Stochastics and Dynamics

Stochastic resonance is an amplification and synchronization effect of weak periodic signals in nonlinear systems through a small noise perturbation. In this paper we study the dynamics of stochastic resonance in a bistable system driven by multiplicative Lévy noise with heavy tails, e.g., $\alpha$-stable Lévy noise. We determine the optimal tuning with respect to a probabilistic synchronization measure for both the jump-diffusion and the reduced two-state Markov chain. These results extend the theory of stochastic resonance to the case of heavy tail Lévy perturbations.

In this paper by using Malliavin calculus we prove derivative formulas of Bismut type for a class of stochastic (functional) differential equations driven by fractional Brownian motions. As applications, the dimensional-free Harnack type inequalities and the strong Feller property are presented.

In this paper, we study the Moderate Deviation Principle for a perturbed stochastic heat equation in the whole space ${ℝ}^d,d\ge 1$. This equation is driven by a Gaussian noise, white in time and correlated in space, and the differential operator is a fractional derivative operator. The weak convergence method plays an important role.

Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a$ and $b$ satisfying $a>-1,$ $-1<b<0,$ $\left|b\right|<1+a$. In this paper, motivated by the asymptotic property

We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized solutions of class $C^1$. The generator $L$ is associated with a Markov process $X$ which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward process is $X$. Since $X$ is a weak solution of the forward SDE, the BSDE appears naturally to be driven by a martingale. In the paper we also discuss the uniqueness of solutions of a BSDE with random terminal time when the driving process is a general càdlàg martingale.

We study harmonic and totally invariant measures in a foliated compact Riemannian manifold. We construct explicitly a Stratonovich differential equation for the foliated Brownian motion. We present a characterization of totally invariant measures in terms of the flow of diffeomorphisms associated to this equation. We prove an ergodic formula for the sum of the Lyapunov exponents in terms of the geometry of the leaves.

We prove that every expanding minimal semigroup action of $C^1$ diffeomorphisms of a compact manifold (resp. $C^{1+\alpha }$ conformal) is robustly minimal (resp. ergodic with respect to the Lebesgue emeasure). We also show how, locally, a blending region yields the robustness of the minimality and implies ergodicity.