Past Stochastic Analysis Seminar

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.