Past Stochastic Analysis Seminar

A representation of Hermite polynomials of degree 2n + 1, as sum of an element in the polynomial ideal generated by the roots of the Hermite polynomial of degree n and of a reminder, suggests a folding of multivariate polynomials over a finite set of points. From this, the expectation of some polynomial combinations of random variables normally distributed is computed. This is related to quadrature formulas and has strong links with designs of experiments.

This is joint work with G. Pistone

We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 1/2 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying H\"ormander's condition. We show that such systems satisfy a suitable version of the strong Feller property and we conclude that they admit a unique stationary solution that is physical in the sense that it does not "look into the future".

The main technical result required for the analysis is a bound on the moments of the inverse of the Malliavin covariance matrix, conditional on the past of the driving noise.

The main aim is to incorporate the nonlinear term into non-Markovian Master equations for a continuous time random walk (CTRW) with non-exponential waiting time distributions. We derive new nonlinear evolution equations for the mesoscopic density of reacting particles corresponding to CTRW with arbitrary jump and waiting time distributions. We apply these equations to the problem of front propagation in the reaction-transport systems of KPP-type.

We find an explicit expression for the speed of a propagating front in the case of subdiffusive transport.

TBA

By means of a series of examples (Korteweg-de Vries equation, non-

linear stochastic heat equations and Navier-Stokes equation) we will show how it is possible to apply rough path ideas in the study of the Cauchy problem for PDEs with and without stochastic terms.

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.