Freidlin-Wentzell Large Deviation Research Articles

A Selection Principle for Weak KAM Solutions via Freidlin–Wentzell Large Deviation Principle of Invariant Measures

A Selection Principle for Weak KAM Solutions via Freidlin–Wentzell Large Deviation Principle of Invariant Measures

Integrability in the weak noise theory

We consider the variational problem associated with the Freidlin–Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation (SHE). For a general class of initial-terminal conditions, we show that a minimizer of this variational problem exists, and any minimizer solves a system of imaginary-time Nonlinear Schrödinger equations. This system is integrable. Utilizing the integrability, we prove that the formulas from the physics work (see Alexandre Krajenbrink and Pierre Le Doussal [Phys. Rev. Lett. 127 (2021), p. 8]) hold for every minimizer of the variational problem. As an application, we consider the Freidlin–Wentzell LDP for the SHE with the delta initial condition. Under a technical assumption on the poles of the reflection coefficients, we prove the explicit expression for the one-point rate function that was predicted in the physics works (see Pierre Le Doussal, Satya N. Majumdar, Alberto Rosso, and Grégory Schehr [Phys. Rev. Lett. 117 (2016), p. 070403]; Alexandre Krajenbrink and Pierre Le Doussal [Phys. Rev. Lett. 127 (2021), p. 8]). Under the same assumption, we give detailed pointwise estimates of the most probable shape in the upper-tail limit.

Noise-induced transition in the Zeldovich–Semenov model with local and global bifurcations

The noise-induced transition of the Zeldovich–Semenov model in a continuous stirred tank reactor is investigated under small random perturbations. The deterministic model will exhibit mono- and bistable characteristics via local and global bifurcations. In the bistable zone, based on the Freidlin–Wentzell large deviation theory, the stochastic preference is explained by analyzing the required action of the fluctuational path. For the case of monostability, in the weak noise limit, the emergence of the switching line gives rise to the sudden switch of the optimal path and the sliding cycle will appear via the sliding bifurcation, which is verified by numerical methods. In addition, when there is no saddle in phase space, stochastic excitation with large-amplitude spikes is studied. On the quasi-threshold manifold, the point with the minimum quasi-potential plays the same role as the saddle, which means that the optimal path will undergo a large excursion by crossing this special point. These phenomena are verified by employing stochastic simulations.

Short Time Large Deviations of the KPZ Equation

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 $$ \frac{5}{2} $$ law for the deep lower tail. These power laws confirm existing physics predictions (Kolokolov and Korshunov in Phys Rev B 75(14):140201, 2007, Phys Rev E 80(3):031107, 2009; Meerson et al. in Phys Rev Lett 116(7):070601, 2016; Le Doussal et al. in Phys Rev Lett 117(7):070403, 2016; Kamenev et al. in Phys Rev E 94(3):032108, 2016).

Large deviations for stochastic 3D Leray-<inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> model with fractional dissipation

In this paper we establish the Freidlin-Wentzell's large deviation principle for stochastic 3D Leray-$ \alpha $ model with general fractional dissipation and small multiplicative noise. This model is the stochastic 3D Navier-Stokes equations regularized through a smoothing kernel of order $ \theta_1 $ in the nonlinear term and a $ \theta_2 $-fractional Laplacian. The main result generalizes the corresponding LDP result of the classical stochastic 3D Leray-$ \alpha $ model ($ \theta_1 = 1 $, $ \theta_2 = 1 $), and it is also applicable to the stochastic 3D hyperviscous Navier-Stokes equations ($ \theta_1 = 0 $, $ \theta_2\geq\frac{5}{4} $) and stochastic 3D critical Leray-$ \alpha $ model ($ \theta_1 = \frac{1}{4} $, $ \theta_2 = 1 $).

Large deviations for stochastic heat equations with memory driven by Lévy-type noise

For a heat equation with memory driven by a Levy-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 Levy-type noise. For this purpose, we exploit the recently introduced weak convergence approach.

Large deviation principles for 3D stochastic primitive equations

In this paper, we establish the Freidlin–Wentzell's large deviations for 3D stochastic primitive equations with small noise perturbation. The weak convergence approach plays an important role.

Noise induced transitions and topological study of a periodically driven system

Noise induced transitions of an overdamped periodically driven oscillator are investigated theoretically and numerically in the limit of weak noise due to the Freidlin-Wentzell large deviation theory. Heteroclinic trajectories are found to approach the unstable orbit with fluctuational force tending to zeros. The global minimizer of the action functional corresponds to the most probable escape path and it shows a good agreement with statistical results. We then study the origins of singularities from a topological point of view by considering structures of the Lagrangian manifold and action surface. The switching line and cusp point turn out to have physical significance since they may impact the prehistory distributions, making the optimal path invalid.

Perturbative Calculation of Quasi-Potential in Non-equilibrium Diffusions: A Mean-Field Example

In stochastic systems with weak noise, the logarithm of the stationary distribution becomes proportional to a large deviation rate function called the quasi-potential. The quasi-potential, and its characterization through a variational problem, lies at the core of the Freidlin-Wentzell large deviations theory%.~\cite{freidlin1984}.In many interacting particle systems, the particle density is described by fluctuating hydrodynamics governed by Macroscopic Fluctuation Theory%, ~\cite{bertini2014},which formally fits within Freidlin-Wentzell's framework with a weak noise proportional to $1/\sqrt{N}$, where $N$ is the number of particles. The quasi-potential then appears as a natural generalization of the equilibrium free energy to non-equilibrium particle systems. A key physical and practical issue is to actually compute quasi-potentials from their variational characterization for non-equilibrium systems for which detailed balance does not hold. We discuss how to perform such a computation perturbatively in an external parameter $\lambda$, starting from a known quasi-potential for $\lambda=0$. In a general setup, explicit iterative formulae for all terms of the power-series expansion of the quasi-potential are given for the first time. The key point is a proof of solvability conditions that assure the existence of the perturbation expansion to all orders. We apply the perturbative approach to diffusive particles interacting through a mean-field potential. For such systems, the variational characterization of the quasi-potential was proven by Dawson and Gartner%. ~\cite{dawson1987,dawson1987b}. Our perturbative analysis provides new explicit results about the quasi-potential and about fluctuations of one-particle observables in a simple example of mean field diffusions: the Shinomoto-Kuramoto model of coupled rotators%. ~\cite{shinomoto1986}. This is one of few systems for which non-equilibrium free energies can be computed and analyzed in an effective way, at least perturbatively.

Large deviation principle for stochastic heat equation with memory

In this work, using the weak convergence argument, we prove a Freidlin-Wentzell's large deviation principle for a class of stochastic heat equations with memory and Dirichlet boundary conditions, where the nonlinear term is allowed to be of polynomial growth.

The Geometry of Spontaneous Spiking in Neuronal Networks

The mathematical theory of pattern formation in electrically coupled networks of excitable neurons forced by small noise is presented in this work. Using the Freidlin-Wentzell large deviation theory for randomly perturbed dynamical systems and the elements of the algebraic graph theory, we identify and analyze the main regimes in the network dynamics in terms of the key control parameters: excitability, coupling strength, and network topology. The analysis reveals the geometry of spontaneous dynamics in electrically coupled network. Specifically, we show that the location of the minima of a certain continuous function on the surface of the unit n-cube encodes the most likely activity patterns generated by the network. By studying how the minima of this function evolve under the variation of the coupling strength, we describe the principal transformations in the network dynamics. The minimization problem is also used for the quantitative description of the main dynamical regimes and transitions between them. In particular, for the weak and strong coupling regimes, we present asymptotic formulas for the network activity rate as a function of the coupling strength and the degree of the network. The variational analysis is complemented by the stability analysis of the synchronous state in the strong coupling regime. The stability estimates reveal the contribution of the network connectivity and the properties of the cycle subspace associated with the graph of the network to its synchronization properties. This work is motivated by the experimental and modeling studies of the ensemble of neurons in the Locus Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive performance and behavior.

Synchronization and spontaneous dynamics in the Locus Coeruleus network

The mathematical theory of pattern formation in electrically coupled networks of excitable neurons forced by small noise is presented in this work. Using the Freidlin-Wentzell large deviation theory for randomly perturbed dynamical systems and the elements of the algebraic graph theory, we identify and analyze main regimes in the network dynamics in terms of the key control parameters: excitability, coupling strength, and network topology. The analysis reveals the geometry of spontaneous dynamics in electrically coupled network. Specifically, we show that the location of the minima of a certain continuous function on the surface of the unit n-cube encodes the most likely activity patterns generated by the network. By studying how the minima of this function evolve under the variation of the coupling strength, we identify the principal transformations in the network dynamics. The minimization problem is also used for the quantitative description of the main dynamical regimes and transitions between them. In particular, for the weak and strong coupling regimes, we present asymptotic formulae for the activity rate as a function of the coupling strength and the degree of the network. The variational analysis is complemented by the stability analysis of the synchronous state in the strong coupling regime. The stability estimates reveal the contribution of the network connectivity and the properties of the cycle subspace associated with the graph of the network to its synchronization properties. This work is motivated by the experimental and modeling studies of the ensemble of neurons in the Locus Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive performance and behavior [1].

General Freidlin–Wentzell Large Deviations and positive diffusions

We prove Freidlin–Wentzell Large Deviation estimates under rather minimal assumptions. This allows one to derive Wentzell–Freidlin Large Deviation estimates for diffusions on the positive half line with coefficients that are neither bounded nor Lipschitz continuous. This applies to models of interest in Finance, i.e. the CIR and the CEV models, which are positive diffusion processes whose diffusion coefficient is only Hölder continuous.

Uniform large deviations for multivalued stochastic differential equations with Poisson jumps

Based on a variational representation for functionals of a general Poisson random measure plus an independent infinite-dimensional Brownian motion developed by Budhiraja, Dupuis, and Maroulas, the Freidlin-Wentzell large deviation principle is established for multivalued stochastic differential equations with Poisson jumps in this paper.

Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise

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 principle in our framework.

Freidlin–Wentzell's large deviations for stochastic evolution equations

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.

Stochastic Differential Equation Driven by Countably Many Brownian Motions with Non-Lipschitzian Coefficients

We study m-dimensional SDE , where {W i } i≥1 is an infinite sequence of independent standard one-dimensional Brownian motions. Existence and pathwise uniqueness, non-explosion, and a Freidlin-Wentzell large deviations principle of strong solutions to the SDE are established under modulus of continuity of the coefficients is less than , which are different from that results constructed recently by Cao and He [2] and Fang and Zhang [6].

Freidlin–Wentzell's large deviations for homeomorphism flows of non-Lipschitz SDEs

We prove a Freidlin–Wentzell large deviation principle for multi-dimensional stochastic differential equations with non-Lipschitz coefficients and apply it to the Brownian motion on the diffeomorphism group of the disc constructed recently by Airault, Malliavin and Thalmaier.

Stochastic evolution equations with a spatially homogeneous Wiener process

A semilinear parabolic equation on R d with a non-additive random perturbation is studied. The noise is supposed to be a spatially homogeneous Wiener process. Conditions for the existence and uniqueness of the solution in terms of the spectral measure of the noise are given. Applications to population and geophysical models are indicated. The Freidlin-Wentzell large deviation estimates are obtained as well.