Spring 2017

Consider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.

This talk will present a result on existence and uniqueness of solution of Ito type stochastic differential equation ${\scriptsize dx(t)=b(t, x(t))dt+\sigma(t,x(t)) d B(t)}$, where ${\scriptsize B(t)}$ is a fractional Brownian motion of Hurst parameter ${\scriptsize H>1/2}$ and ${\scriptsize dB(t)}$ is the Ito differential defined by using Wick product or divergence operator. The coefficients ${\scriptsize b}$ and ${\scriptsize \sigma}$ are random and anticipative. Using the relationship between the Ito and pathwise integrals we first write the equation as a stochastic differential equation involving pathwise integral plus a Malliavin derivative term. To handle this Malliavin derivative term the equation is then further reduced to a system of characteristic equations without Malliavin derivative, which is then solved by a careful analysis of Picard iteration, with a new technique to replace the Gronwall lemma which is no longer applicable. The solution of this system of characteristic equations is then applied to solve the original Ito stochastic differential equation up to a positive random time. In special linear and quasilinear cases the global solutions are proved to exist uniquely.

The ${\scriptsize L^p}$ boundedness of certain Littlewood-Paley functionals that arise from Lévy processes are investigated. These results are applied to a new class of Fourier multiples which give ${\scriptsize L^p}$ bounds of solutions to non-local operators, including the fractional Laplacian. The latter has been extensively studied in recent years by researchers in analysis, probability and PDE.

Stochastic differential equations arise in modeling of complex biological and environmental systems with uncertainties. Such uncertain sources could come from initial conditions, boundary conditions, forcing terms, and model parameters. Experience suggests that such uncertainties often play an important role in quantifying the performance of complex systems. Therefore, stochastic differential equations need to be treated as a core element in modeling, simulation and optimization of complex systems. The field of uncertainty quantification (UQ) by solving stochastic differential equations has received an increasing amount of attention recently. Extensive research efforts have been devoted to it and many novel numerical techniques have been developed. These techniques aim to solve stochastic differential equations. In this talk, I will present an efficient numerical method for solving stochastic differential equations - generalized polynomial chaos and some effective new ways of dealing with the discontinuities and high random dimensions. We will illustrate the main idea of our developed numerical algorithms for solving stochastic differential equations using groundwater flow in an aquifer problem.

Stochastic differential equations arise in modeling of complex biological and environmental systems with uncertainties. Such uncertain sources could come from initial conditions, boundary conditions, forcing terms, and model parameters. Experience suggests that such uncertainties often play an important role in quantifying the performance of complex systems. Therefore, stochastic differential equations need to be treated as a core element in modeling, simulation and optimization of complex systems. The field of uncertainty quantification (UQ) by solving stochastic differential equations has received an increasing amount of attention recently. Extensive research efforts have been devoted to it and many novel numerical techniques have been developed. These techniques aim to solve stochastic differential equations. In this talk, I will present an efficient numerical method for solving stochastic differential equations - generalized polynomial chaos and some effective new ways of dealing with the discontinuities and high random dimensions. We will illustrate the main idea of our developed numerical algorithms for solving stochastic differential equations using groundwater flow in an aquifer problem.

This talk will highlight--at a level accessible to graduate students--some ways that generating functions can be utilized in probability theory and in the study of the asymptotic behavior of sequences. I will "chalk and talk," so that the pace is reasonable, and so we can have a little bit of discussion about the ways that GF's are broadly applicable. To underscore the use of generating functions in my own research, I will also discuss a couple of problems that I am working on at present, related to the precise asymptotic growth of sequences, especially those in which oscillations play a key role.

Reaction diffusion equations (RDE) are an important and popular tool for modeling complex spatial-temporal patterns including Turing patterns, traveling waves and periodic switching. These models, however, ignore the stochasticity and discreteness of many complex systems in nature. Recognizing these discrepancies, scientists are developing individual-based models for model selection purposes. The latter models are sometimes studied under the framework of interacting particle systems (IPS) by mathematicians, who prove scaling limit theorems to connect various IPS with RDE. In this talk, I will present some new limiting objects including stochastic partial differential equations (SPDE) on metric graphs and coupled SPDE. These SPDE not only interpolate between IPS and RDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and novel duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of certain population dynamics. In particular, I will present rigorous results about the lineage dynamics of a biased voter model introduced by Hallatschek and Nelson (2007).

Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. We have made the rigorous connection between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss recent progress on understanding fluctuations in quantum systems, and a couple of quantum central limit theorems. Joint work with Gerard Ben Arous, Michael Brannan, Benjamin Schlein, and Gigliola Staffilani.

Random sup-measures are natural objects when investigating extremes of stochastic processes. A new family of stationary and self-similar random sup-measures are introduced. The representation of this family of random sup-measures is based on intersections of independent stable regenerative sets. These random sup-measures arise in limit theorems for extremes of a family of stationary infinitely divisible processes with long-range dependence. The talk will first review of the role of random sup-measures in extremal limit theorems, and then focus on the representation of the new family of random sup-measures. Joint work with Gennady Samorodnitsky.

We study a class of conditional mean-field SDEs (CMFSDEs, for short), in which the reference dynamics also involves the conditional law of the solution, hence closed-loop in nature. We investigate the well-posedness of the CMFSDEs in two particular forms, along with a mean-field type stochastic control problems with partial observations and an extended form of the so-called Kyle-Back strategic insider trading equilibrium problem. In the former we study a McKean-Vlasov type stochastic control problem, with added features of path-dependence and partial observation, and prove the corresponding Pontryagin's Stochastic Maximum Principle. In the latter we seek a rigorous theoretical basis for the general Kyle-Back strategic insider trading equilibrium model, in the case when the insider is allowed to have dynamic information of the underlying asset rather than only the static one. Our results will tie some loose ends of the heuristic arguments in the literature of this problem. This talk is based on the joint works with Rainer Buckdahn, Juan Li, Yonghui Zhou, and Rentao Sun.

In this talk, some of recent developments on the parabolic Anderson model, which is described by a class of linear stochastic partial differential equations (SPDEs) with multiplicative Gaussian noise, will be reviewed. Then we consider fractional parabolic equation of the form ${\scriptsize \frac{\partial u}{\partial t}=-(-\Delta)^{\alpha/2}u+u\dot W(t,x)}$, where ${\scriptsize -(-\Delta)^{\alpha/2}}$ with ${\scriptsize \alpha\in(0,2]}$ is a fractional Laplacian and ${\scriptsize \dot W}$ is a Gaussian noise colored in space and time. The precise moment Lyapunov exponents of the Stratonovich solution and the Skorohod solution are obtained. The result is based on the work joint with X. Chen, Y. Hu and X. Song.

For a zero-sum stochastic game which does not satisfy the Isaacs condition, we provide a value function representation for an Isaacs-type equation whose Hamiltonian lies in between the lower and upper Hamiltonians, as a convex combination of the two. For the general case (i.e. the convex combination is time and state dependent) our representation amounts to a random change of the rules of the game, to allow each player at any moment to see the other player's action or not, according to a coin toss with probabilities of heads and tails given by the convex combination appearing in the PDE. If the combination is state independent, then the rules can be set all in advance, in a deterministic way. This means that tossing the coin along the game, or tossing it repeatedly right at the beginning leads to the same value. The representations are asymptotic, over time discretizations. Space discretization is possible as well, leading to similar results. The talk is based on joint work with Daniel Hernandez-Hernandez.

Operator-scaling random fields satisfy an anisotropic self-similarity property, which extends the classical self-similarity property. Hence they generalize the fractional Brownian field, which is the most famous isotropic Gaussian self-similar random field. In this talk we focus on operator Gaussian random fields with stationary increments and with variograms defined as anisotropic deformations of the fractional Brownian field variogram. Stein has proposed a fast and exact method of simulation of fractional Brownian fields, which is based on a locally stationary periodic representation and uses the fast Fourier transform. We adapt this method to get a fast and exact synthesis of the class of operator-scaling fields under consideration.

Abstract coming soon.