Purdue Probability Seminar, Fall 2022

Assuming the heat kernel on a doubling Dirichlet metric measure space has sub-Gaussian bounds, we prove an asymptotically sharp spectral upper bound on the tail probability of exit times of the associated diffusion process from a domain. As a consequence, we can show that the supremum of the mean exit time over all starting points is finite if and only if the bottom of the spectrum is positive. Moreover, the spectral upper bound on the survival probability implies a bound for the Hot Spots constant for Riemannian manifolds. We also prove a new characterization of sub-Gaussian heat kernel upper bounds that gives a partial answer to a conjecture of Grigor'yan, Hu and Lau. Our results apply to many interesting examples including Carnot groups, sub-Riemannian manifolds with transverse symmetries, and fractals. This talk is based on joint works with Rodrigo Bañuelos, Hugo Panzo and Jing Wang.

We consider the stochastic behavior of a class of local U-statistics of Poisson processes−which include subgraph and simplex counts as special cases, and amounts to quantifying clustering behavior−for point clouds lying in diverging halfspaces. We provide limit theorems for distributions with light and heavy tails. In particular, we prove finite-dimensional central limit theorems. In the light tail case we investigate tails that decay at least as slow as exponential and at least as fast as Gaussian. These results also furnish as a corollary that U-statistics for halfspaces diverging at different angles are asymptotically independent, and that there is no asymptotic independence for heavy-tailed densities. Using state-of-the-art bounds derived from recent breakthroughs combining Stein's method and Malliavin calculus, we quantify the rate of this convergence in terms of Kolmogorov distance. We also investigate the behavior of local U-statistics of a Poisson Process conditioned to lie in diverging halfspace and show how the rate of convergence in the Kolmogorov distance is faster the lighter the tail of the density is.

Sinai's random walk is a standard model of 1-dimensional random walk in random environment. Brox diffusion is its continuous counterpart, that is a Brownian diffusion in a Brownian environment. The convergence in law of a properly rescaled version of Sinai’s walk to Brox diffusion has been established 20 years ago. In this talk, I will explain a strategy which yields the convergence of Sinai’s walk to Brox diffusion thanks to an explicit coupling. This method, based on rough paths techniques, opens the way to rates of convergence in this demanding context. Notice that I'll try to give a maximum of background about the objects I'm manipulating, and will keep technical considerations to a minimum.

A Littlewood-Offord type inequality is an upper bound on the number of sub-sums of a set of vectors that can fall in a given convex set. On the real line this problem has a simple probabilistic interpretation as an upper bound on the concentration of a weighted sum of iid Bernoulli(1/2) variables. Erdos gave a celebrated sharp upper bound in this problem using Sperner theory when the convex set in question was a singleton. Recently Fox, Kwan, and Sauerman asked what bounds could be derived for general iid Bernoulli(p). In this talk we will investigate a generalization of this question in the language of information theory, and give a sharp upper bound on the \(\alpha\)-Renyi entropy of a sum of "weighted" iid Bernoulli(p) when \(\alpha \geq\) 2, and time permitting discuss its relationship to a discrete "min-entropy power" inequality.

Recently, a precise intermittency for the hyperbolic Anderson model \(\frac{\partial^2 u}{\partial t^2}(t,x) = \Delta u(t,x) + \dot{W}(x)u(t,x) \) has been established in Ito-Skorohod regime. In this talk, we discuss the same problem in Stratonovich regime. Our approach provides new ingridient on representation and comp- utation for Stratanovich moments.

The work is based on a collaborative project with Hu, Yaozhong.

In this talk we discuss the excursion sets and the location and type of the critical points of isotropic Gaussian random fields satisfying certain conditions over high thresholds. We show that for these Gaussian random fields, when the threshold tends to infinity and the searching area expands with a matching speed, both the location of the excursion sets and the location of the local maxima above the threshold converge weakly to a Poisson point process. We will further discuss the possibility to approximate these locations when the threshold is high but not extremely high, by studying the local behavior of the critical points above the threshold of the random field. It is shown that a pair of close critical points in R^n, both above a high threshold, predominantly consist of one local maxima and one saddle point with index n-1. This is a joint work with Paul Marriott and Weinan Qi.

The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to have stationary distributions modulo additive constants given by Brownian motion with drift. In this talk, we will discuss some results-in-progress which show that these distributions are totally ergodic and present some progress toward the conjecture that these are the only ergodic stationary distributions of the KPZ equation. The talk will discuss our coupling of Hopf-Cole solutions, which enables us to study the KPZ equation started from any measurable function valued initial condition. Through this coupling, we give a sharp characterization of when such solutions explode, show that all non-explosive functions become instantaneously continuous, and then study the problem of ergodicity on a natural topology on the space of non-explosive continuous functions (mod constants) in which the equation defines a Feller process. We show that any ergodic stationary distribution on this space is either a Brownian motion with drift or a process of a very peculiar form which will be described in the talk. Based on joint works with Tom Alberts, Firas Rassoul-Agha, and Timo Seppäläinen.

The stationary horizon (SH) is a stochastic process of coupled Brownian motions indexed by their real-valued drifts. It was first introduced Busani as the diffusive scaling limit of the Busemann process of exponential last-passage percolation. It was independently discovered as the Busemann process of Brownian last-passage percolation by the Seppäläinen and Sorensen. We show that SH is the unique invariant distribution and an attractor of the KPZ fixed point under conditions on the asymptotic spatial slopes. It follows that SH describes the Busemann process of the directed landscape. This gives control of semi-infinite geodesics simultaneously across all initial points and directions. The countable dense set \(\Xi\) of directions of discontinuity of the Busemann process is the set of directions in which not all geodesics coalesce and in which there exist at least two distinct geodesics from each initial point. This creates two distinct families of coalescing geodesics in each \(\Xi\) direction. In \(\Xi\) directions, the Busemann difference profile is distributed as Brownian local time. We describe the point process of directions \(\xi\in\Xi\) and spatial locations where the \(\xi\pm\) Busemann functions separate. Based on joint work with Ofer Busani and Timo Seppäläinen.

It is common in probability theory and statistics to study distributional convergences of sums of random variables conditioned on another such sum. In this talk I will present a novel approach using Stein's method for exchangeable pairs that allows to derive a conditional central limit theorem of the form $(X_n|Y_n = k)$ with explicit rate of convergence as well as its extensions to multidimensional setting. We will apply these results to particular models including pattern count in a random binary sequence and subgraph count in Erdos-Renyi random graph. This talk is based on joint work with Partha S. Dey.

Stochastic reaction-diffusion equations are important mathematical models for spreading behaviors in physical and biological systems. Fundamental questions include quantifying the combined effect of noise and spatial interactions on the dynamical behaviors of these systems. While these equations driven by space-time white noise naturally arise as the scaling limit of discrete systems such as interacting particle models, the ill-posedness issue in higher spatial dimensions significantly hinder the analysis and the applications of these equations. In this talk, I will discuss about methods to compute the extinction probabilities, the quasi-stationary distributions, the asymptotic speeds and other long-time behaviors for stochastic reaction-diffusion equations of Fisher-KPP type. Importantly, we consider these equations on general metric graphs that flexibly parametrize the underlying space. This enables us to bypass the ill-posedness issue of these equations, while still assess the impact of space and stochasticity on the coexistence and the genealogies of interacting populations. The latter applications are useful in spatial population genetics. Based on joint work with Rick Durrett, Wenqing Hu, Greg Terlov, and ongoing work with Yifan (Johnny) Yang and Oliver Tough.

With applications in the analysis of political districtings, Markov chains have become and essential tool for studying contiguous partitions of geometric regions. Nevertheless, there remains a dearth of rigorous results on the mixing times of the chains employed for this purpose. In this talk we'll discuss a sub-exponential bound on the mixing time of the Glauber dynamics chain for the case of bounded-size contiguous partition classes on certain grid-like classes of graphs.

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 \((x_0, t_0)\), 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 spacetime 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 :
\(\lim_{r\to\infty} \lim_{t\to\infty} P(\text{Red leads at time } t|\mathcal{F}_r)=L\)
Here \(\mathcal{F}_r\) is the sigma-field generated by the red and blue processes up through time r.