Past Stochastic Analysis Seminar

We study a generalized version of the signaling processoriginally introduced and studied by Argiento, Pemantle, Skyrms and Volkov(2009), which models how two interacting agents learn to signal each other andthus create a common language.

We show that the process asymptotically leads to the emergence of a graph ofconnections between signals and states which has the property that nosignal-state correspondance could be associated both to a synonym and aninformational bottleneck.

We consider the time average of the (renormalized) current fluctuation field in one-dimensional weakly asymmetric simple exclusion.

The asymmetry is chosen to be weak enough such that the density fluctuation field still converges in law with respect to diffusive scaling. Remark that the density fluctuation field would evolve on a slower time scale if the asymmetry is too strong and that then the current fluctuations would have something to do with the Tracy-Widom distribution. However, the asymmetry is also chosen to be strong enough such that the density fluctuation field does not converge in law to an infinite-dimensional Ornstein-Uhlenbeck process, that is something non-trivial is happening.

We will, at first, motivate why studying the time average of the current fluctuation field helps to understand the structure of this non-trivial scaling limit of the density fluctuation field and, second, show how one can replace the current fluctuation field by a certain functional of the density fluctuation field under the time average. The latter result provides further evidence for the common belief that the scaling limit of the density fluctuation field approximates the solution of a Burgers-type equation

Abstract: The goal of this talk is to discuss threeproblems on fractional and related stochastic fields, in which wavelet methodshave turned out to be quite useful.

  The first problemconsists in constructing optimal random series representations of Lévyfractional Brownian field; by optimal we mean that the tails of the seriesconverge to zero as fast as possible i.e. at the same rate as the l-numbers.Note in passing that there are close connections between the l-numbers of aGaussian field and its small balls probabilities behavior.

  The secondproblem concerns a uniform result on the local Hölder regularity (the pointwiseHölder exponent) of multifractional Brownian motion; by uniform we mean thatthe result is satisfied on an event with probability 1 which does not depend onthe location.

  The third problemconsists in showing that multivariate multifractional Brownian motion satisfiesthe local nondeterminism property. Roughly speaking, this property, which wasintroduced by Berman, means that the increments are asymtotically independentand it allows to extend to general Gaussian fields many results on the localtimes of Brownian motion.

ABSTRACT "We give a short introduction to randomwalk in random environment

(RWRE) and some open problems connected to RWRE.

Then, in dimension larger than or equal to four we studyballisticity conditions and their interrelations. For this purpose, we dealwith a certain class of ballisticity conditions introduced by Sznitman anddenoted $(T)_\gamma.$ It is known that they imply a ballistic behaviour of theRWRE and are equivalent for parameters $\gamma \in (\gamma_d, 1),$ where$\gamma_d$ is a constant depending on the dimension and taking values in theinterval $(0.366, 0.388).$ The conditions $(T)_\gamma$ are tightly interwovenwith quenched exit estimates.

As a first main result we show that the conditions are infact equivalent for all parameters $\gamma \in (0,1).$ As a second main result,we prove a conjecture by Sznitman concerning quenched exit estimates.

Both results are based on techniques developed in a paperon slowdowns of RWRE by Noam Berger.

(joint work with Alejandro Ram\'{i}rez)"

We investigate a class of weakly interactive particle systems with absorption. We assume that the coefficients in our model depend on an "absorbing" factor and prove the existence and uniqueness of the proposed model. Then we investigate the convergence of the empirical measure of the particle system and derive the Stochastic PDE satisfied by the density of the limit empirical measure. This result can be applied to credit modelling. This is a joint work with Dr. Ben Hambly.

Abstract: There has in the last year been much progresson the universality problem for the spectra of a Wigner random matrices, i.e.Hermitian or symmetric random matrices with independent elements. I will givesome background on this problem and also discuss what can be said when we onlyassume a few moments of the matrix elements to be finite.

A copolymer is a chain of repetitive units (monomers) that

are almost identical, but they differ in their degree of

affinity for certain solvents. This difference leads to striking

phenomena when the polymer fluctuates

in a non-homogeneous medium, for example made up by two solvents

separated by an interface.

One may observe, for exmple, the localization of the polymer at the

interface between the two solvents.

Much of the literature on the subject focuses on the most basic model

based on the simple symmetric random walk on the integers, but

E. Bolthausen and F. den Hollander (AP 1997) pointed out

the convergence of the (rescaled) free energy of such a discrete model

toward

the free energy of a continuum model, based on Brownian motion,

in the limit of weak polymer-solvent coupling. This result is

remarkable because it strongly suggests

a universal feature for copolymer models. In this work we prove that

this is indeed the case. More precisely,

we determine the weak coupling limit for a general class of discrete

copolymer models, obtaining as limits

a one-parameter (alpha in (0,1)) family of continuum models, based on

alpha-stable regenerative sets.

In this talk, we will focus on the asymptotic behavior in time of the solution of a model equation for Bose-Einstein condensation, in the case where the trapping potential varies randomly in time.

The model is the so called Gross-Pitaevskii equation, with a quadratic potential with white noise fluctuations in time whose amplitude tends to zero.

The initial condition is a standing wave solution of the unperturbed equation We prove that up to times of the order of the inverse squared amplitude the solution decomposes into the sum of a randomly modulatedmodulation parameters.

In addition, we show that the first order of the remainder, as the noise amplitude goes to zero, converges to a Gaussian process, whose expected mode amplitudes concentrate on the third eigenmode generated by the Hermite functions, on a certain time scale, as the frequency of the standing wave of the deterministic equation tends to its minimal value.

Abstract: In this talk we will review some recentadvances in order to construct geometric or weakly geometric rough paths abovea multidimensional fractional Brownian motion, with a special emphasis on thecase of a Hurst parameter H<1/4. In this context, the natural piecewiselinear approximation procedure of Coutin and Qian does not converge anymore,and a less physical method has to be adopted. We shall detail some steps ofthis construction for the simplest case of the Levy area.

TBA

The subject of distributional symmetries and invarianceprinciples yields deep results on the structure of the underlying randomobjects. So it is of general interest to investigate if such an approach turnsout to be also fruitful in the quantum world. My talk will report recentprogress in the transfer of de Finetti's pioneering work to noncommutativeprobability. More precisely, an infinite sequence of random variables isexchangeable if its distribution is invariant under finite permutations. The deFinetti theorem characterizes such sequences as conditionally i.i.d. Recentlywe have proven a noncommutative analogue of this celebrated theorem. We willdiscuss the new symmetries `braidability'

and `quantum exchangeability' emerging from our approach.In particular, this brings our approach in close contact with Jones' subfactortheory and Voiculescu's free probability. Finally we will address that ourmethods give a new proof of Thoma's theorem on the general form of charactersof the infinite symmetric group. Quite surprisingly, Thoma's theorem turns outto be the spectral analysis of the tail algebra coming from a certainexchangeable sequence of transpositions. This is in part joint work with RolfGohm and Roland Speicher.

REFERENCES:

[1] C. Koestler. A noncommutative extended de Finettitheorem 258 (2010) 1073-1120.

[2] R. Gohm & C. Kostler. Noncommutativeindependence from the braid group $\mathbb{B}_\infty$. Commun. Math. Phys.289(2) (2009), 435-482.

[3] C. Koestler & R. Speicher. A noncommutative deFinetti theorem:

Invariance under quantum permutations is equivalent tofreeness with amalgamation. Commun. Math. Phys. 291(2) (2009), 473-490.

[4] R. Gohm & C. Koestler: An application ofexchangeability to the symmetric group $\mathbb{S}_\infty$. Preprint.

Numerous studies of asset returns reveal excess kurtosis as fat tails, often characterized by power law behaviour. A hybrid of arithmetic and geometric Brownian motion is proposed as a model for short-term asset returns, and its equilibrium and dynamical properties explored. Some exact solutions for the time-dependent behaviour are given, and we demonstrate the existence of a stochastic bifurcation between mean- reverting and momentum-dominated markets. The consequences for risk management will be discussed.

We prove that when a sequence of Lévy processes $X(n)$ or a normed sequence of random walks $S(n)$ converges a.s. on the Skorokhod space toward a Lévy process $X$, the sequence $L(n)$ of local times at the supremum of $X(n)$ converges uniformly on compact sets in probability toward the local time at the supremum of $X$. A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and

descending) converges jointly in law towards the ladder processes of $X$. As an application, we show that in general, the sequence $S(n)$ conditioned to stay positive converges weakly, jointly with its local time at the future minimum, towards the corresponding functional for the limiting process $X$. From this we deduce an invariance principle for the meander which extends known results for the case of attraction to a stable law.

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.

This talk is based on a joint work with Zhan Shi: We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou (2005). Our method applies furthermore to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn (1988). Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

Given a foliation of a compact manifold, leaves of which are equipped with a Riemannian metric, one can consider the associated "leafwise"

Brownian motion, and study its asymptotic properties (such as asymptotic distribution, behaviour of holonomy maps, etc.).

Lucy Garnet studied such measures, introducing the notion of a harmonic measure -- stationary measure of this process; the name "harmonic" comes from the fact that a measure is stationary if and only if with respect to it integral of every leafwise Laplacian of a smooth function equals zero (so, the measure is "harmonic" in the sense of distributions).

It turns out that for a transversally conformal foliation, unless it possesses a transversally invariant measure (which is a rather rare case), the associated random dynamics can be described rather precisely. Namely, for every minimal set in the foliation there exists a unique harmonic measure supported on it -- and this gives all the possible ergodic harmonic measures (in particular, there is a finite number of them, and they are always supported on the minimal sets).

Also, the holonomy maps turn out to be (with probability one) exponentially contracting -- so, the Lyapunov exponent of the dynamics is negative. Finally, for any initial point almost every path tends to one of the minimal sets and is asymptotically distributed with respect to the corresponding harmonic measure -- and the functions defining the probabilities of tending to different sets form a base in the space of continuous leafwise harmonic functions.

An interesting effect that is a corollary of this consideration is that for transversally conformal foliations the number of the ergodic harmonic measures does not depend on the choice of Riemannian metric on the leaves. This fails for non-transversally conformal foliations:

there is an example, recently constructed in a joint with S.Petite (following B.Deroin's technique).

Stochastic geometry gradually becomes a necessary theoretical tool to model and analyse modern telecommunication systems, very much the same way the queuing theory revolutionised studying the circuit switched telephony in the last century. The reason for this is that the spatial structure of most contemporary networks plays crucial role in their functioning and thus it has to be properly accounted for when doing their performance evaluation, optimisation or deciding the best evolution scenarios.  The talk will present some stochastic geometry models and tools currently used in studying modern telecommunications.  We outline specifics of wired, wireless fixed and ad-hoc systems and show how the stochastic geometry modelling helps in their analysis  and optimisation.