Stochastics and Dynamics

We consider one-dimensional stochastic differential equations driven by white noises and Poisson random measure. We introduce new techniques based on local time prove new results on pathwise uniqueness and comparison theorems. Our approach is very easy to handle and do not need any approximation approach. Similar equations without jumps were studied in the same context by [8, 12] and other authors.

After defining non-Gaussian Lévy processes for two-sided time, stochastic differential equations with such Lévy processes are considered. Solution paths for these stochastic differential equations have countable jump discontinuities in time. Topological equivalence (or conjugacy) for such an Itô stochastic differential equation and its transformed random differential equation is established. Consequently, a stochastic Hartman–Grobman theorem is proved for the linearization of the Itô stochastic differential equation. Furthermore, for Marcus stochastic differential equations, this topological equivalence is used to prove the existence of global random attractors.

We consider the stochastic non-Newtonian fluids defined on a two-dimensional Poincaré unbounded domain, and prove that it generates an asymptotically compact random dynamical system. Then, we establish the existence of random attractor for the corresponding random dynamical system. Random attractor is invariant and attracts every pullback tempered random set.

Given a controlled diffusion with jumps, it is shown under some conditions that there exists a Markov controlled diffusion which has the same cost under any criterion which only depends on the one-dimensional distributions. This result is then used to prove that the finite dimensional marginal distribution of a controlled diffusion with jumps at a prescribed set of time instants can also be attained by using a control from a much smaller class of controls called "nearly Markov controls", extending the work of Borkar [4] to the discontinuous setting.

The goal of this paper is to define the coset topological pressure for sub-additive potentials via separated sets on a compact metric group. Analogues of basic properties for topological pressure hold. This study also reveals a variational principle for the coset topological pressure. The process of the proof is quite similar to that of Cao, Feng and Huang's approximations, but the analysis needs more techniques of ergodic theory and topological dynamics.

We study the behavior of solutions to a Schrödinger equation with large, rapidly oscillating, mean zero, random potential with Gaussian distribution. We show that in high dimension d > 𝔪, where 𝔪 is the order of the spatial pseudo-differential operator in the Schrödinger equation (with 𝔪 = 2 for the standard Laplace operator), the solution converges in the L² sense uniformly in time over finite intervals to the solution of a deterministic Schrödinger equation as the correlation length ε tends to 0. This generalizes to long times the convergence results obtained for short times and for the heat equation in [2]. The result is based on the decomposition of multiple scattering contributions introduced in [6]. In dimension d < 𝔪, the random solution converges to the solution of a stochastic partial differential equation; see [1, 13].

The aim of this paper is to investigate the distribution of a continuous homopolymer in the presence of an attractive finitely supported potential. The most intricate behavior can be observed if we simultaneously vary two parameters: the temperature, which approaches the critical value, and the length of the polymer, which tends to infinity. As the main result, we identify the distributions that appear in the limit (after a diffusive scaling of the original polymer measures) and depend on the relation between the two parameters.

We show that the Gibbs states (known from [10] to be unique) of Hölder continuous potentials and random distance expanding maps coincide with relative equilibrium states of those potentials, proving in particular that the latter exist and are unique. In the realm of conformal expanding random maps, we prove that given an ergodic (globally) invariant measure with a given marginal, for almost every fiber the corresponding conditional measure has dimension equal to the ratio of the relative metric entropy and the Lyapunov exponent. Finally we show that there is exactly one invariant measure whose conditional measures are of full dimension. It is the canonical Gibbs state.