Spring 2017
Tuesdays in REC 315 from 3:304:20pm, unless otherwise noted.
Please send comments and suggestions to the seminar organizer, Samy Tindel.
Please send comments and suggestions to the seminar organizer, Samy Tindel.
Date 
Speaker 
Title 

1/10/17 
Toby Johnson
New York University 
GaltonWatson fixed points, tree automata, and interpretations
Abstract
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 GaltonWatson 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 GaltonWatson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.

1/17/17 
Yaozhong Hu
University of Kansas 
Ito stochastic differential equations
driven by fractional Brownian motions of Hurst parameter ${\small H>1/2}$
Abstract
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is then solved by a careful analysis of Picard iteration, with a new technique to replace the Gronwall lemma which is no longer applicable.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.

1/24/17 
Rodrigo Bañuelos
Purdue University 
LittlewoodPaley estimates for Lévy processes
Abstract
The ${\scriptsize L^p}$ boundedness of certain LittlewoodPaley 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 nonlocal operators, including the fractional Laplacian. The latter has been extensively studied in recent years by researchers in analysis, probability and PDE.

1/31/17 
Guang Lin
Purdue University 
Fast numerical methods for solving stochastic differential equations I
Abstract
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.

2/7/17 
Guang Lin
Purdue University 
Fast numerical methods for solving stochastic differential equations II
Abstract
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.

2/14/17 
Mark Ward
Purdue University 
Applications of generating functions in probability and asymptotics
Abstract
This talk will highlightat a level accessible to graduate studentssome 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.

2/21/17 
Louis Fan
University of Wisconsin 
Particle representations for deterministic and stochastic reactiondiffusion equations
Abstract
Reaction diffusion equations (RDE) are an important and popular tool for modeling complex spatialtemporal 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 individualbased 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 timeasymptotic 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).

2/28/17 
Kay Kirkpatrick
UIUC 
On the validity of a large deviation principle for some nonlinear SPDEs with rough noise.
This is a Math colloquium, at 4:30 in Math 175 Abstract
Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called BoseEinstein condensation (BEC), that behaves like a giant quantum particle. We have made the rigorous connection between the physics of the microscopic manybody 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.

3/7/17 
Yizao Wang
University of Cincinnati 
A new family of random supmeasures
Abstract
Random supmeasures are natural objects when investigating extremes of stochastic processes. A new family of stationary and selfsimilar random supmeasures are introduced. The representation of this family of random supmeasures is based on intersections of independent stable regenerative sets. These random supmeasures arise in limit theorems for extremes of a family of stationary infinitely divisible processes with longrange dependence. The talk will first review of the role of random supmeasures in extremal limit theorems, and then focus on the representation of the new family of random supmeasures. Joint work with Gennady Samorodnitsky.

3/21/17 
Jin Ma
University of South California 
Conditional Meanfield SDEs and Some Related Stochastic Optimization Problems
Abstract
We study a class of conditional meanfield SDEs (CMFSDEs, for short), in which the reference dynamics also involves the conditional law of the solution, hence closedloop in nature. We investigate the wellposedness of the CMFSDEs in two particular forms, along with a meanfield type stochastic control problems with partial observations and an extended form of the socalled KyleBack strategic insider trading equilibrium problem. In the former we study a McKeanVlasov type stochastic control problem, with added features of pathdependence and partial observation, and prove the corresponding Pontryagin's Stochastic Maximum Principle. In the latter we seek a rigorous theoretical basis for the general KyleBack 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.

3/28/17 
Jian Song
University of HongKong 
Temporal asymptotics for fractional parabolic Anderson model
Abstract
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.

4/4/17 
Mihai Sîrbu
University of Texas at Austin 
Zerosum stochastic differential games without the Isaacs condition:random
rules of priority and intermediate Hamiltonians
Abstract
For a zerosum stochastic game which does not satisfy the
Isaacs condition, we provide a value function representation for an
Isaacstype 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 HernandezHernandez.

4/18/17 
Céline Lacaux
Université d'Avignon 
Simulation of Gaussian Operatorscaling random fields
Abstract
Operatorscaling random fields satisfy an anisotropic selfsimilarity property, which extends the classical selfsimilarity property. Hence they generalize
the fractional Brownian field, which is the most famous isotropic Gaussian selfsimilar 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 operatorscaling fields
under consideration.

4/25/17 
Paul Bourgade
New York University 
This is a Math colloquium, at 4:30 in Math 175
Abstract
Abstract coming soon.
