Stochastics and Dynamics

We prove a conditional local limit theorem for discrete-time fractional Brownian motions (dfBm) with Hurst parameter $\frac{3}{4}<H<1$. Using results from infinite ergodic theory, it is then shown that the properly scaled occupation time of dfBm converges to a Mittag-Leffler distribution.

Stochastic periodic type solution is a powerful tool for studying qualitative analysis of stochastic dynamical systems. In this paper, we will establish the bounded solutions, stationary solutions, periodic solutions, almost periodic solutions, almost automorphic solutions for stochastic linearly coupled complex cubic-quintic Ginzburg–Landau equations under suitable conditions. The main novelty of this paper is dealing with cubic nonlinear terms and the quintic nonlinear terms which are not Lipschitz. We overcome this difficulty by the semigroup approach, stochastic analysis techniques, energy estimate method and refined inequality technique.

We study, in this paper, a stochastic version of a coupled Allen–Cahn–Navier–Stokes model in a two-dimensional (2D) bounded domain. The model consists of the Navier–Stokes equations (NSEs) for the velocity, coupled with a Allen–Cahn model for the order (phase) parameter. We prove the existence and the uniqueness of a variational solution.

In this paper, we consider two classes of backward stochastic differential equations (BSDEs). First, under a Lipschitz-type condition on the generator of the equation, which can also be unbounded, we give sufficient conditions for the existence of a unique solution pair. The method of proof is that of Picard iterations and the resulting conditions are new. We also prove a comparison theorem. Second, under the linear growth and continuity assumptions on the possibly unbounded generator, we prove the existence of the solution pair. This class of equations is more general than the existing ones.

In this paper, we investigate the averaging principle for SDEs of neutral type driven by $G$-Brownian motion. The solutions of convergence in the sense of $p$th moment and convergence in capacity between standard form and the corresponding averaged form are considered. Two examples are presented to demonstrate the applications of the averaging principle.

Let $f:{𝕊}^1\to {𝕊}^1$ be a $C^{2+𝜖}$ expanding map of the circle and let $v:{𝕊}^1\to ℝ$ be a $C^{1+𝜖}$ function. Consider the twisted cohomological equation $v=\alpha \circ f-Df\cdot \alpha ,$ which has a unique bounded solution $\alpha$. We show that $\alpha$ is either $C^{1+𝜖}$ or continuous but nowhere differentiable. If $\alpha$ is nowhere differentiable then the Newton quotients of $\alpha$, after an appropriated normalization, converges in distribution (with respect to the unique absolutely continuous invariant probability of $f$) to the normal distribution. In particular, $\alpha$ is not a Lipschitz continuous function on any subset with positive Lebesgue measure.

We present a general form of renormalization operator ${ℛ}$ acting on potentials $V:{\{0,1\}}^{\mathbb{N}}\to ℝ$. We exhibit the analytical expression of the fixed point potential $V$ for such operator ${ℛ}$. This potential can be expressed in a natural way in terms of a certain integral over the Hausdorff probability on a Cantor type set on the interval [0,1]. This result generalizes a previous one by Baraviera, Leplaideur and Lopes where the fixed point potential $V$ was of Hofbauer type.

For the potentials of Hofbauer type (a well-known case of phase transition) the decay is like $n^{-\gamma }$, $\gamma >0$.

Among other things we present the estimation of the decay of correlation of the equilibrium probability associated to the fixed potential $V$ of our general renormalization procedure. In some cases we get polynomial decay like $n^{-\gamma }$, $\gamma >0$, and in others a decay faster than $n{e}^{-\sqrt{n}}$, when $n\to \infty$.

The potentials $g$ we consider here are elements of the so-called family of Walters’ potentials on ${\{0,1\}}^{\mathbb{N}}$ which generalizes a family of potentials considered initially by Hofbauer. For these potentials some explicit expressions for the eigenfunctions are known.

In a final section we also show that given any choice $d_n\to 0$ of real numbers varying with $n\in \mathbb{N}$ there exists a potential $g$ on the class defined by Walters which has a invariant probability with such numbers as the coefficients of correlation (for a certain explicit observable function).

We provide a particle picture representation for the non-symmetric Rosenblatt process and for Hermite processes of any order, extending the result of Bojdecki, Gorostiza and Talarczyk in [4]. We show that these processes can be obtained as limits of certain functionals of a system of particles evolving according to symmetric stable Lévy motions. In the case of $k$-Hermite processes the corresponding functional involves $k$-intersection local time of symmetric stable Lévy processes.