Stochastics and Dynamics

With the crucial estimation of random Stieltjes-type convolution, the Lyapunov–Perron method allows us to establish the existence of invariant dynamics for stochastic evolution equation with non-dense domain. In this paper, we develop the conditions for $C^k$-smoothness of center manifold for such equation with linear multiplicative noise, unstable manifold and stable set for the one with nonlinear multiplicative noise as well. Our result may apply to a broad range of stochastic systems with non-dense domain beyond Brownian noise.

A finite-state Markov chain is introduced in the noise terms of the three-dimensional stochastic Navier–Stokes equations in order to allow for transitions between two types of multiplicative noises. We call such systems as stochastic Navier–Stokes equations with Markov switching. To solve such a system, a family of regularized stochastic systems is introduced. For each such regularized system, the existence of a unique strong solution (in the sense of stochastic analysis) is established by the method of martingale problems and pathwise uniqueness. The regularization is removed in the limit by obtaining a weakly convergent sequence from the family of regularized solutions, and identifying the limit as a solution of the three-dimensional stochastic Navier–Stokes equation with Markov switching.

The intermittency phenomenon is the occurrence of very high but rare peaks in the density, which despite their rarity influence the asymptotic behavior of the underlying system. Mathematically this can be characterized with the asymptotics of moments. In this paper, we show the existence of intermittency phenomena for SDEs with interaction with dissipative coefficients by showing uniform convergence of their Lyapunov exponents.

This paper concerns the large deviations regime and the consequent solution of the Kramers problem for a two-time scale stochastic system driven by a common jump noise signal perturbed in small intensity $𝜀>0$ and with accelerated jumps by intensity $\frac{1}{𝜀}$. We establish Freidlin–Wentzell estimates for the slow process of the multiscale system in the small noise limit $𝜀\to 0$ using the weak convergence approach to large deviations theory. The core of our proof is the reduction of the large deviations principle to the establishment of a stochastic averaging principle for auxiliary controlled processes. As consequence we solve the first exit time/exit locus problem from a bounded domain containing the stable state of the averaged dynamics for the family of the slow processes in the small noise limit.

We study two types of one-dimensional stochastic Burgers equations: Burgers equations with stochastic fluxes and Burgers equations perturbed by transport noise. Using the methods of stochastic characteristics and the characteristics perturbed by noise, respectively, we obtain the probabilities for the occurrence of shocks before and after a given time. Compared with the deterministic equations, in case of small initial data the occurrence time of shocks for Burgers equations with stochastic fluxes will be delayed with large probability, but this conclusion is only true for Burgers equations perturbed by the transport noise with large initial data.

It is known that a tempered exponential dichotomy (which is the version of nonuniform exponential dichotomy for random dynamical systems) can be described by admissibility of a pair of function classes defined in terms of Lyapunov norms. Moreover, for MET-systems (i.e. random dynamical systems which satisfy the assumptions of the Multiplicative Ergodic Theorem) over an ergodic base system, tempered exponential dichotomy can be described by admissibility of a pair of function spaces which are defined in terms of the original norm. In present paper, we find an admissibility description of tempered exponential dichotomies for MET-systems in terms of new output and input classes. These classes can be regarded as a critical case of admissibility classes in the known result. Finally, we use our new pair of admissible spaces to prove the roughness of tempered exponential dichotomies for parametric MET-systems and give a Hölder continuous dependence of the associated projections on the parameter.

This paper is concerned with the general theme of relating the Large Deviation Principle (LDP) for the invariant measures of stochastic processes to the associated trajectorial LDP. It is shown that if the trajectorial deviation function has certain structure and if the invariant measures are exponentially tight, then the LDP for the invariant measures is implied by the trajectorial LDP, no other properties of the stochastic processes in question being material. As an application, we obtain an LDP for the stationary distributions of jump diffusions. Methods of large deviation convergence and idempotent probability play an integral part.

In this paper, we study a large deviation principle for a reflected stochastic partial differential equation on infinite spatial domain. A new sufficient condition for the weak convergence criterion proposed by Matoussi, Sabbagh and Zhang [A. Matoussi, W. Sabbagh and T.-S. Zhang, Large deviation principles of obstacle problems for quasilinear stochastic PDEs, Appl. Math. Optim. 83(2) (2021) 849–879] plays an important role in the proof.