RIT on Stochastics and Probability Archives for Academic Year 2020


Statistical properties of uniformly expanding maps: decay of correlations and the CLT

When: Mon, October 7, 2019 - 11:00am
Where: Kirwan Hall 0301
Speaker: Alex Blumenthal (UMD) -
Abstract: Hyperbolicity is the infinitesimal dynamical mechanism responsible for chaotic behavior of trajectories x_n, n \geq 0. Hyperbolic systems tend to have the property that for Holder regular observables \phi, the time-n observations \phi(x_n) tend to behave like weakly correlated random variables and satisfy (i) fast decay of correlations and (ii) a central limit theorem (the latter only for "typical" observables). In my two talks at the stochastics RIT, I will prove properties (i) and (ii) in the slightly simpler setting of uniformly expanding maps, describing briefly how these ideas pertain to uniformly hyperbolic systems.

Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms

When: Mon, October 21, 2019 - 11:00am
Where: Kirwan Hall 0301
Speaker: Jacob Bedrossian (UMD) -
Abstract: We will discuss the 1989 paper of the same name by Baxendale.

Relative entropy and stochastic flows of diffeomorphisms

When: Mon, October 28, 2019 - 11:00am
Where: Kirwan Hall 0301
Speaker: Jacob Bedrossian (UMD) -
Abstract: Continuation of discussion of Baxendale's 1989 paper of the same name

Long-Time Influence of Small Perturbations and Motion on the Simplex of Invariant Probability measures

When: Mon, November 4, 2019 - 11:00am
Where: Kirwan Hall 0301
Speaker: Mark Freidlin (UMD) -
Abstract: I will present a general approach to various asymptotic problems concerning long time influence of small deterministic or stochastic perturbations of dynamical systems, semi-flows defined by evolutionary PDEs, and diffusion processes. Long time evolution of the perturbed system can be described by a motion on the
simplex M of invariant probability measures of the non-perturbed system. The
simplex is, in a sefanse, an analogy to the action coordinates for completely
integrable Hamiltonian systems, and the measure corresponding to a point of the simplex M characterizes the generalized action coordinates.
As examples of such an approach, regularized averaging principle, random
perturbations of systems with asymptotically stable attractors, wave fronts in
Reaction-Diffusion Equations, homogenization problems, and some other
asymptotic problems for PDEs will be considered.
The limiting motion on the simplex M, in an appropriate time scale, is determined
by laws of large numbers, diffusion approximation, or by the limit theorems for
large deviations.

TBD

When: Mon, November 25, 2019 - 11:00am
Where: 0301
Speaker: Sandra Cerrai (UMD) -


Long - time asymptotics for PDE's with a small parameter and a motion on related simplex of invariant probability measures

When: Mon, December 2, 2019 - 11:00am
Where: 0301
Speaker: Mark Freidlin (UMD) -
Abstract: I will present a general approach to various asymptotic problems concerning long time influence of small deterministic or stochastic perturbations of dynamical systems, semi-flows defined by evolutionary PDEs, and diffusion processes. Long time evolution of the perturbed system can be described by a motion on the simplex M of invariant probability measures of the non-perturbed system. The simplex is, in a sense, an analogy to the action coordinates for completely integrable Hamiltonian systems, and the measure corresponding to a point of the simplex M characterizes the generalized action coordinates.
As examples of such an approach, regularized averaging principle, random
perturbations of systems with asymptotically stable attractors, wave fronts in
Reaction-Diffusion Equations, homogenization problems, and some other
asymptotic problems for PDEs will be considered. The limiting motion on the simplex M, in an appropriate time scale, is determined by laws of large numbers, diffusion approximation, or by the limit theorems for large deviations.