Stochastics and Dynamics

Nearly-elastic model systems with one or two degrees of freedom are considered: the system is undergoing a small loss of energy in each collision with the "wall". We show that instabilities in this purely deterministic system lead to stochasticity of its long-time behavior. Various ways to give a rigorous meaning to the last statement are considered. All of them, if applicable, lead to the same stochasticity which is described explicitly. So that the stochasticity of the long-time behavior is an intrinsic property of the deterministic systems.

We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet W^{α, β} with Hurst parameter (α, β) ∈ (0, 1)². When or a central limit theorem holds for the renormalized Hermite variations of order q ≥ 2, while for we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time (1, 1).

In this paper, stochastic flows driven by Kunita-type stochastic differential equations are studied, focusing on support theorems (ST) and large deviation principles (LDP). We establish a new ST and LDP for Brownian flows with respect to a fine Hölder topology. Our approach is based on recent advances in rough paths theory, which is the natural framework for proving ST and LDP. Nevertheless, while rigorous, our presentation stays rather clear from the rough paths technicalities and is accessible for readers not familiar with them. We view the localized Brownian stochastic flow as a projection of the solution of a rough path differential equation implying the ST and LDP. In a second step the results are generalized for the global flow.

We provide conditions which yield a strong law of large numbers for expressions of the form where X(n), n ≥ 0's is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polinomial growth and certain regularity properties and q_i, i > m are positive functions taking on integer values on integers with some growth conditions. Applying these results we study certain multifractal formalism type questions concerning Hausdorff dimensions of some sets of numbers with prescribed asymptotic frequencies of combinations of digits at places q₁(n), …, q_ℓ(n).

In this paper, we study the asymptotic behavior of solutions of a second order stochastic lattice differential equation with multiplicative white noise in weighted spaces. We first provide some sufficient conditions for the existence and uniqueness of solutions, and then establish the existence of tempered random bounded absorbing sets and global random attractors.

The goal of this paper is to solve backward doubly stochastic differential equations (BDSDEs, in short) under weak assumptions on the data. The first part is devoted to the development of some new technical aspects of stochastic calculus related to this BDSDEs. Then we derive a priori estimates and prove the existence and uniqueness of solution in L^p, p ∈ (1, 2), extending the work of Pardoux and Peng (see [12]).

Let (X_n) denote an independent and identically distributed random sequence. Let and M_n = max{X₁, …, X_n} be its partial sum and maximum. Suppose that some of the random variables of X₁, X₂,… can be observed and denote by the maximum of observed random variables from the set {X₁, …, X_n}. In this paper, we consider the joint limiting distribution of and the almost sure central limit theorems related to the random vector . Furthermore, we extend related results to weakly dependent stationary Gaussian sequences.

In earlier work the second author investigated the multifractal q Rényi dimensions of Polish spaces for q ≥ 1. In this paper we complement those results by investigating the multifractal q Rényi dimensions of Polish spaces for q < 1.