Schedule for: 16w2696 - Retreat for Young Researchers in Stochastics

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

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. Note that BIRS does not pay for meals for 2-day workshops.

I will discuss self-avoiding polygons in a restricted geometry, namely an infinite $L\times M$ tube in $\mathbb Z^3$. These polygons can be viewed as a simple model of DNA molecules in viral capsids. The model can be equipped with a Boltzmann weight corresponding to a stretching or compressing force, and I will discuss the behaviour of the polygons as the magnitude of the force becomes large. I will also present some related results regarding the topological properties of confined polygons, that is, when and how they can be knotted. As the knotting of DNA can affect its ability to replicate correctly, this is of great importance from a biological standpoint.

I will present a general approach for constructing a Markov process that describes the dynamics of a process when one or more observables of this process are observed to fluctuate in time away from their typical values. This will extend some well known notions as bridge and Q-quasi-stationary process. Articles : Phys. Rev. Lett. 2013, Ann Henri Poincaré 2014, J. Stat. Mech 2015

We construct two diffusions on a space of partitions of the unit interval. These are stationary with the law of the complement of the zero sets of Brownian motion and Brownian bridge, respectively. Our construction is based on decorating the jumps of a spectrally positive L\'evy process with independent continuous excursions. The processes of ranked interval lengths of our partitions belong to a two parameter family of diffusions introduced by Ethier and Kurtz (1981) and Petrov (2009). These are continuum limits of up-down Markov chains on Chinese restaurant processes. Our construction works towards building a diffusion on the space of real trees whose existence has been conjectured by Aldous.

We present the CORSING (COmpRessed SolvING) method for the numerical approximation of PDEs. Establishing an analogy between the bilinear form associated with the weak formulation of a PDE and the signal acquisition process, CORSING combines the classical Petrov-Galerkin method with Compressed Sensing.  Considering the advection-diffusion-reaction equation of fluid dynamics as a case study, we discuss some MATLAB numerical examples and analyze the method from a theoretical perspective.

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. Note that BIRS does not pay for meals for 2-day workshops.

We will start with the De Giorgi iterations scheme on SPDEs and the a priori Holder continuity results on parabolic SPDEs. We will then explain the extension of such scheme on a Harnack inequality for the parabolic SPDE and introduce a strong maximum principle. Some variants of the method will also be introduced for SPDEs with random drifts.

We consider a super-critical branching particle system in which the combined trajectory of each particle and its ancestors follows a path Gaussian process in $\mathbb{R}^d$. Unlike branching diffusion systems, such  model is not necessary Markov. Assuming that the offspring distribution has finite second moment and some mild conditions on the underlying Gaussian process, we show a strong law of large numbers with the limit object characterized in terms of asymptotic behavior between the mean and variance of the Gaussian process. Long memory processes, like fractional Brownian motions and fractional Ornstein-Uhlenbeck processes with Hurst parameter greater than 1/2, as well as rough processes, like fractional processes with Hurst parameter smaller than 1/2 are included as important examples. Our techniques include moment computations and conditional probabilities. This is a joint work with Mike Kouritzin and Deniz Sezer.

Graduate student presentations and presentations of open problems for discussion. Chaired by Omer Angel.

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. Note that BIRS does not pay for meals for 2-day workshops.

Presenters are Katharine Cera and Julia Schmidt. Our talk gives an introduction to Limit Order Books and summarizes the model proposed by Cont and de Larrard in their paper “Price Dynamics in a Markovian Limit Order Market” (SIAM J. Finan. Math (2013)). We present the generalizations done by Swishchuk and Vadori in “A Semi-Markovian Modeling of Limit Order Markets” (e.g., arXiv (2016)) and the evidence found in our data that justifies their approach. The talk closes with highlighting our own work on extending this model to the general semi-Markov model for stock price process in the Limit Order Book including gained numerical results for diffusion limits.

In this talk, a model for a level-1 Limit Order Book, in the same spirit as the model discussed by Cont and deLarrard (2012), will be presented. However, as empirical evidence shows, the intensity function of the price process driving the arrivals of orders is non-constant. To account for that fact, the assumption of homogeneous Poisson arrivals is relaxed and more general non-homogeneous Poisson arrivals are considered. This becomes part of an effort to understand the macroscopic long-run price dynamics and the different limiting price processes when the orderbook is analyzed from the market micro-structure.

Kaufman's dimension doubling theorem states that for a planar Brownian motion $\{\mathbf{B}(t): t\in [0,1]\}$ we have $$P(\dim \mathbf{B}(A)=2\dim A \textrm{ for all } A\subset [0,1])=1,$$ where $\dim$ may denote Hausdorff and packing dimension, too. Our goal is to prove similar uniform dimension results in the one-dimensional case. Let $B\colon [0,1]\to \mathbb{R}$ be a fractional Brownian of Hurst index $\alpha$. We try to characterize the Borel sets $D\subset [0,1]$ for which \[P(\dim B(A)=(1/\alpha) \dim A \textrm{ for all } A\subset D)=1.\] In order to do so, we introduce a new concept of dimension, the modified Assouad dimension. This is a joint work with Yuval Peres. \bigskip Let $B\colon \mathbb{R}^n\to \mathbb{R}^d$ be a fractional Brownian motion of Hurst index $\alpha$ and let $f\colon \mathbb{R}^n\to \mathbb{R}^d$ be a Borel map. For every Borel set $A\subset \mathbb{R}^n$ we determine the almost sure value of the packing dimension of the image and graph of $B+f$ restricted to $A$. In order to do so, we assign different notions of dimension to jointly measurable stochastic processes. This generalizes a result of Xiao, who calculated the packing dimension of $B(A)$ by using the concept of packing dimension profiles introduced by Falconer and Howroyd.

We shall see some motivating applications behind studying the boundary Harnack principle. Then, I will present a recent generalization of boundary Harnack principle, that provides new results even in R^n. This talk is based on joint work with Martin Barlow.

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.

I will discuss the influence of the seed in models of randomly growing trees; in particular, I will focus on the preferential attachment and uniform attachment models. In both of these models, different seeds lead to different distributions of limiting trees from a total variation point of view. I will discuss the differences and similarities in proving this for the two models. This is based on joint work with Sebastien Bubeck, Ronen Eldan, and Elchanan Mossel.