Past Stochastic Analysis Seminar

We consider a process $X_t\in\R^d$, $t\ge0$, introduced by Durrett and Rogers in 1992 in order to model the shape of a growing polymer; it undergoes a drift which depends on its past trajectory, and a Brownian increment. Our work concerns two conjectures by these authors (1992), concerning repulsive interaction functions $f$ in dimension $1$ ($\forall x\in\R$, $xf(x)\ge0$).

We showed the first one with T. Mountford (AIHP, 2008, AIHP Prize 2009), for certain functions $f$ with heavy tails, leading to transience to $+\infty$ or $-\infty$ with probability $1/2$. We partially proved the second one with B. T\'oth and B. Valk\'o (to appear in Ann. Prob. 2011), for rapidly decreasing functions $f$, through a study of the local time environment viewed from the

particule: we explicitly display an associated invariant measure, which enables us to prove under certain initial conditions that $X_t/t\to_{t\to\infty}0$ a.s., that the process is at least diffusive asymptotically and superdiffusive under certain assumptions.

Thanks to a Last Passage Percolation model, 3 colored sources are in competition to fill all the positive quadrant N2. There is coexistence when the 3 souces have infected an infinite number of sites.
A coupling between the percolation model and a particle system -namely, the TASEP- allows us to compute the coexistence probability.

A very general model of evolving graphs was introduced by Cooper and Frieze in 2003, and further analysed by Cooper. At each stage of the process, either a new edge is added
between existing vertices, or a new vertex is added and joined to some number of existing vertices. Each vertex gaining a new neighbour may be chosen either uniformly, or by preferential attachment, i.e., with probability proportional to the current degree.
It is known that the degrees of vertices in any such model follow a ``power law''. Here we study in detail the degree sequence of a graph obtained from such a procedure, looking at the vertices of large degree as well as the numbers of vertices of each fixed degree.
This is joint work with Graham Brightwell.

The talk will explain how a geometric principle gave rise to a new variational description of information-theoretic entropy and how this led to the solution of a problem dating back to the 50's: whether the the central limit theorem is driven by an analogue of the second law of thermodynamics.

Abstract: A random polytope $K_n$ is, by definition, the convex hull of $n$ random independent, uniform points from a convex body $K subset R^d$. The investigation of random polytopes started with Sylvester in 1864. Hundred years later R\'enyi and Sulanke began studying the expectation of various functionals of $K_n$, for instance number of vertices, volume, surface area, etc. Since then many papers have been devoted to deriving precise asymptotic formulae for the expectation of the volume of $K \setminus K_n$, for instance. But with few notable exceptions, very little has been known about the distribution of this functional. In the last couple of years, however, two breakthrough results have been proved: Van Vu has given tail estimates for the random variables in question, and M. Reitzner has obtained a central limit theorem in the case when $K$ is a smooth convex body. In this talk I will explain these new results and some of the subsequent development: upper and lower bounds for the variance, central limit theorems when $K$ is a polytope. Time permitting, I will indicate some connections lattice polytopes.

Abstract: We consider the inverse problem of finding the diffusion coefficient of a linear uniformly elliptic partial differential equation in divergence form, from noisy measurements of the forward solution in the interior. We adopt a Bayesian approach to the problem. We consider the prior measure on the diffusion coefficient to be either a Besov or Gaussian measure. We show that if the functions drawn from the prior are regular enough, the posterior measure is well-defined and Lipschitz continuous with respect to the data in the Hellinger metric. We also quantify the errors incurred by approximating the posterior measure in a finite dimensional space. This is joint work with Stephen Harris and Andrew Stuart.

The signature of the path is an essential object in rough path theory which takes value in tensor algebra and it is anticipated that the expected signature of Brownian motion might characterize the rough path measure of Brownian path itself. In this presentation we study the expected signature of a Brownian path in a Bananch space E stopped at the first exit time of an arbitrary regular domain, although we will focus on the case E=R^{2}. We prove that such expected signature of Brownian motion should satisfy one particular PDE and using the PDE for the expected signature and the boundary condition we can derive each term of expected signature recursively. We expect our method to be generalized to higher dimensional case in R^{d}, where d is an integer and d >= 2.

Abstract: We construct solutions to Burgers type equations perturbed by a multiplicative

space-time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation. We show that our solutions are stable under smooth approximations of the driving noise. A more general class of approximations will also be discussed. This is joint work with Martin Hairer and Jan Maas.

We analyse stability properties of stochastic Lagrangian Navier stokes flows on compact Riemannian manifolds.

Abstract: In this talk, we first introduce the notion of ergodic BSDE which arises naturally in the study of ergodic control. The ergodic BSDE is a class of infinite-horizon BSDEs:
Y_{t}^{x}=Y_{T}^{x}+∫_{t}^{T}[ψ(X^{x}_{σ},Z^{x}_{σ})-λ]dσ-∫_{t}^{T}Z_{σ}^{x}dB_{σ}, P-<K1.1/>, ∀0≤t≤T<∞,
<K1.1 ilk="TEXTOBJECT" > <screen-nom>hbox</screen-nom> <LaTeX>\hbox{a.s.}</LaTeX></K1.1> where X^{x} is a diffusion process. We underline that the unknowns in the above equation is the triple (Y,Z,λ), where Y,Z are adapted processes and λ is a real number. We review the existence and uniqueness result for ergodic BSDE under strict dissipative assumptions.
Then we study ergodic BSDEs under weak dissipative assumptions. On the one hand, we show the existence of solution to the ergodic BSDE by use of coupling estimates for perturbed forward stochastic differential equations. On the other hand, we show the uniqueness of solution to the associated Hamilton-Jacobi-Bellman equation by use of the recurrence for perturbed forward stochastic differential equations.
Finally, applications are given to the optimal ergodic control of stochastic differential equations to illustrate our results. We give also the connections with ergodic PDEs.

Probability measures in infinite dimensional spaces especially that induced by stochastic processes are the main objects of the talk. We discuss the role played by measures on analysis on path spaces, Sobolev inequalities, weak formulations and local versions of such inequalities related to Brownian bridge measures.

We relate the partition function associated with a certain Brownian directed polymer model to a diffusion process which is closely related to a quantum integrable system known as the quantum Toda lattice. This result is based on a `tropical' variant of a combinatorial bijection known as the Robinson-Schensted-Knuth (RSK) correspondence and is completely analogous to the relationship between the length of the longest increasing subsequence in a random permutation and the Plancherel measure on the dual of the symmetric group.

We study a simple heat-bath type dynamic for a simple model of
polymer interacting with an interface. The polymer is a nearest neighbor path in
Z, and the interaction is modelised by energy penalties/bonuses given when the
path touches 0. This dynamic has been studied by D. Wilson for the case without
interaction, then by Caputo et al. for the more general case. When the interface
is repulsive, the dynamic slows down due to the appearance of a bottleneck in the
state space, moreover, the systems exhibits a metastable behavior, and, after time
rescaling, behaves like a two-state Markov chain.

We consider a directed random polymer interacting with an interface
that carries random charges some of which attract while others repel
the polymer. Such a polymer can be in a localized or delocalized
phase, i.e., it stays near the interface or wanders away respectively.
 The phase it chooses depends on the temperature and the average bias
of the disorder. At a given temperature, there is a critical bias
separating the two phases. A question of particular interest, and
which has been studied extensively in the Physics and Mathematics
literature, is whether the quenched critical bias differs from the
annealed critical bias. When it does, we say that the disorder is
relevant.

Using a large deviations result proved recently by Birkner, Greven,
and den Hollander, we derive a variational formula for the quenched

critical bias. This leads to a necessary and sufficient condition for
disorder relevance that implies easily some known results as well as
new ones.

The talk is based on joint work with Frank den  Hollander.

In this talk, we will look at the diffusive scaling limit of a class of

one-dimensional random walks in a random space-time environment. In the

scaling limit, this gives rise to a so-called stochastic flow of kernels as

introduced by Le Jan and Raimond and generalized by Howitt and Warren. We will

prove several new results about these stochastic flows of kernels by making

use of the theory of the Brownian web and net. This is joint work with R. Sun

and E. Schertzer.

Abstract:  An instance of the Potts model is defined by a graph of interactions and a number, q, of  different ``spins''.  A configuration in this model is an assignment of spins to vertices. Each configuration has a weight, which in the ferromagnetic case is greater when more pairs of adjacent spins are alike.  The classical Ising model is the special case

of q=2 spins.  We consider the problem of computing an approximation to the partition function, i.e., weighted sum of configurations, of

an instance of the Potts model.  Through the random cluster formulation it is possible to make sense of the partition function also for non-integer q.  Yet another equivalent formulation is as the Tutte polynomial in the positive quadrant.

About twenty years ago, Jerrum and Sinclair gave an efficient (i.e., polynomial-time) algorithm for approximating the partition function of a ferromagnetic Ising system. Attempts to extend this result to q≠2 have been unsuccessful. At the same time, no convincing evidence has been presented to indicate that such an extension is impossible.  An interesting feature of the random cluster model when q>2 is that it exhibits a first-order phase transition, while for 1≤q≤2 only a second-order phase transition is apparent.  The idea I want to convey in this talk is that this first-order phase transition can be exploited in order to encode apparently hard computational problems within the model.  This provides the first evidence that the partition function of the ferromagnetic Potts model may be hard to compute when q>2.

This is joint work with Leslie Ann Goldberg, University of Liverpool.

The sequence {nα}, where α is an irrational number and {.} denotes fractional part, plays

a fundamental role in probability theory, analysis and number theory. For suitable α, this sequence provides an example for "most uniform" infinite sequences, i.e. sequences whose discrepancy has the

smallest possible order of magnitude. Such 'low discrepancy' sequences have important applications in Monte Carlo integration and other problems of numerical mathematics. For rapidly increasing n_k the behaviour of {n_kα} is similar to that of independent random variables, but its asymptotic properties depend strongly also on the number theoretic properties of n_k, providing a simple example for pseudorandom behaviour. Finally, for periodic f the sequence f(nα) provides a generalization of the trig-onometric system with many interesting  properties. In this lecture, we give a survey of the field  (going back more than 100 years) and formulate new results.

We consider a non linear Schrödinger equation on a compact manifold of dimension d subject to some multiplicative random perturbation. Using some stochastic Strichartz inequality, we prove the existence and uniqueness of a maximal solution in H^1 under some general conditions on the diffusion coefficient. Under stronger conditions on the noise, the nonlinearity and the diffusion coefficient, we deduce the existence of a global solution when d=2. This is a joint work with Z. Brzezniak.

We consider a dilute stationary system of N particles uniformly distributed in space and interacting pairwise according to a compactly supported potential, which is repellent at short distances and attractive at moderate distances. We are interested in the large-N behaviour of the system. We show that at a certain scale there are phase transitions in the temperature parameter and describe the energy and ground states explicitly in terms of a variational problem

 Rough path theory, invented by T. Lyons, is a successful and general method for solving ordinary or stochastic differential equations driven by irregular H\"older paths, relying on the definition of a finite number of substitutes of iterated integrals satisfying definite algebraic and regularity properties.

Although these are known to exist, many questions are still open, in

particular:  (1) "how many" possible choices are there ? (2) how to construct one explicitly ?  (3) what is the connection to "true" iterated integrals obtained by an approximation scheme ?

  In a series of papers, we (1) showed that "formal" rough paths (leaving aside

regularity) were exactly determined by so-called "tree data"; (2) gave several explicit constructions, the most recent ones relying on quantum field renormalization methods; (3) obtained with J. Magnen (Laboratoire de Physique Theorique, Ecole Polytechnique)  a L\'evy area for fractional Brownian motion with Hurst index <1/4 as the limit in law of  iterated integrals of a non-Gaussian interacting process, thus calling for a redefinition of the process itself.  The latter construction belongs to the field of high energy physics, and as such established by using constructive field theory and renormalization; it should extend to a general rough path (work in progress).

We review the problem of determining the high frequency asymptotics of the spectrum of the Laplacian and its relationship to the geometry of a domain. We then establish these asymptotics for some continuum random trees as well as the scaling limit of the critical random graph.

For a superprocess in a random environment in one dimensional space, a nonlinear stochastic partial differential equation is derived for its density by Dawson-Vaillancourt-Wang (2000). The joint continuity was left as an open problem. In this talk, we will give an affirmative answer to this problem.

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 &gt; 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.