Probability Archives for Fall 2017 to Spring 2018

An SDE approximation for stochastic differential delay equations with colored state-dependent noise

When: Wed, November 9, 2016 - 2:00am
Where: 1313.0
Speaker: Scott Hottovy (USNA) -
Abstract: In this talk I will introduce a stochastic differential delay equation with state-dependent colored noise which arises from a noisy circuit experiment. In the experimental paper, a small delay and correlation time limit was performed by using a Taylor expansion of the delay. However, a time substitution was first used to obtain a good match with experimental results. I will discuss how this limit can be proved without the use of a Taylor expansion by using a theory of convergence of stochastic processes developed by Kurtz and Protter. To obtain a necessary bound, the theory of sums of weakly dependent random variables is used. This analysis leads to the explanation of why the time substitution was needed in the previous case.

Synchronization of coupled dynamical systems: Relating network structure to dynamics

When: Wed, December 7, 2016 - 2:00pm
Where: Kirwan Hall 1313
Speaker: Georgi Medvedev (Drexel University) -
Abstract: We will discuss dynamics in two representative models of coupled
dynamical systems
on graphs: chaotic maps and excitable oscillators forced by small
white noise. For both models, we analyze synchronization and explain what
structural features of the network favor synchronization. For the second model,
we also describe other dynamical regimes such as spontaneous oscillations and
formation of clusters. The role of network topology in shaping each of these
patterns will be explained.

Fluctuations in stochastic homogenization

When: Mon, January 9, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Yu Gu (Stanford University) -
Abstract: To analyze multiscale PDEs and their asymptotic behaviors, a formal expansion can be used to construct correctors and prove the convergence. It is nevertheless unclear whether those terms appearing in the expansion indeed represent the first and higher order errors. In other words, the expansion sometimes stays only on the formal level.
In this talk, we will discuss the classical problem of stochastic homogenization of elliptic operators in divergence form, and identity the first and higher order random fluctuations. It turns out that the formal expansion may or may not indicate the right answer depending on the scales on which we make the measurement. Part of the talk is based on joint work with Jean-Christophe Mourrat.

Randomness in convection-diffusion problems

When: Mon, January 23, 2017 - 2:00pm
Where: Kirwan Hall 1308
Speaker: Martina Hofmanova (Technical University Berlin) -
Abstract: In this talk, I will consider quasi-linear parabolic PDEs subject to stochastic or rough perturbation and explain how various assumptions on coefficients and roughness of the noise naturally ask for different notions of solution with different regularity properties and different techniques of the proofs. On the one hand, the problems under consideration will be stochastic second order parabolic PDEs with noise smooth in space, either with a possible degeneracy in the leading order operator, where only low regularity holds true, or under the uniform ellipticity assumption, where arbitrarily high regularity can be proved under suitable assumptions on the coefficients. On the other hand, I will discuss a rough pathwise approach towards these problems based on tools from paracontrolled calculus.


When: Mon, March 13, 2017 - 11:00am
Where: Kirwan Hall 3206
Speaker: Andrey Sarantsev (University of California, Santa Barbara) -
Abstract: Systems of interacting Brownian particles with rank-dependent drift and diffusion coefficients attracted considerable attention recently, due to their applications in finance and otherwise. We survey recent work on these systems by the speaker and others.

Stationary diffusions on a space of interval partitions

When: Wed, March 15, 2017 - 11:00am
Where: Kirwan Hall 3206
Speaker: Noah Forman (University of Washington) -
Abstract: We construct two diffusions on a space of partitions of the unit interval. These are stationary with the law of the complement of the zero sets of Brownian motion and Brownian bridge, respectively. Our construction is based on decorating the jumps of a spectrally positive L\'evy process with independent continuous excursions. The processes of ranked interval lengths of our partitions belong to a two parameter family of diffusions introduced by Ethier and Kurtz (1981) and Petrov (2009). These are continuum limits of up-down Markov chains on Chinese restaurant processes. Our construction works towards building a diffusion on the space of real trees whose existence has been conjectured by Aldous.

Rare event simulation via importance sampling for linear SPDEs

When: Wed, March 29, 2017 - 11:00am
Where: Kirwan Hall 3206
Speaker: Michael Salins (Boston University) -
Abstract: We develop provably efficient importance sampling methods for estimating rare events for linear stochastic partial differential equations exposed to small noise. We use a spectral method to identify a one-dimensional linear span where the rare event likely occurs and we project our change of measure onto that direction. The scheme we develop works well for a wide variety of different intensities of noise, time horizons, and finite dimensional Galerkin approximations of the infinite dimensional system. Simulations support the theoretical results.

On Stochastic Brinkman-Forchheimer Anisotropic 3D Navier-Stokes Equations

When: Wed, April 26, 2017 - 11:00am
Where: Kirwan Hall 3206
Speaker: Annie Millet (Université Paris 1 Panthéon Sorbonne) -