Organizer: David Lipshutz. Click on a title to display the associated abstract. Non-standard seminar times and locations are emphasized in **bold**.

Tuesday, February 4, 2014, 11 a.m., Room 110:

Maria Gordina, University of Connecticut.

Stochastic analysis and geometric functional inequalities.

Our starting point is the heat flow operator which is used to describe a number of physical phenomena. Recall that its generator is the Laplace operator. A probabilistic point of view comes from a path integral representation of the heat flow for stochastic differential equations driven by a Brownian motion. In particular, a well-known connection between the spectrum of the Laplacian and the speed of heat diffusion leads to several functional inequalities such as Poincare, Nash etc. The geometry of the underlying space plays an important role in such an analysis. Another example of a functional inequality is the log-Sobolev inequality which is used to describe entropic convergence of the heat flow to an equilibrium. The talk will review recent advances in the field including elliptic and hypo-elliptic settings over both finite- and infinite-dimensional spaces.

Tuesday, February 11, 2014, 11 a.m., Room 110:

Prasad Tetali, Georgia Institute of Technology.

*Displacement convexity of entropy and related inequalities on graphs*.

In recent years, Optimal Transport and its link with the Ricci curvature in Riemannian geometry attracted a considerable amount of attention. While a lot is now known in the Riemannian setting (and more generally in geodesic spaces), little is known so far in discrete spaces (such as finite graphs or finite Markov chains), with the notable exception of some notions of (discrete) Ricci curvature proposed recently by several authors.

Unfortunately there is not yet a satisfactory (universally agreed upon) resolution. In particular, the notions of transport inequalities, interpolating paths on the measure space, displacement convexity of entropy, are yet to be properly introduced, analyzed and understood in discrete spaces. The chief aim of the lecture is to motivate the problem and mention some recent developments in this direction by the speaker and his collaborators, including Nathael Gozlan, Cyril Roberto, Paul-Marie Samson. Time permitting, relation to classical sumset inequalities will also be mentioned.

Tuesday, March 4, 2014, 11 a.m., Room 110:

Ioannis Karatzas, Columbia University & INTECH.

Competing Brownian particle systems.

We consider systems of diffusing particles, whose local characteristics are assigned in terms of their ranks. We construct the associated multidimensional diffusions and discuss questions of strength and pathwise uniqueness for the stochastic equations that realize them. Multi-dimensional, and perturbed, versions of the Tanaka equation arise, and are studied in some detail -- as are triple (or higher-order) collisions, which play here an important role. When we allow for elastic collisions between particles, the ranked versions of the resulting diffusive motions arise as scaled limits of asymmetric, exclusion-type interacting particle systems. (Survey of joint works with E. R. Fernholz, T. Ichiba, S. Pal, V. Prokaj and M. Shkolnikov.)

Tuesday, April 1, 2014, 11 a.m., Room 110:

Lea Popovic, Concordia University.

Stochastic dynamics in intracellular systems.

Cellular functions in biological organisms comprise of complex interactions of many intracellular species such as proteins, DNA, mRNA molecules, and others. There are many aspects of intracellular interactions in which stochasticity plays an integral role. Sources of stochasticity in cells are multiple - some are intrinsic to the system due to inherent randomness of biochemical reactions between the species, while others are extrinsic and are due to variations in the cellular composition, noisy cellular division mechanisms, etc. In this talk I will present a number of different mathematical results that characterize phenomena observed in intracellular systems for which stochasticity is responsible. I will discuss stochastic switching, heterogeneous systems, and rigorous approximations of models of intracellular stochastic dynamics.

Tuesday, April 15, 2014, 11 a.m., Room 110:

Hye-Won Kang, University of Maryland, Baltimore County.

*Multiscale approximations in stochastic biochemical networks*.

I will talk about stochastic modeling and approximations of chemical reaction networks. Stochastic effects may play an important role in biological and chemical processes in case the copy number of some species involved in the system is small. Chemical reaction networks can be modeled in terms of continuous-time Markov jump processes. Since chemical reaction networks are generally large in size and since systems may be nonlinear, it is hard to analyze large-sized systems and simulation of those systems requires long time. In this talk, I will suggest multiscale approximation methods which help to reduce the network complexity using various scales in species numbers and reaction rate constants. A limiting model with simple structure is derived in each time scale of interest, which is used to approximate the behavior of the full model during a specific time interval. Then, asymptotic behavior of the error between the full model and the limiting model is approximated. This is a joint work with Thomas G. Kurtz and Lea Popovic.

Tuesday, April 22, 2014, 11 a.m., Room 110:

Zhen-Qing Chen, University of Washington.

*Anomalous diffusions and fractional order differential equations*.

Anomalous diffusion phenomenon has been observed in many natural systems, from the signalling of biological cells, to the foraging behaviour of animals, to the travel times of contaminants in groundwater. I will first discuss the connections between anomalous diffusions and differential equations of fractional order, and then present some recent results on heat kernels for non-local operators of fractional order.

Tuesday, April 29, 2014, 11 a.m., Room 110:

Martin Hairer, Warwick University.

Taming diverging stochastic PDEs.

Several very natural classes of nonlinear stochastic partial differential equations are ill-posed when considered in a "naïve" way. Very recently, a general theory was developed, allowing to tame these divergencies by compensating them with suitable counterterms, in a way reminiscent of the renormalisation procedures known from quantum field theory. I will give an overview of some of the main results obtained so far, as well as a glimpse into the techniques of proof.

Tuesday, May 13, 2014, 11 a.m., Room 110:

Harry Chang, US Army.

*Ergodicity and statistical stability of quantum Markov semigroups*.

In this talk, we give a brief review and survey recent results on invariance, ergodicity, and statistical stability of normal quantum states for a general quantum Markov semigroups of operators on a C*- or von Neumann algebra. It is known that GNS representation has been used heavily in establishing mean ergodicity results for quantum states in recent years. In this talk, we extend some tools in classical Markov processes and provide an alternate and yet more useful and more intuitive approach to these problems.

**Wednesday**, May 28, 2014, 11 a.m., Room 110:

Philippe Robert, INRIA.

*An asymptotic study of stochastic networks with failures*.

The qualitative behavior of a large scale storage network of non-reliable file servers is investigated. In such systems a fraction of the processing capacity can be used to duplicate files on new servers when necessary. Due to random losses, with probability 1 all files will be lost eventually. When the size of the network gets large it is shown, via a simplified model, that there is a critical value for the value of the mean number of files per server such that below this quantity, the system stays away from the undesirable absorbing state in a quasi-stationary state where most files have a maximum number of copies. Above this value, the network quickly loses a significant number of files until some quasi-equilibrium is reached. When the network is stable, it is shown that, with a convenient set of time scales, the evolution of the network towards degradation can be described in terms of a stochastic averaging principle.