Recent Advances in Probability

Organizer: Thomas Selke, Professor of Statistics, Purdue University

Speakers

Steve Lalley, Professor, Department of Statistics and the College, University of Chicago
Takashi Owada, Associate Professor of Statistics, Purdue University
Thomas Selke, Professor fo Statistics, Purdue University

Speaker	Title
Steve Lalley	Critical Branching Brownian Motion with Killing
Abstract: We obtain sharp asymptotic estimates for hitting probabilities of a critical branching Brownian motion in one dimension with killing at 0. We also obtain sharp asymptotic formulas for the tail probabilities of the number of particles killed at 0. In the special case of double-or-nothing branching, we give exact formulas for both the hitting probabili- ties, in terms of elliptic functions, and the distribution of the number of killed particles.
Takashi Owada	Large deviation principle for geometric and topological functionals and associated point processes
Abstract: We prove a large deviation principle for the point process associated to k-element connected components in R^d with respect to the connectivity radii decaying to 0. The random points are generated from a homogeneous Poisson point process or the corresponding binomial point process, so that the connecting radius is of the sparse regime. The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function. If times allows, we also discuss the large deviation principle for the volume of k-nearest neighbor balls. This is joint work with Christian Hirsch.
Thomas Sellke	Limit Theorems for the Frontier of a One-Dimensional Time-Inhomogeneous Branching Diffusion
Abstract: Suppose we have a time-inhomogeneous, one-dimensional branching diffusion environment. In terms of the movement and reproduction of particles, this means that if a particle is placed into the environment at space-time location (x₀, t₀), then the particle moves and reproduces in a strongly Markovian manner along a random continuous path in a way that can depend locally on both space and time. Offspring particles behave in the same way, beginning at their space- time locations of birth, and thereafter independently of other particles currently in the environment. Branching Brownian motion is an example in which both movement and reproduction are spatially and temporally homogeneous. Suppose a red particle is placed into the environment at space-time location (x_r, t_r), initiating a red-particle process. Suppose a blue particle is placed into the environment at space-time location (x_b, t_b), initiating a blue-particle process independent of the red-particle process. Say that Red is in the lead at time t if the right-most particle at time t is red. Then the probability that Red is in the lead at time t converges to a limit l as t goes to infinity, with the value of l of course depending on the initial positions (x_r, t_r) and (x_b, t_b). Furthermore, the conditional probability at time r that Red is in the lead at a time t in the distant future converges to a limiting random variable L : lim lim P {Red leads at time t\|F_r} = L r→∞ t→∞ Here F_r is the sigma-field generated by the red and blue processes up through time r.