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

  Speaker: Alex Blumenthal (UMD) -
  

  When: Mon, October 7, 2019 - 11:00am
  Where: Kirwan Hall 0301
  

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  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

  Speaker: Jacob Bedrossian (UMD) -
  

  When: Mon, October 21, 2019 - 11:00am
  Where: Kirwan Hall 0301
  

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  Abstract: We will discuss the 1989 paper of the same name by Baxendale.
  
• Relative entropy and stochastic flows of diffeomorphisms

  Speaker: Jacob Bedrossian (UMD) -
  

  When: Mon, October 28, 2019 - 11:00am
  Where: Kirwan Hall 0301
  

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  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

  Speaker: Mark Freidlin (UMD) -
  

  When: Mon, November 4, 2019 - 11:00am
  Where: Kirwan Hall 0301
  

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  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.