Department of

Mathematics


Seminar Calendar
for Probability Seminar events the year of Thursday, September 14, 2017.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, March 7, 2017

2:00 pm in 347 Altgeld Hall,Tuesday, March 7, 2017

Gromov-Hausdorff -Prohorov convergence of vertex cut-trees of n-leaf Galton-Watson trees

Hui He (Beijing Normal University)

Abstract: We study the vertex cut-tree of Galton-Watson trees conditioned to have n leaves. This notion is a slight variation of Dieuleveut's vertex cut-tree of Galton-Watson trees conditioned to have n vertices. Our main result is a joint Gromov-Hausdorff -Prohorov convergence in the finite variance case of the Galton-Watson tree and its vertex cut-tree to Bertoin and Miermont's joint distribution of the Brownian CRT and its cut-tree. The methods also apply to the infinite variance case, but the problem to strengthen Dieuleveut's and Bertoin and Miermont's Gromov-Prohorov convergence to Gromov-Hausdorff-Prohorov remains open for their models conditioned to have n vertices. This is a joint work with Matthias Winkel.

Tuesday, March 14, 2017

2:00 pm in 347 Altgeld Hall,Tuesday, March 14, 2017

On random multiplicative functions

Joseph Najnudel (University of Cincinnati)

Abstract: In this talk, we give a presentation of random multiplicative functions on the integers and their main properties. At the end of the talk, we state some results, proven in a preprint recently submitted, on the behaviour and the independence properties of such functions taken at consecutive integers.

Tuesday, March 28, 2017

2:00 pm in 347 Altgeld Hall,Tuesday, March 28, 2017

Hardy-Stein identity and square functions for pure jump Levy processes

​Daesung Kim (Purdue University)

Abstract: In the recent paper of R. Banuelos, K. Bogdan and T. Luks (2016), the authors prove $L^{p}$ bounds of square function for non-local operators and then applied them to prove $L^{p}$ bounds for certain Fourier multipliers. The key to the proof in that paper is a Hardy-Stein identity which is proved from properties of the semigroup. Using Ito’s formula for processes with jumps, we give a simple direct proof of the Hardy-Stein identity. Also, we extend the proof given in that paper to non-symmetric Levy-Fourier multipliers.

Tuesday, April 4, 2017

2:00 pm in 347 Altgeld Hall,Tuesday, April 4, 2017

Statistical physics of exponential random graphs

Mei Yin (University of Denver )

Abstract: The exponential random graph model has been a topic of continued research interest. The past few years especially has witnessed (exponentially) growing attention in exponential models and their variations. Emphasis has been made on the variational principle of the limiting free energy, concentration of the limiting probability distribution, phase transitions, and asymptotic structures. This talk with focus on the phenomenon of phase transitions in large exponential random graphs. The main techniques that we use are variants of statistical physics but the exciting new theory of graph limits, which has rich ties to many parts of mathematics and beyond, also plays an important role in the interdisciplinary inquiry. Some open problems and conjectures will be presented.

Friday, April 7, 2017

3:00 pm in 343 Altgeld Hall ,Friday, April 7, 2017

Time fractional equations and probabilistic representation

Zhen-Qing Chen (University of Washington)

Abstract: Time-fractional diffusion equation can be used to model the anomalous diffusions exhibiting subdiffusive behavior, due to particle sticking and trapping phenomena. In this talk, I will discuss general fractional-time derivatives and probabilistic representation of solutions of the corresponding parabolic equations in terms of the corresponding inverse subordinators with or without drifts. An explicit relation between occupation measure for Markov processes time-changed by inverse subordinator in open sets and that of the original Markov process in the open set will also be given.

Tuesday, April 11, 2017

2:00 pm in 347 Altgeld Hall,Tuesday, April 11, 2017

Concentration of measure without independence: a unified approach via the martingale method

Maxim Raginsky (UIUC)

Abstract: The concentration of measure phenomenon may be summarized as follows: a function of many weakly dependent random variables that is not too sensitive to any of its individual arguments will tend to take values very close to its expectation. This phenomenon is most completely understood when the arguments are mutually independent random variables, and there exist several powerful complementary methods for proving concentration inequalities, such as the martingale method, the entropy method, and the method of transportation inequalities. The setting of dependent arguments is much less well understood. This talk, based on joint work with Aryeh Kontorovich, will focus on the martingale method for deriving concentration inequalities without independence assumptions. In particular, I will show how the machinery of so-called Wasserstein matrices together with the Azuma-Hoeffding inequality can be used to recover and sharpen several known concentration results for nonproduct measures.

Tuesday, April 18, 2017

2:00 pm in 347 Altgeld Hall,Tuesday, April 18, 2017

Path transformations of excursion landscape

Ju-Yi Yen (University of Cincinnati)

Abstract: In this talk, we study the process obtained from a Brownian bridge after excising all the excursions below the waterline level which reach zero. Three variables of interest are the maximum of this process, the value where this maximum is attained, and the total length of the excursions which are excised. Our analysis relies on some interesting transformations connecting Brownian path fragments and the 3-dimensional Bessel process.

Tuesday, April 25, 2017

2:00 pm in 347 Altgeld Hall,Tuesday, April 25, 2017

A new family of random sup-measures

Yizao Wang (University of Cincinnati)

Abstract: 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 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.

Tuesday, May 2, 2017

2:00 pm in 347 Altgeld Hall,Tuesday, May 2, 2017

Random data Cauchy theory for power type nonlinear wave equations

Dana Mendelson (University of Chicago)

Abstract: In this talk, I will discuss the random data Cauchy theory for some power type nonlinear wave equations. Local well-posedness for these equations is by now well understood for initial data of subcritical or critical regularities, but techniques break down for initial data in the supercritical regime. In recent years, probabilistic methods have been used to investigate the behavior of solutions in regimes where deterministic techniques fail. I will present an almost sure global existence result in the supercritical regime for these equations, and a recent result on scattering for the energy critical equation in 4D with randomized radial data. I will discuss the main aspects of the proof, in particular, the randomization procedure for initial data in Sobolev spaces of low regularity, some new large deviation estimates, and energy estimates for a forced wave equation. This talk is based on several joint works with Jonas Luhrmann, and work with Ben Dodson and Jonas Luhrmann.

Tuesday, September 5, 2017

2:00 pm in 347 Altgeld Hall,Tuesday, September 5, 2017

Spectral heat content for Levy processes

Hyunchul Park (SUNY at New Paltz)

Abstract: In this talk, we study a short time asymptotic behavior of spectral heat content for Levy processes. The spectral heat content of a domain D can be interpreted as the amount of heat if the initial temperature on D is 1 and temperature outside D is identically 0 and the motion of heat particle is governed by underlying Levy processes. We study spectral heat content for arbitrary open sets with finite Lebesgue measure in a real line under some growth condition on the characteristic expo- nents of the L ́evy processes. We observe that the behavior is very different from the classical heat content for Brownian motions. We also study the spectral heat content in general dimensions when the processes are of bounded varia- tion. Finally we prove that asymptotic expansion of spectral heat content is stable under integrable perturbation when heat loss is sufficiently large. This is a joint work with Renming Song and Tomasz Grzywny.

Tuesday, September 12, 2017

2:00 pm in 347 Altgeld Hall,Tuesday, September 12, 2017

On the fast convergence of random perturbations of the gradient flow

Wenqing Hu (Missouri S&T)

Abstract: We consider in this talk small random perturbations (of multiplicative noise type) of the gradient flow. We rigorously prove that under mild conditions, when the potential function is a Morse function with additional strong saddle condition, the perturbed gradient flow converges to the neighborhood of local minimizers in O(ln(\eps^{−1})) time on the average, where \eps>0 is the scale of the random perturbation. Under a change of time scale, this indicates that for the diffusion process that approximates the stochastic gradient method, it takes (up to logarithmic factor) only a linear time of inverse stepsize to evade from all saddle points and hence it implies a fast convergence of its discrete--time counterpart.

Tuesday, September 19, 2017

2:00 pm in 347 Altgeld Hall,Tuesday, September 19, 2017

Derivative formula for Mean-field SDEs with jumps

Yulin Song (Nanjing University and University of Illinois)

Abstract: By using Malliavin calculus for jump processes, we study the Bismut type derivative formula for Mean-field stochastic differential equations driven by L\'evy processes. Both of the derivatives with respect to a fixed initial value $x\in R^d$ and the ones with respect to an initial distribution are considered.