## Jonathan Mattingly : Noise induced stabilization of dynamical systems

- Probability ( 208 Views )We investigate an example of noise-induced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell,Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the addition of additive noise in the vertical direction leads to a unique invariant probability measure to which the system converges at a uniform, exponential rate. These facts are established primarily through the construction of a Lyapunov function which we generate as the solution to a sequence of Poisson equations. Unlike a number of other works, however, our Lyapunov function is constructed in a systematic way, and we present a meta-algorithm we hope will be applicable to other problems. We conclude by proving positivity properties of the transition density by using Malliavin calculus via some unusually explicit calculations. arXiv:1111.175v1 [math.PR]

## Maria Gordina : Gaussian type analysis on infinite-dimensional Heisenberg groups

- Probability ( 183 Views )This is a joint work with B.Driver. The groups in question are modeled on an abstract Wiener space. Then a group Brownian motion is defined, and its properties are studied in connection with the geometry of this group. The main results include quasi-invariance of the heat kernel measure, log Sobolev inequality (following a bound on the Ricci curvature), and the Taylor isomorphism to the corresponding Fock space. The latter is a version of the Ito-Wiener expansion in the non-commutative setting.

## James Johndrow : Error Bounds for Approximations of Markov Chains

- Probability ( 140 Views )We give results on the properties of Markov kernels that approximate another Markov kernel. The basic idea is that when the original kernel satisfies a contraction condition in some metric, the long-time dynamics of the two chains -- as well as the invariant measures, when they exist -- will be close in that metric, so long as the approximating kernel satisfies a suitable approximation error condition. We focus on weighted total variation and Wasserstein metrics, and motivate the results with applications to scalable Markov chain Monte Carlo algorithms. This is joint work with Jonathan Mattingly.

## Rick Durrett : Evolutionary Games on the Torus

- Probability ( 117 Views )We study evolutionary games on the torus with N points in dimensions \(d\ge 3\) with matrices of the form \(\bar G = {\bf 1} + w G\), where **1** is a matrix that consists of all 1's, and *w* is small. We show that there are three weak selection regimes (i) \(w \gg N^{-2/d}\), (ii) \(N^{-2/d} \gg w \gg N^{-1}\), and (iii) there is a mutation rate \(\mu\) so that \(\mu \gg N^{-1}\) and \(\mu \gg w\) where in the last case
we have introduced a mutation rate \(\mu\) to make it nontrivial.
In the first and second regimes the rescaled process converges to a PDE and an ODE respectively. In the third, which is the classical weak selection regime of population genetics, we give a new derivation of Tarnita's formula which describes how the
equilibrium frequencies are shifted away from uniform due to
the spatial structure.

## Jasmine Foo : Modeling diversity in tumor populations and implications for drug resistance

- Probability ( 115 Views )In this talk I will discuss a branching process model developed to study intra-tumor diversity (i.e. the variation amongst the cells within a single tumor). This variation arises due to heritable (epi)genetic alterations which can confer changes in cellular fitness and drug response. In the asymptotic (t-> infinity) regime, we study the growth rates of the population as well as some ecological measures of diversity in the tumor. In the second half of the talk I will discuss applications of this model to studying the emergence of drug resistant populations in Chronic Myeloid Leukemia (CML). (Joint work w/K. Leder, J. Mayberry, R. Durrett, F. Michor)

## Ted Cox : Cutoff for the noisy voter model

- Probability ( 112 Views )Given a continuous time Markov Chain \( q(x,y)\) on a
finite set *S*, the associated noisy voter model is the
continuous time Markov chain on \(\{0,1\}^S\) which evolves
by (i) for each two sites x and y in *S*, the state at
site x changes to the value of the state at site
y at rate \( q(x,y) \) and (ii) each site rerandomizes
its state at rate 1. We show that if there is a uniform
bound on the rates \(q(x,y)\) and the corresponding
stationary distributions are ``almost'' uniform, then the
mixing time has a sharp cutoff at time \(\log |S|/2\) with a
window of order 1. Lubetzky and Sly proved cutoff with a
window of order 1 for the stochastic Ising model on
toroids: we obtain the special case of their result for
the cycle as a consequence of our result.

## Hao Shen : Stochastic quantization of gauge theories

- Probability ( 108 Views )Quantum field theory is the theoretical framework for studying fundamental interactions. "Stochastic quantizationÂ? refers to a formulation of quantum field theory as stochastic PDEs. The recent years witnessed interesting progress in understanding solutions of these stochastic PDEs, one of the remarkable examples being Hairer and Mourrat-Weber's results on the Phi^4_3 equation.

In this talk we will discuss stochastic quantization of quantum field theory with gauge symmetries, with focus on an Abelian example but also provide prospects of non-Abelian Yang-Mills theories. We address issues regarding WilsonÂ?s lattice regularization, dynamical gauge fixing, renormalization, Ward identities, and construction of dynamical loop and string observables.

## Paul Bressloff : Stochastic models of intracellular transport: a PDE perspective

- Probability ( 108 Views )The efficient delivery of proteins and other molecular products to their correct location within a cell (intracellular transport) is of fundamental importance to normal cellular function and development. Moreover, the breakdown of intracellular transport is a major contributing factor to many degenerative diseases. There are two major types of transport. (I) Passive diffusion within the cytosol or the surrounding plasma membrane of the cell. Since the aqueous environment (cytosol) of a cell is highly viscous at the length and velocity scales of macromolecules (low Reynolds number), a diffusing particle can be treated as an overdamped Brownian particle where inertial effects are ignored. (II) Active motor-driven transport along polymerized filaments such as microtubules and F-actin that comprise the cytoskeleton. At appropriate length and time scales, active transport can either be modeled as a velocity-jump process or as an advection-diffusion process. In this talk I present various PDE models of active and passive transport within cells. The bulk of the talk will focus on three examples: synaptic democracy and vesicular transport in axons and dendrites; stochastically gated diffusion in bounded domains; cytoneme-based transport of morphogens during embryogenesis. (A cytoneme is a thin actin-rich filament that forms direct contacts between cells and is thought to provide an alternative to diffusion-based morphogen gradient formation.) Other applications include cellular length control, cell polarization, and synaptogenesis in C. elegans.

## Jingchen Liu : Rare-event Analysis and Monte Carlo Methods for Gaussian Processes

- Probability ( 107 Views )Gaussian processes are employed to model spatially varying errors in various stochastic systems. In this talk, we consider the analysis of the extreme behaviors and the rare-event simulation problems for such systems. In particular, the topic covers various nonlinear functionals of Gaussian processes including the supremum norm and integral of convex functions. We present the asymptotic results and the efficient simulation algorithms for the associated rare-event probabilities.