Forthcoming events in this series
Joint work with C. Donati-Martin and D. Feral
Condition supercritical percolation so that the origin is enclosed by a dual circuit whose interior traps an area of n^2.
The Wulff problem concerns the shape of the circuit. We study the circuit's fluctuation. A well-known measure of this fluctuation is maximum local roughness (MLR), which is the greatest distance from a point on the circuit to the boundary of circuit's convex hull. Another is maximum facet length (MFL), the length of the longest line segment of which this convex hull is comprised.
In a forthcoming article, I will prove that
for various models including supercritical percolation, under the conditioned measure,
MLR = \Theta(n^{1/3}\log n)^{2/3}) and MFL = \Theta(n^{2/3}(log n)^{1/3}).
An important tool is a result establishing the profusion of regeneration sites in the circuit boundary. The talk will focus on deriving the main results with this tool
We consider the analysis for a class of random differential equations driven by rough noise and with a trajectory that is influenced by its own law. Having described the mathematical setup with great precision, we will illustrate how such equations arise naturally as the limits of a cloud of interacting particles. Finally, we will provide examples to show the ubiquity of such systems across a range of physical and economic phenomena and hint at possible extensions.
Diffusion limits of MCMC methods in high dimensions provide a useful theoretical tool for studying efficiency.
In particular they facilitate precise estimates of the number of steps required to explore the target measure, in stationarity, as a function of the dimension of the state space. However, to date such results have only been proved for target measures with a product structure, severely limiting their applicability to real applications. The purpose of this talk is to desribe a research program aimed at identifying diffusion limits for a class of naturally occuring problems, found by finite dimensional approximation of measures on a Hilbert space which are absolutely continuous with respect to a Gaussian reference measure.
The diffusion limit to a Hilbert space valued SDE (or SPDE) is proved.
Joint work with Natesh Pillai (Warwick) and Jonathan Mattingly (Duke)
In a quantum network model, unitary matrices are assigned to each edge and node of a graph. The quantum amplitude for a particle to propagate from node A to node B is the sum over all random walks (Feynman paths) from A to B, each walk being weighted by the ordered product of matrices along the path. In most cases these models are too difficult to solve analytically, but I shall argue that when the matrices are random elements of SU("), independently drawn from the invariant measure on that group, then averages of these quantum amplitudes are equal to the probability that a certain kind of self-avoiding *classical* random walk reaches B when started at A. This leads to various conjectures about the generic behaviour of such network models on regular lattices in two and three dimensions.
My goal is to estimate unknown parameters in the vector field of a rough differential equation, when the expected signature for the driving force is known and we estimate the expected signature of the response by Monte Carlo averages.
I will introduce the "expected signature matching estimator" which extends the moment matching estimator and I will prove its consistency and asymptomatic normality, under the assumption that the vector field is polynomial. Finally, I will describe the polynomial system one needs to solve in order to compute this estimatior.
In this talk we will review some recent results on the long-time/large-scale, weak-friction asymptotics for the one dimensional Langevin equation with a periodic potential. First we show that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We also show that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. Furthermore we prove that the same result is valid for a whole one parameter family of space/time rescalings. We also present a new numerical method for calculating the diffusion coefficient and we use it to study the multidimensional problem and the problem of Brownian motion in a tilted periodic potential.
If a distribution, say F, has all moments finite, then either F is unique (M-determinate) in the sense that F is the only distribution with these moments, or F is non-unique (M-indeterminate). In the latter case we suggest a method for constructing a Stieltjes class consisting of infinitely many distributions different from F and all having the same moments as F. We present some shocking examples involving distributions such as N, LogN, Exp and explain what and why. We analyse conditions which are sufficient for F to be M-determinate or M-indeterminate. Then we deal with recent problems from the following areas:
(A) Non-linear (Box-Cox) transformations of random data.
(B) Distributional properties of functionals of stochastic processes.
(C) Random sums of random variables.
If time permits, some open questions will be outlined. The talk will be addressed to colleagues, including doctoral and master students, working or having interests in the area of probability/stochastic processes/statistics and their applications.
A new class of relativistic diffusions encompassing all the previously studied examples has recently been introduced by C. Chevalier and F Debbasch, both in a heuristic and analytic way. Roughly speaking, they are characterised by the existence at each (proper) time (of the moving particle) of a (local) rest frame where the random part of the acceleration of the particle (computed using the time of the rest frame) is brownian in any spacelike direction of the frame.
I will explain how the tools of stochastic calculus enable us to give a concise and elegant description of these random paths on any Lorentzian manifiold. A mathematically clear definition of the the one-particle distribution function of the dynamics will emerge from this definition, and whose main property will be explained. This will enable me to obtain a general H-theorem and to shed some light on links between probablistic notions and the large scale structure of the manifold.
All necessary tools from stochastic calculus and geometry will be explained.
Numerical computer codes implementing physics based models are the backbone of today's mechanical/aerospace engineering analysis and design methods. Such computational codes can be extremely expensive consisting of several millions of degrees of freedom. However, large models even with very detailed physics are often not enough to produce credible numerical results because of several types of uncertainties which exist in the whole process of physics based computational predictions. Such uncertainties include, but not limited to (a) parametric uncertainty (b) model inadequacy; (c) uncertain model calibration error coming from experiments and (d) computational uncertainty. These uncertainties must be assessed and systematically managed for credible computational predictions. This lecture will discuss a random matrix approach for addressing these issues in the context of complex structural dynamic systems. An asymptotic method based on eigenvalues and eigenvectors of Wishart random matrices will be discussed. Computational predictions will be validated against laboratory based experimental results.
The parabolic Anderson model is the Cauchy problem for the heat equation with random potential. It offers a case study for the possible effects that a random, or irregular environment can have on a diffusion process. In this talk I review results obtained for an extreme case of heavy-tailed potentials, among the effects we discuss our intermittency, strong localisation and ageing.