Abstract: I will discuss a classical example featuring a metastable
behavior: finite-dimensional diffusion processes in the vanishing
noise limit. Exponential estimates have been introduced fifty years
ago by Freidlin and Wentzell. Recent developments in potential theory
and variational convergence allowed a refinement of those results. I
will focus on the non-reversible case, with motivations coming from
MCMC and open problems in the context of wKAM theory. The talk is
based on recent papers with G.DiGesu (TU Wien), C.Landim (IMPA) and
I.Seo (UC Berkeley/Seoul NU).
Abstract: I will talk about extending my earlier work on the vanishing noise limit of diffusions in noisy heteroclinic networks to longer time scales. In this field, the results are based on sequential analysis of exit locations and exit times for neighborhoods of unstable equilibria. The new results on exit times and the emergent hierarchical structure are joint with Zsolt Pajor-Gyulai.
Abstract: Diffusion processes are closely related to PDEs in the sense that the transition probability of diffusion processes is the fundamental solution to the corresponding parabolic equation. In this talk, I will present the Aronson type estimate for a family of parabolic equations with singular divergence-free drift in Lebesgue spaces. All these results rely critically on the divergence-free condition, which was motivated for applications in fluid dynamics.
Abstract: I will discuss diffusion in the high contrast, highly packed composites in which inclusions are irregularly (randomly) distributed in a hosting medium so that a significant fraction of them may not participate in the conducting spanning cluster.
Abstract: I will discuss the problem of characterizing exit time and exit shape asymptotics from a domain of attraction for a stochastically perturbed partial differential equation (SPDE). This analysis requires proving that the sample paths of the SPDE satisfy certain large deviations principles that are uniform over bounded subsets of initial conditions (in a specified function space). Budhiraja, Dupuis, and Maroulas (2008) demonstrated that a weak-convergence approach can be used to prove uniform large deviations that are uniform over compact sets of initial conditions, but compact sets in infinite dimensional function spaces are too degenerate to be useful for these problems. I will discuss recent advances that allow us to use similar methods to prove uniform large deviations principles over (non-compact) bounded sets and in some cases we can prove uniformity over the entire function space.
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