相关论文: Noise-Induced Stabilization of Planar Flows I

This paper proposes a notion of viscosity weak supersolutions to build a bridge between stochastic Lyapunov stability theory and viscosity solution theory. Different from ordinary differential equations, stochastic differential equations…

The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker-Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting diffusions with…

We consider proper orthogonal decomposition (POD) methods to approximate the incompressible Navier-Stokes equations. We study the case in which one discretization for the nonlinear term is used in the snapshots (that are computed with a…

The stabilization by noise for parabolic equations in perforated domains, i.e. domains with small holes, is investigated. We show that when the holes are small enough, one can stabilize the unstable equations using suitable multiplicative…

We study the kinetic Fokker-Planck equation perturbed by a stochastic Vlasov force term. When the noise intensity is not too large, we solve the Cauchy Problem in a class of well-localized (in velocity) functions. We also show that, when…

This paper studies the 1D stochastic Allen--Cahn equation on a bounded domain driven by localized white noise. We prove that the associated Markov process admits a unique invariant measure and is exponential mixing. The main challenge lies…

In this article, we consider the following class of stochastic partial differential equations (SPDE): \begin{equation*} \left\{\begin{aligned}\mathrm{d} \mathbf{X}(t)&=\mathrm{A}(t,\mathbf{X}(t))\mathrm{d}…

For the nonlinear stochastic partial differential equation which is driven by multiplicative noise of the form \[D_t^\beta u = \left[ { - {{\left( { - \Delta } \right)}^s}u + \zeta \left( u \right)} \right]dt + A\sum\limits_{m \in Z_0^d}…

The present article is devoted to well-posedness by noise for the continuity equation. Namely, we consider the continuity equation with non-linear and partially degenerate stochastic perturbations in divergence form. We prove the existence…

This paper presents a nonlinear model predictive control strategy for stochastic systems with general (state and input dependent) disturbances subject to chance constraints. Our approach uses an online computed stochastic tube to ensure…

We present recent results on noise-induced transitions in a nonlinear oscillator with randomly modulated frequency. The presence of stochastic perturbations drastically alters the dynamical behaviour of the oscillator: noise can wash out a…

The stabilisation by noise on the boundary of the Chafee-Infante equation with dynamical boundary conditions subject to a multiplicative It\^o noise is studied. In particular, we show that there exists a finite range of noise intensities…

This paper is concerned with fully discrete finite element methods for approximating variational solutions of nonlinear stochastic elastic wave equations with multiplicative noise. A detailed analysis of the properties of the weak solution…

Stochastic approximation is a class of algorithms that update a vector iteratively, incrementally, and stochastically, including, e.g., stochastic gradient descent and temporal difference learning. One fundamental challenge in analyzing a…

First-order methods are often analyzed via their continuous-time models, where their worst-case convergence properties are usually approached via Lyapunov functions. In this work, we provide a systematic and principled approach to find and…

The paper deals with the problem of long-time asymptotic behaviour of solutions for classes of ODEs and PDEs, perturbed by stationary noises. The latter are not assumed to be $\delta$-correlated in time, so that the evolution in question is…

Intrinsic instability of trajectories characterizes chaotic dynamical systems. We report here that trajectories can exhibit a surprisingly high degree of stability, over a very long time, in a chaotic dynamical system. We provide a detailed…

This paper continues the study of [11, 13] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a…

We present a stability analysis of the standard nonautonomous systems type for a recently introduced generalized Lane-Emden equation which is shown to explain the presence of some of the structures observed in the atomic spatial…

We present an analytical study of a nonlinear oscillator subject to an additive Ornstein-Uhlenbeck noise. Known results are mainly perturbative and are restricted to the large dissipation limit (obtained by neglecting the inertial term) or…