space of Young measures. The associated rate functional vanishes on a wide set, the so-called set of measure-valued solutions to the limiting conservation law. A second order large deviations principle is therefore investigated, however, this can be only partially proved. The second order rate functional provides a generalization for non-convex fluxes of the functional introduced by Jensen and Varadhan in a stochastic particles system setting.     

Large deviations for stochastic generalized porous media equations

The large deviation principle is established for the distributions of a class of generalized stochastic porous media equations for both small noise and short time.     

A large deviations principle for stochastic flows of viscous fluids

We study the well-posedness of a stochastic differential equation on the two dimensional torus ${\mathbb{T}}^2$, driven by an infinite dimensional Wiener process with drift in the Sobolev space $L^2(0,T;H^1\left({\mathbb{T}}^2\right))$. The solution corresponds to a stochastic Lagrangian flow in the sense of DiPerna Lions. By taking into account that the motion of a viscous incompressible fluid on the torus can be described through a suitable stochastic differential equation of the previous type, we study the inviscid limit. By establishing a large deviations principle, we show that, as the viscosity goes to zero, the Lagrangian stochastic Navier–Stokes flow approaches the Euler deterministic Lagrangian flow with an exponential rate function.     

Large deviations for the stochastic shell model of turbulence

In this work, we first prove the existence and uniqueness of a strong solution to stochastic GOY model of turbulence with a small multiplicative noise. Then using the weak convergence approach, Laplace principle for solutions of the stochastic GOY model is established in certain Polish space. Thus a Wentzell–Freidlin type large deviation principle is established utilizing certain results by Varadhan and Bryc.      

Large deviations for the 3D stochastic Navier-Stokes-Voight equations

In this paper, using the weak convergence method, a large deviation principle for 3D stochastic Navier–Stokes–Voight equations is proved.     

Large deviations for stochastic PDE with Lévy noise

We prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small Lévy noise. We use general large deviations theorems of Varadhan and Bryc coupled with the techniques of Feng and Kurtz (2006) [15], viscosity solutions of integro-partial differential equations in Hilbert spaces, and deterministic optimal control methods. The Laplace limit is identified as a viscosity solution of a Hamilton-Jacobi-Bellman equation of an associated control problem. We also establish exponential moment estimates for solutions of stochastic evolution equations driven by Lévy noise. General results are applied to stochastic hyperbolic equations perturbed by subordinated Wiener process.     

Embedding smooth diffeomorphisms in flows

In this paper we study the problem on embedding germs of smooth diffeomorphisms in flows in higher dimensional spaces. First we prove the existence of embedding vector fields for a local diffeomorphism with its nonlinear term a resonant polynomial. Then using this result and the normal form theory, we obtain a class of local Ck diffeomorphisms for k∈N∪{∞,ω} which admit embedding vector fields with some smoothness. Finally we prove that for any k∈N∪{∞} under the coefficient topology the subset of local Ck diffeomorphisms having an embedding vector field with some smoothness is dense in the set of all local Ck diffeomorphisms.     

Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems

A classical damping Hamiltonian system perturbed by a random force is considered. The locally uniform large deviation principle of Donsker and Varadhan is established for its occupation empirical measures for large time, under the condition, roughly speaking, that the force driven by the potential grows infinitely at infinity. Under the weaker condition that this force remains greater than some positive constant at infinity, we show that the system converges to its equilibrium measure with exponential rate, and obeys moreover the moderate deviation principle. Those results are obtained by constructing appropriate Lyapunov test functions, and are based on some results about large and moderate deviations and exponential convergence for general strong-Feller Markov processes. Moreover, these conditions on the potential are shown to be sharp.     

Large deviations for stochastic evolution equations in duals of nuclear frechet spaces

This paper presents the large deviation principle for stochastic evolution equation driven by gaussian martingales taking values in duals of nuclear Frechet spaces