Spring 2019
Time & Location: All talks are on Wednesdays in Dinwiddie 102 at 3:00 PM unless otherwise noted.
Organizer: Scott McKinley and Swati Patel
January 11
Influence of Receptor Recharge on the Statistics of Captured Particles
Greg Handy | University of Utah
Abstract:
We consider a setup where particles are released into a domain and diffuse freely. Part of the boundary is absorbing, where the particles can escape the domain, another part is reflecting. The rest of boundary consists of capture regions that switch between being reflecting and absorbing. After capturing a particle, the capture region becomes reflecting for an exponentially distributed amount of time. This non-zero recharge time correlates the particles' paths, complicating the mathematical analysis of this system. We are interested in the distribution of the number of particles that are captured before they escape.
Our results are derived from considering our system in several ways: as a full spatial diffusion process with recharging traps on the boundary; as a continuous-time Markov process approximating the original system; and lastly as a system of ODEs in a mean-field approximation. Considering the full spatial diffusion process, we prove that the total expected number of the captured particles has an upper-bound of the order of log n. We then apply our approximations to investigate time courses for the expected number and higher ordered statistics of captured particles. We find that the amount of variation observed in the total number of captured particles varies non-monotonically with the mean recharge time. Lastly, we combine these results together to predict stochastic properties of intracellular signals resulting from receptor activation.
February 13
Diffusive search for diffusing receptors
Chris Miles | NYU
Abstract:
Cells send and receive signals in the form of diffusing particles that search for target sites called receptors. This mechanism has received great theoretical interest for over 40 years, but a more recent fact that receptors themselves diffuse along a surface has largely been neglected. This raises the natural question: does the target diffusing help or hurt the ability of a diffusing particle to locate it? We'll modify a classical PDE model into a PDE with stochastic boundary conditions, which we'll study using matched asymptotic analysis and Monte Carlo simulations.
March 13
A non-central limit theorem on heavy-tailed chaos
Ray Bai | Institution
Abstract:
We introduce the partial-sum limit theorems for a class of stationary sequences exhibiting both heavy tails and long-range dependence. The stationary sequences are constructed using heavy-tailed multiple stochastic integrals and conservative null dynamical systems. The limits are represented by multiple stables integrals, where the integrands involve the local times of the intersections of stationary stable regenerative sets. This is a joint work with Takashi Owada and Yizao Wang.
March 20
Recent issues and discoveries on stochatically modeled reaction networks
Jinsu Kim | UC Irvine
Abstract:
A reaction network is a graphical configuration that describes an interaction between species (molecules). If the abundances of the network system is small, then the randomness inherent in the molecular interactions is important to the system dynamics, and the abundances are modeled stochastically as a jump by jump fashion continuous-time Markov chain. One of challenging issues facing researchers who study biological systems is the often extraordinarily complicated structure of their interaction networks. Thus, how to characterize network structures that induce characteristic behaviors of the system dynamics is one of the major open questions in this literature. In this talk, I will provide recent issues and discoveries on stochastically modeled reaction networks. Specifically, we will focus on existence of stationary distributions and the convergence rate for the process to its stationary distribution.
March 27
Application of large deviations to genetic evolution of bacterial populations
Ilya Timofeyev | University of Houston
Abstract:
Radical shifts in the genetic composition of large cell populations are rare events with quite low probabilities, which direct numerical simulations generally fail to evaluate accurately. We develop an applicable large deviations framework for a class of Markov chains used to model genetic evolution of bacterial populations. We illustrate this framework by computing the most likely evolutionary paths describing emergence of genotypes with lower fitness in realistic parameter settings.
April 3
Stochastic processes with semi-long range dependence
Farzad Sabzikar | Iowa State University
Abstract:
Stochastic processes with long range dependence correlation have been proven useful in many areas from engineering to science in both theory and applications. This class includes fractional Brownian motion, fractional Gaussian noise, and fractional ARIMA time series. One of the main properties of long range dependence is the fact that the spectral density is unbounded at the origin. However, in many applications, data fit with this spectral density model only up to a low frequency cutoff, after which the observed spectral density remains bounded. In this talk, we present a novel modification of these models that involves tempering the power law correlation function with an exponential. This results in a tempered fractional Brownian motion, tempered fractional Gaussian noise, and tempered fractional ARIMA time series. These processes have semi-long range dependence: Their autocovariance function resembles that of a long memory model for moderate lags but eventually diminishes exponentially fast according to the presence of a decay factor governed by a tempering parameter. Several applications of these new models in finance, geophysics, turbulence will be presented. Finally, some theoretical problems related to tempered processes will be discussed.
