Schedule for: 18w2239 - Retreat for Young Researchers in Stochastics

Note: the Lecture rooms are available after 16:00.

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. This should be free.

Beverages and a small assortment of snacks are available in the lounge on a cash honour system.

A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. This should be free; please confirm when you check in.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Co-authors: Alexander E. Holroyd, Avi Levy and Joel Spencer The existential monadic second order (EMSO) language on rooted trees defines a class of properties where only existential, and no universal, quantification over subsets of vertices are allowed, and the root is considered a special symbol. It is straightforward to show that survival (i.e. infiniteness) of rooted trees is expressible as an EMSO sentence. We show that the negation of infiniteness, i.e. finiteness of rooted trees is not expressible as an EMSO sentence. So far, the discussion does not involve probability. After this, we focus on rooted Galton-Watson (GW) trees with Poisson offspring distribution (though our results apply to much more general rooted random trees). Let $P_{\lambda}$ denote the Poisson$(\lambda)$ GW measure. We show that it is not possible to find a subset $\mathcal{N}$ of rooted infinite trees with $P_{\lambda}(\mathcal{N}) = 0$, and an EMSO sentence $A$, such that for every finite tree, $A$ holds, and for every infinite tree that is not in $\mathcal{N}$, $\neg A$ holds. Thus, we show that finiteness is not even almost surely expressible as an EMSO sentence.

The study of the scaling limit of the uniform spanning tree has been fruitful in the planar case. However, scaling limits of uniform spanning trees in higher dimensions are not as well understood. This talk discusses challenges in the description of these scaling limits and recent existence results in the three-dimensional case. This work is part of ongoing joint work with Omer Angel, David Croydon, and Daisuke Shiraishi.

In this talk, we characterize synchronization for discrete-time, discrete-state random dynamical systems, with random and probabilistic Boolean networks as particular examples. By studying multiplicative ergodic properties of the induced linear cocycle, we show such a random dynamical system with a finite state set synchronizes if and only if the Lyapunov exponent $0$ has simple multiplicity. For the case of countable state set, characterization of synchronization is provided in term of the spectral subspace corresponding to the Lyapunov exponent $-\infty$. In addition, for both cases of finite and countable state sets, the mechanism of partial synchronization is described by partitioning the state set into synchronized subsets. This is a joint work with W. Huang, H. Qian, F. Ye and Y. Yi.

Finitary codings for random fields Abstract: Let $X$ be a translation-invariant random field on $\mathbb{Z}^d$ (i.e., a stationary $\mathbb{Z}^d$-process). We say that $X$ can be coded by an i.i.d. process if there is a deterministic and translation-invariant way to construct a realization of $X$ from i.i.d. variables associated to the sites of $\mathbb{Z}^d$. That is, if there is an i.i.d. process $Y$ and a measurable map F from the underlying space of $Y$ to that of $X$, which commutes with translations of $\mathbb{Z}^d$ and satisfies that $F(Y)=X$ in distribution. Such a coding is called finitary if in order to determine the value of $X$ at a given site, one only needs to look at a finite (but random) region of $Y$. We discuss various conditions which guarantee that $X$ can be finitarily coded by an i.i.d. process. Based on work with Matan Harel.

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!

A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. This should be free.

We prove almost sure site recurrence for coalescing random walk (CRW) on any unimodular random graph for which the root has finite expected degree. The proof relies on a linear (in time) bound on the annealed second moment of the cluster size in the dual process, namely the voter model. In turn, this bound is achieved through a first moment estimate on the size-biased cluster, by controlling the adhesion rate to a tagged particle in the CRW. Joint work with Tom Hutchcroft and Matt Junge.

The main goal is the study of the convergence to equilibrium for a family of Ornstein-Uhlenbeck processes when the underlying noise is given by a Lévy process. Under some log-moment condition on the associated Lévy measure of the noise, when the magnitude of the perturbation is fixed, the stochastic dynamics goes to its equilibrium as the time goes by. We show that the convergence is actually abrupt: as the magnitude of the noise goes to zero, the total variation distance between the law of the stochastic dynamics and its equilibrium in a time window around the mixing time comes abruptly from one to zero, and only after this time window the convergence starts to be exponentially fast. This fact is known as cut-off phenomenon in the context of Markov chains. This a joint work with Juan Carlos Pardo.

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. This should be free.

2-day workshop participants are welcome to use BIRS facilities (Corbett Hall Lounge, TCPL, Reading Room) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 12 noon. There is no coffee break service on Sunday afternoon, but self-serve coffee and tea are always available in the 2nd floor lounge, Corbett Hall.

A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. This should be free; please confirm when you check in.

This talk is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.

This talk is an introduction to the mathematical theory of front propagation phenomena in fluctuating environments with the focus on the spreading speeds. I will first introduce some backgrounds and classical results in homogeneous environments. It is followed by the presentation of some developments in heterogeneous but deterministic environments. Finally, I will introduce the problem in random/stochastic environments by presenting some mathematical models and related mathematical problems.

In '99-'00, David Aldous conjectured that a certain natural "random walk" on the space of binary combinatorial trees should have a continuum analogue, which would be a diffusion on the space of continuum trees, and which would project down, via certain maps, to Wright-Fisher diffusions. This talk discusses ongoing work by F-Pal-Rizzolo-Winkel that has recently yielded a construction of a continuum-tree-valued process, which we claim is the conjectured process. This construction combines our work on dynamics of certain projections of the combinatorial tree with our previous construction of interval-partition-valued diffusions.

In this paper, we study the stochastic differential equation in $R^d$: $dX_t=b(t, X_t)dt+\sigma(t, X_{t-})dZ_t,\ X_0=x$, where $Z$ is a Levy process. We show that for a class of Levy processes $Z$ and Holder continuous drift $b$ and Lipchitz continous  and  uniformly elliptic diffusion $\sigma$, the above SDE has a unique strong solution for every starting point $x$ by the Zvonkin's transform.

A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. This should be free.