Schedule for: 17w5119 - Stochastic Analysis and its Applications

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider singular stochastic PDEs in simple square domains with the usual Neumann / Dirichlet / mixed boundary data. We show that in some circumstances, renormalisation effects appear in the boundary data and we discuss the significance of this effect.

In this talk we address a question posed several years ago by G. Zaslovski: what is the effect of heavy tails of one-dimensional random potentials on the standard objects of localization theory: Lyapunov exponents, density of states, statistics of eigenvalues, etc. We'll consider several models of potentials constructed by the use of $iid$ random variables which belong to the domain of attraction of the stable distribution with parameter $\alpha<1.$ In order to put our results in context, we'll recall the "regular theory" as presented in Carmona-Lacroix or Figotin-Pastur. We consider the one-dimensional Schr\"{o}dinger operator on the half line with boundary condition: \begin{eqnarray}\label{seqn} H^{\theta_0}\psi(x)=-\psi''(x)+V(x,\omega)\psi(x),\,\psi(0)\cos\theta_0-\psi'(0)\sin\theta_0=0. \end{eqnarray} where for each $x\in [0,\infty),\,V(x,\cdot)$ is a random variable on a basic probability space $(\Omega,\mathcal{F}, P)$ and $\theta_0\in[0,\pi]$ is fixed. Our potentials $V(x,\omega)$ will be piecewise constant, these are the so-called Kr\"{o}nig-Penny type potentials. As opposed to the regular theory, the large tails of the probability distribution of the potential $V$ will lead to random Lyapunov exponents and a different rate of decay of eigenfunctions from the standard case. The talk is based on joint work with S. Molchanov and N. Squartini.

I will describe a simple queueing model and argue that, at the diffusion scale, its state process converges to a Walsh Brownian motion. This is joint work with Asaf Cohen (Haifa U.).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

I will begin by reviewing the general relations that exist between the cover time of graphs by random walk and the Gaussian free field on the graph, and explain the strength and limitations of these general methods. I will then discuss recent results concerning the cover time of the binary tree of depth $n$ by simple random walk, and in particular sharp fluctuation results for the cover time, mirroring those for the maximal displacement of branching random walk; certain barrier estimates for Bessel processes play a crucial role. Finally, I will describe how these technique can be applied to the study of the cover time of the 2-sphere by the Brownian sausage. Based on joint work with David Belius and Jay Rosen.

An infinite sequence of real random variables $(\xi_1, \xi_2, \ldots)$ is said to be rotatable if every finite subsequence $(\xi_1, \ldots, \xi_n)$ has a spherically symmetric distribution. A classical theorem of David Freedman says that $(\xi_1, \xi_2, \ldots)$ is rotatable if and only if $\xi_j = \sigma \eta_j$ for all $j$, where $(\eta_1, \eta_2, \ldots)$ is a sequence of independent standard Gaussian random variables and $\sigma$ is an independent nonnegative random variable. We establish the analogue of Freedman's result for sequences of random variables taking values in arbitrary locally compact, nondiscrete fields other than the field of real numbers or the field of complex numbers. This is joint work with Daniel Raban, a Berkeley undergraduate.

The dimer model on a finite bipartite graph is a uniformly chosen perfect matching, i.e., a set of edges which cover every vertex exactly once. It is a classical model of mathematical physics, going back to work of Kasteleyn and Temeperley/Fisher in the 1960s. A central object for the dimer model is a notion of height function introduced by Thurston, which turns the dimer model into a random discrete surface. I will discuss a series of recent results with Benoit Laslier and Gourab Ray where we establish the convergence of the height function to a scaling limit in a variety of situations. This includes simply connected domains of the plane with arbitrary linear boundary conditions for the height, in which case the limit is the Gaussian free field, and Temperleyan graphs drawn on Riemann surfaces. In all these cases the scaling limit is universal and conformally invariant. A key new idea in our approach is to exploit "imaginary geometry" couplings between the Gaussian free field and SLE.

Let $Z$ be a subordinate Brownian motion in $\mathbf{R}^d$, $d\ge 3$, via a subordinator with Laplace exponent $\phi$. We kill the process $Z$ upon exiting a bounded open set $D\subset {\mathbb R}^d$ to obtain the killed process $Z^D$, and then we subordinate the process $Z^D$ by a subordinator with Laplace exponent $\psi$. The resulting process is denoted by $Y^D$. Both $\phi$ and $\psi$ are assumed to satisfy certain weak scaling conditions at infinity. In this talk, I will present some recent results on the potential theory, in particular the boundary theory, of $Y^D$. First, in case that $D$ is a $\kappa$-fat bounded open set, we show that the Harnack inequality holds. If, in addition, $D$ satisfies the local exterior volume condition, then we prove the Carleson estimate. In case $D$ is a smooth open set and the lower weak scaling index of $\psi$ is strictly larger than $1/2$, we establish the boundary Harnack principle with explicit decay rate near the boundary of $D$. On the other hand, when $\psi(\lambda)=\lambda^{\gamma}$ with $\gamma\in (0,1/2]$, we show that the boundary Harnack principle near the boundary of $D$ fails for any bounded $C^{1,1}$ open set $D$. Our results give the first example where the Carleson estimate holds true, but the boundary Harnack principle does not. We also prove a boundary Harnack principle for non-negative functions harmonic in a smooth open set $E$ strictly contained in $D$, showing that the behavior of $Y^D$ in the interior of $D$ is determined by the composition $\psi\circ \phi$. This talk is based a joint paper with Renming Song and Zoran Vondracek.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We develop an excursion theory for Brownian motion indexed by the Brownian tree, which in many respects is analogous to the classical Ito theory for linear Brownian motion. Each excursion is associated with a connected component of the complement of the zero set of the tree-indexed Brownian motion. Each such connected component is itself a continuous tree, and we introduce a quantity measuring the length of its boundary. The collection of boundary lengths coincides with the collection of jumps of a certain continuous-state branching process. Furthermore, conditionally on the boundary lengths, the different excursions are independent, and we determine their conditional distribution in terms of an excursion measure which is the analog of the Ito measure of Brownian excursions. If time permits, we will describe applications to the random metric spaces called Brownian disks. This is based in part on a joint work in collaboration with C. Abraham.

We study the heat equation on time-dependent metric measure spaces (being a dynamic forward gradient flow for the energy) and its dual (being a dynamic backward gradient flow for the Boltzmann entropy). Monotonicity estimates for transportation distances and for squared gradients will be shown to be equivalent to the so-called dynamical convexity of the Boltzmann entropy on the Wasserstein space which is the define property of super-Ricci flows. Moreover, we show equivalence with monotone coupling property of backward Brownian motion as well as with log Sobolev, local Poincare and dimension free Harnack inequalities.

In this talk, we will discuss heat kernel estimates, parabolic and elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We will present equivalent characterizations of heat kernel estimates, parabolic Harnack inequalities in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincaré inequalities, and we will also establish connections among heat kernel estimates, parabolic Harnack inequalities and elliptic Harnack inequalities. As an application, we will study the quenched invariance principle for random conductance models with long rang jumps of stable-like type. The talk is mainly based on joint work with Zhen-Qing Chen and Takashi Kumagai.

Following Burkholder’s seminal work on sharp martingale inequalities, there has been considerable interest in their applications to various problems in analysis. Unlike many sharp inequalities in analysis and geometry (Sobolev, Log-Sobolev, Hardy-Littlewood-Sobolev, Nash, Housdorff-Young, Isoperimetric, Faber-Krahn, etc.), extremals for the martingale inequalities do not exist and equality is never attained except for trivial cases. Motivated by applications, this talk discusses the nature and stability of the “almost extremals’’ in Burkholder’s inequalities and consequences for certain singular integrals. This talk is based on join work with Adam Adam Oscekowski of the University of Warsaw.

We give sharp two-sided estimates of the semigroup generated by the fractional Laplacian plus the Hardy potential, including the case of the critical constant. This is a joint project with Tomasz Grzywny, Tomasz Jakubowski and Dominika Pilarczyk from WUST.

Hausdorff measure is often used to measure fractal sets. However, there is a more natural quantity, Minkowski content, which more closely matches the scaling limit of discrete counting measures and is closely related to the idea of local time. I will discuss this in the context of several sets for which Chris Burdzy made fundamental contributions: cut points for Brownian paths and outer boundary of two-dimensional Brownian motion. The latter is closely related to the Schramm-Loewner evolution (SLE). I will include joint work with Mohammad Rezaei and recent work with Nina Holden, Xinyi Li, and Xin Sun.

Consider a symmetric game with $n$ players in which each player incurs a cost function and chooses a strategy that depends on its own state and on the state of the other players only through the empirical distribution of their states. The Nash equilibria of such symmetric $n$-player games are hard to analyze or even compute, but their limit, as the number of players goes to infinity, can be characterized in terms of a certain stochastic differential game with a continuum of players, referred to as a mean-field game. We show how properties of solutions to an infinite-dimensional partial differential equation for the value function of the mean-field game can be used to establish central limit theorems and large deviation principles for the sequence of empirical measures of Nash equilibria in $n$-player games. This is joint work with Francois Delarue and Dan Lacker.

We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by \begin{eqnarray*} X^{\varepsilon}_t &=& x_0 + \int_0^t b(X^{\varepsilon}_s, Y^{\varepsilon}_s)ds + \varepsilon^{\alpha}B_t, \label{ex} \\ Y^{\varepsilon}_t &=& y_0 - \frac{1}{\varepsilon} \int_0^t \nabla_yU(X^{\varepsilon}_s, Y^{\varepsilon}_s)ds+ \frac{s(\varepsilon)}{\sqrt{\varepsilon}} W_t,\label{wye} \end{eqnarray*} where $x_0 \in {\mathbb R}^d, y_0 \in {\mathbb R}^m$, $B_t, W_t$ are independent Brownian motions, $b : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}^d$, $U : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}$, and $s : (0, \infty) \rightarrow (0, \infty).$ One observes that there is a time scale separation between $X$ and $Y$. Under suitable assumptions on $b, U$, for $0 < \alpha < \frac{1}{2}$, if $s(\epsilon) \rightarrow 0$ goes to zero at a prescribed slow enough rate then we establish all weak limits points of $X^{\epsilon}$, as $\epsilon \rightarrow 0$, as Fillipov solutions to a differential inclusion. This is joint work with V. Borkar, S. Kumar and R. Sundaresan.

We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph $G$ are either open or closed and refresh their status at rate $\mu$, while at the same time a random walker moves on $G$ at rate 1, but only along edges which are open. On the $d$-dimensional torus with side length $n$, when the bond parameter is subcritical, we determined (with A. Stauffer and J. Steif) the mixing times for both the full system and the random walker. The supercritical case is harder, but using evolving sets we were able (with J. Steif and P. Sousi) to analyze it for p sufficiently large. The critical and moderately supercritical cases remain open.

We prove a quenched invariance principle and local limit theorem for a random walk in an ergodic balanced time dependent environment on the lattice. Our proof relies on the parabolic Harnack inequality for the adjoint operator. This is joint work with X. Guo.

Consider a binary tree with n labeled leaves. Randomly select a leaf for removal and then reinsert it back on an edge selected at random from the remaining structure. This produces a Markov chain on the space of n-leaved binary trees whose invariant distribution is the uniform distribution. David Aldous, who introduced and analyzed this Markov chain, conjectured the existence of a continuum limit of this process if we remove labels from leaves, scale edge- length and time appropriately with n, and let n go to infinity. The conjectured diffusion will have an invariant distribution given by the so-called Brownian Continuum Random Tree. In a series of papers, co-authored with N. Forman, D. Rizzolo, and M. Winkel, we construct this continuum limit. This talk will be an overview of our construction and describe the path behavior of this limiting object.

We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger, $L^{2+\epsilon}$ (rather than $L^2$) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the non-reversible, divergence-free drift case.

Recently Ozawa gave a functional analysis proof of Gromov’s polynomial growth theorem. A key element in the proof is to establish that a group of polynomial growth has Shalom’s property $H_F D$. Ozawa’s work renewed interest in this somewhat mysterious property. We extend Ozawa’s approach to show that a class of locally-finite-by-Z groups have property $H_FD$. As examples we show that there are groups with property $H_FD$, where the speed of simple random walk can follow any prescribed function and all sub-linear harmonic functions are constant. Joint with Jeremie Brieussel.

Following the work of Moser, as well as de Giorgi and Nash, Harnack inequalities have proved to be a powerful tool in PDE as well as in the study of the geometry of spaces. In the early 1990s Grigor'yan and Saloff-Coste gave a characterisation of the parabolic Harnack inequality (PHI). This characterisation implies that the PHI is stable under bounded perturbation of weights, as well as rough isometries. In this talk we prove the stability of the EHI. This is joint work with Mathav Murugan (UBC).

We develop a new approach to studying the asymptotic behavior of fluid models for critically loaded processor sharing queues, using a certain relative entropy. Joint work with Amber Puha.

The Skorokhod map on the half-line has proved to be a useful tool for studying processes with non-negativity constraints. In this work we introduce a measure-valued analog of this map that transforms each element ? of a certain class of c`adl`ag paths that take values in the space of signed measures on [0, ?) to a c\'ad\'ag path that takes values in the space of non-negative measures on $[0, \infty)$ in such a way that for each $x>0$, the path $t\to \zeta_t [0, x]$ is transformed via a Skorokhod map on the half-line, and the regulating functions for different $x>0$ are coupled. We establish regularity properties of this map and show that the map provides a convenient tool for studying queueing systems in which tasks are prioritized according to a continuous parameter. One such priority assignment rule is the well known earliest-deadline-first priority rule. We study it both for the single and the many server queueing systems. We show how the map provides a framework within which to form fluid model equations, prove uniqueness of solutions to these equations and establish convergence of scaled state processes to the fluid model. In particular, for these models, we obtain new convergence results in time-inhomogeneous settings, which appear to fall outside the purview of existing approaches and is essential when studying the EDF policy for many servers queues. Based on Joint work with Rami Atar, Anup Biswas and Kavita Ramanan.

In the area of Dirichlet forms, positive continuous additive functionals (PCAFs), together with the Revuz correspondence between PCAFs and smooth measures, constitute an active subject and play important roles in stochastic analysis. In the case of positivity preserving forms, there is no single measure on the path space, therefore we introduce a notion of Optional measurable positive continuous additive functionals (O-PCAFs). Employing the structure of Optional sigma-field and the structure of Predictable sigma-field, we have established the Revuz correspondence between O-PCAFs and smooth measures. Thus O-PCAFs may play similar roles as PCAFs do in the framework of Dirichlet forms. The talk is based on a joint research with Xian Chen and Xue Peng.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.