Large Deviation Principle and Inviscid Shell Models (original) (raw)

A.: Large deviation principle and inviscid shell models

2009

A LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient ν converges to 0 and the noise intensity is multiplied by √ ν, we prove that some shell models of turbulence with a multiplicative stochastic perturbation driven by a H-valued Brownian motion satisfy a LDP in C([0, T ], V) for the topology of uniform convergence on [0, T ], but where V is endowed with a topology weaker than the natural one. The initial condition has to belong to V and the proof is based on the weak convergence of a family of stochastic control equations. The rate function is described in terms of the solution to the inviscid equation.

Large deviations for the stochastic shell model of turbulence

Nonlinear Differential Equations and Applications NoDEA, 2009

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 shell model of turbulence perturbed by Lévy noise

Communications on Stochastic Analysis, 2013

The Laplace principle for the strong solution of the stochastic shell model of turbulence perturbed by Lévy noise is established in a suitable Polish space using weak convergence approach. The large deviation principle is proved using the well known results of Varadhan and Bryc.

Large deviation principle for stochastic evolution equations

Probability Theory and Related Fields, 1994

The large deviation principle obtained by Freidlin and Wentzell for measures associated with finite-dimensional diffusions is extended to measures given by stochastic evolution equations with non-additive random perturbations. The proof of the main result is adopted from the Priouret paper concerning finite-dimensional diffusions. Exponential tail estimates for infinite-dimensional stochastic convolutions are used as main tools.

Large deviations for stochastic PDE with Lévy noise

Journal of Functional Analysis, 2011

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, 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.

Large deviations principles for stochastic scalar conservation laws

Probability Theory and Related Fields, 2009

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.

On Large Deviations of Stochastic Integrodifferential Equations with Brownian Motion

In this paper, a Freidlin-Wentzell type large deviation principle is established for the stochastic integrodifferential equation driven by finite dimensional Brownian motion. Both the additive and multiplicative noise cases are considered here. Large deviation principle for additive noise case is established via contraction principle whilst weak convergence approach is employed to obtain the same for the multiplicative noise case.

Large deviation principles for stochastic dynamical systems with a fractional Brownian noise

arXiv (Cornell University), 2020

We study small noise large deviation asymptotics for stochastic differential equations with a multiplicative noise given as a fractional Brownian motion B H with Hurst parameter H > 1 2. The solutions of the stochastic differential equations are defined pathwise under appropriate conditions on the coefficients. The ingredients in the proof of the large deviation principle, which include a variational representation for nonnegative functionals of fractional Brownian motions and a general sufficient condition for a LDP for a collection of functionals of a fractional Brownian motions, have a broader applicability than the model considered here.

Shell model of turbulence perturbed by Lévy noise

Nonlinear Differential Equations and Applications NoDEA, 2011

In this work we prove the existence and uniqueness of the strong solution of the shell model of turbulence perturbed by Lévy noise. The local monotonicity arguments have been exploited in the proofs.

Large deviations for infinite dimensional stochastic dynamical systems

The Annals of Probability, 2008

The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process.