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Stochastic Analysis and its Applications (17w5119)

The Banff International Research Station will host the "Stochastic Analysis and its Applications" workshop from October 22nd to October 27th, 2017. Stochastic Analysis takes a central place in modern probability theory. It has numerous interactions with other areas of mathematics and sciences. For example, one of the main founders and contributors, K. Ito, was awarded the first Gauss prize in 2006. The Gauss prize recognizes scientists in the world whose mathematical research has had an impact outside mathematics either in technology, in business, or simply in people's everyday lives. Over the years, stochastic analysis included various specific topics, such as the general theory of Markov processes, the general theory of stochastic integration, the theory of martingales, Malliavin calculus, the martingale-problem approach to Markov processes, the Dirichlet form approach to Markov processes, Schramm-Loewner equations, and stochastic partial differential equations. Recently, Wendelin Werner and Martin Hairer were awarded a Fields medal for their work in SLE and SPDE in 2006 and 2014, respectively. The Fields Medal is awarded every four years on the occasion of the International Congress of Mathematicians to recognize up to four scientists in the world under 40 years of age for their outstanding mathematical achievement for existing work and for the promise of future achievement. The Fields Medal is widely viewed as one of the highest honor for mathematicians. The scientific goal of the proposed workshop is to bring together top experts in stochastic analysis representing its various branches, with the common theme of developing new foundational methods and their applications to specific areas of probability. We plan to stress geographic diversity and also to invite some of the most promising junior mathematicians. The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).

Kein Logo verfügbarBanff International Research Station (BIRS) for Mathematical Innovation and Discovery
DOI (Serie): 10.5446/s_1509
11
2017
6
8 Stunden 13 Minuten
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11 Ergebnisse
Vorschaubild
Boundary renormalisation of singular SPDEs
46:05
1Hairer, Martin
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.
2017Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
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Heavy tails and one-dimensional localization
45:40
Cranston, Michael
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 α<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: Hθ0ψ(x)=−ψ′′(x)+V(x,ω)ψ(x),ψ(0)cosθ0−ψ′(0)sinθ0=0. where for each x∈[0,∞),V(x,⋅) is a random variable on a basic probability space (Ω,F,P) and θ0∈[0,π] is fixed. Our potentials V(x,ω) 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.
2017Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
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Rotatable random sequences in local fields
45:48
Evans, Steve N.
An infinite sequence of real random variables (ξ1,ξ2,…) is said to be rotatable if every finite subsequence (ξ1,…,ξn) has a spherically symmetric distribution. A classical theorem of David Freedman says that (ξ1,ξ2,…) is rotatable if and only if ξj=σηj for all j, where (η1,η2,…) is a sequence of independent standard Gaussian random variables and σ 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.
2017Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
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Minkowski content and exceptional sets for Brownian paths
42:51
Lawler, Greg F.
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.
2017Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
Vorschaubild
Small noise limit for singularly perturbed diffusion
45:45
1Athreya, Siva
We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by XεtYεt==x0+∫t0b(Xεs,Yεs)ds+εαBt,y0−1ε∫t0∇yU(Xεs,Yεs)ds+s(ε)ε√Wt, where x0∈Rd,y0∈Rm, Bt,Wt are independent Brownian motions, b:Rd×Rm→Rd, U:Rd×Rm→R, and s:(0,∞)→(0,∞). One observes that there is a time scale separation between X and Y. Under suitable assumptions on b,U, for 0<α<12, if s(ϵ)→0 goes to zero at a prescribed slow enough rate then we establish all weak limits points of Xϵ, as ϵ→0, as Fillipov solutions to a differential inclusion. This is joint work with V. Borkar, S. Kumar and R. Sundaresan.
2017Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
Vorschaubild
Random walks on dynamical percolation
46:06
Peres, Yuval
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 μ, 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.
2017Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
Vorschaubild
Aldous diffusion on continuum trees
39:35
Pal, Soumik
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.
2017Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
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Quenched CLT for random walk in divergence-free random drift field
48:45
Toth, Balint
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, L2+ϵ (rather than L2) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the non-reversible, divergence-free drift case.
2017Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
Vorschaubild
Stability of the elliptic Harnack inequality
43:28
Barlow, Martin
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.
2017Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
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A Skorokhod map on measure valued paths with applications to priority queues
46:15
Kaspi, Haya
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,∞) in such a way that for each x>0, the path t→ζt[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.
2017Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
Vorschaubild
Universality for the dimer model
43:06
4Berestycki, Nathanael
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.
2017Banff International Research Station (BIRS) for Mathematical Innovation and Discovery