Probability Archives for Academic Year 2017


Quasi-linear parabolic PDE's with singular inputs

When: Wed, September 27, 2017 - 11:00am
Where: Kirwan Hall 3206
Speaker: Scott Andrew Smith (Max Plank Institute, Leipzig) -
Abstract: The present talk is concerned with quasi-linear parabolic equations which are ill-posed in the classical distributional sense. In the semi-linear context, the theory of regularity structures provides a solution theory which applies to a large class of equations with suitably randomized inputs. The quasi-linear setting has seen recent advances, but a general theory remains open. We will present some partial progress in this direction based on joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.


Edgeworth expansions for weakly dependent random variables

When: Wed, October 11, 2017 - 11:00am
Where: Kirwan Hall 3206
Speaker: Kasun Fernando (UMD, Mathematics) - https://www.math.umd.edu/~abkf/
Abstract: We discuss sufficient conditions that guarantee the existence of asymptotic expansions for the CLT for weakly dependent random variables including observations arising from sufficiently chaotic dynamical systems like piece-wise expanding maps and strongly ergodic Markov chains. We primarily use spectral techniques to obtain the results. This is a joint work with Carlangelo Liverani.

Well-posedness for Stochastic Continuity Equations with Rough Coefficients

When: Wed, November 1, 2017 - 11:00am
Where: Kirwan Hall 3206
Speaker: Sam Punshon-Smith (University of Maryland, Mathematics Department) - http://www2.math.umd.edu/~punshs/
Abstract: According to the theory of Diperna/ Lions, the continuity equation associated to a Sobolev (or BV) vector field with bounded divergence has a unique weak solution in L^\infty. Under the addition of a white in time stochastic perturbation to the characteristics of the continuity equation, it is known that existence and uniqueness can be obtained under a relaxation of the regularity conditions and the requirement of bounded divergence. In this talk, we consider the general stochastic continuity equation associated to an Itô diffusion with irregular drift and diffusion coefficients. This equation has applications in the modeling of turbulent flows and complex fluids. We will discuss two types of probabilistically strong, analytically weak solutions to this equation and give conditions under which the equation has a unique solution. Using these results, we outline a proof of uniqueness of solutions to the stochastic transport equation with drift in L^q_t L^p_x, satisfying the sub-critical Ladyzhenskaya–Prodi–Serrin criterion 2/q + d/p < 1. Connections to existence, uniqueness and regularity of the stochastic flow associated to the SDE will be discussed.

Challenges and Implications of Autocorrelation in Animal Movement Studies

When: Wed, November 8, 2017 - 11:00am
Where: Kirwan Hall 3206
Speaker: Bill Fagan (Department of Biology, University of Maryland) - http://biology.umd.edu/william-fagan.html
Abstract: The interface between basic ecology and applied mathematics is robust, and results from this interface are often critical to effective conservation. In this Probability/Applied Math seminar, I will focus on one part of this interface whereby ecological observations and datasets have created new opportunities for a variety of mathematical tools and approaches. For instance, datasets derived from efforts to track the movements of wild animals (e.g., using GPS-satellite devices) have presented new applications for research on stochastic processes. As technology has improved and datasets have expanded, autocorrelations in both animal position and animal velocity have become key features that can no longer be ignored. Instead there is a need to embrace the information content of the autocorrelation structure of tracking datasets and use that information to obtain biological understanding. Examples include applications of semi-variograms, which identify multiple movement modes and solve the sampling rate problem for tracking data, and autocorrelated kernel density estimators, which provide valuable new approaches for delineating animal home ranges.

Burgers Turbulence: a Model Case for the Kolmogorov Theory

When: Wed, December 6, 2017 - 11:00am
Where: Kirwan Hall 3206
Speaker: Alexandre Boritchev (University of Lyon) - http://math.univ-lyon1.fr/~boritchev/
Abstract: The Kolmogorov 1941 theory (K41) is the starting point for all
models of turbulence. In particular, K41 and corrections to it provide estimates of
small-scale quantities such as increments and energy spectrum for a 3D turbulent flow.
However, because of the well-known difficulties involved in studying 3D turbulent flows, there are no rigorous results confirming or infirming those predictions.

Here, we consider a well-known simplified model for 3D turbulence, the 1D or
multi-dimensional potential Burgers equation, with spatially smooth stochastic forcing.
We give sharp estimates for small-scale quantities such as increments and energy spectrum, and then we say a few words about the dynamical features of this equation, and in particular the speed of convergence to the stationary measure.

Weak Solutions of a Stochastic Landau-Lifshitz-Gilbert Equation Driven by Pure Jump Noise

When: Fri, December 8, 2017 - 10:30am
Where: Kirwan Hall 3206
Speaker: Zdzislaw Brzezniak (University of York) - https://www.york.ac.uk/maths/staff/zdzislaw-brzezniak/
Abstract: In this work we study a stochastic three-dimensional
Landau-Lifschitz-Gilbert equation perturbed by pure jump noise in the
Marcus canonical form. We show existence of weak martingale solutions
taking values in a two-dimensional sphere $\mathbb{S}^2$ and discuss
certain regularity results. The construction of the solution is based
on the classical Faedo-Galerkin approximation, the compactness method
and the Jakubowski version of the Skorokhod Theorem for nonmetric
spaces. This is a joint work with Utpal Manna (Trivandrum).

The Stochastic Gray Scott system

When: Fri, December 8, 2017 - 11:30am
Where: Kirwan Hall 3206
Speaker: Erika Hausenblas (Montanuniversität Leoben) - http://institute.unileoben.ac.at/amat/personal.html
Abstract: Reaction and diffusion of chemical species can produce a variety of patterns, reminiscent of those often seen in nature. The Gray Scott system is a coupled equation of reaction diffusion type, modelling these kind of patterns. Depending on the parameter, stripes, waves, cloud streets, or sand ripples may appear.

These systems are the macroscopic model of microscopic dynamics.
Here, in the derivation of the equation the random fluctuation of the molecules are neglected.
Adding a stochastic noise, the inherit randomness
of the microscopic behaviour is modelled. In particular, we add a time homogenous spatial Gaussian random field with given spectral measure.

In the talk we present our main result about the stochastic Gray Scott system.
In addition, we introduce and explain an algorithm for its numerical approximation by a Operator splitting method. Finally we present some examples illustrating the dynamical behaviour of the stochastic Gray Scott system.

Small random perturbations unlock hyperbolicity

When: Wed, April 18, 2018 - 11:00am
Where: Kirwan Hall 3206
Speaker: Alex Blumenthal (UMD) - http://math.umd.edu/~alexb123/
Abstract: For smooth dynamical systems, hyperbolicity (strong expansion and contraction at infinitesimal level) is the main mechanism by which deterministic dynamical systems have chaotic asymptotic regimes. Uniformly hyperbolic systems, those exhibiting hyperbolicity on all of phase space or on the entirety of a stable attractor, are well-known to be asymptotically chaotic. The hyperbolicity mechanism is quite sensitive, however, in the sense that even for systems which are “mostly” hyperbolic except for a small exceptional set in phase space, the problem of determining the asymptotic regime (chaotic versus `ordered') is notoriously challenging.

In this talk I will discuss some of the inherent difficulties (coexistence phenomena, cone twisting) in studying “mostly” hyperbolic systems. Then, I will put forward the view that the addition of some IID randomness at each timestep has the effect of “unlocking” hyperbolicity, greatly simplifying the study of these systems. I will discuss results on classes of 1D and 2D models which are “mostly” hyperbolic, including multimodal maps of the circle and the well-studied Chirikov standard map family on the torus.

This work is joint with Yun Yang, Lai-Sang Young and Jinxin Xue.