Past Stochastic Analysis Seminar

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.

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel-Riesz capacity, respectively. We apply these results to a system of stochastic wave equations in spatial dimension k≥1 driven by a d-dimensional spatially homogeneous additive Gaussian noise that is white in time and coloured in space.

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.

Random polymers are used to model various physical ( Ising inter- faces, wetting, etc.) and biological ( DNA denaturation, etc.) phenomena They are modeled as a one dimensional random walk (Xn), with excursion length distribution

P(E1 = n) = (n)=nc, c > 1, and (n) a slowly varying function. The polymer gets a random reward, whenever it visits or crosses an interface. The random rewards are realised as a sequence of i.i.d. variables (Vn). Depending on the relation be- tween the mean value of the disorder Vn and the temperature, the polymer might prefer to stick on the interface (pinning) or undergo a long excursion away from it (depinning).

In this talk we will review some aspects of random polymer models. We will also discuss in more detail the pinning-depinning transition of the 'Pinning' model and also its relation to other directed polymer models

We study the class of Azema-Yor processes which are of the form F(M_t)-f(M_t)(X_t-M_t), where F'=f, X_t is a semimartingale with no positive jumps and M_t is its running maximum. We show that these processes arise as unique strong solutions to the Bachelier SDE which we also show is equivalent to the DrawDown SDE. The proofs are greatly simplified thanks to (algebraic) group property of the set of AY processes indexed by functions. We then restrict our attention to the case when X is a martingale. It turns out that the AY martingales are the only local martingales of the form H(X_t,M_t) for a Borel function H. Furthermore, they can also be characterised by their optimal

properties: all uniformly integrable martingales whose maximum dominates a given target are dominated by an AY martingale in the concave ordering of terminal values. We mention how these results find direct applications in portfolio optimisation/insurance theory.

Joint work with Laurent Cararro and Nicole El Karoui

In this talk, we give the asymptotics estimates for the heat kernel and its gradient estimates on H-type groups. Moreover, we get gradient estimates for the heat semi-group.

We discuss some connections between various notions of rationality in the face of uncertainty and the theory of convex risk measures, both in a static and a dynamic setting.

/*-->*/ /*-->*/

We study a class of Markovian optimal stochastic control problems in which the controlled process $Z^\nu$ is constrained to satisfy an a.s.~constraint $Z^\nu(T)\in G\subset \R^{d+1}$ $\Pas$ at some final time $T>0$.  When the set is of the form $G:=\{(x,y)\in \R^d\x \R~:~g(x,y)\ge 0\}$, with $g$ non-decreasing in $y$, we provide a Hamilton-Jacobi-Bellman  characterization of the associated value function. It gives rise to a state constraint problem where the constraint can be expressed in terms of an auxiliary value function $w$ which characterizes the set $D:=\{(t,Z^\nu(t))\in [0,T]\x\R^{d+1}~:~Z^\nu(T)\in G\;a.s.$ for some $ \nu\}$. Contrary to standard state constraint problems, the domain $D$ is not given a-priori and we do not need to impose conditions on its boundary. It is naturally incorporated in the auxiliary value function $w$ which is itself a viscosity solution of a non-linear parabolic PDE.  Applying ideas recently developed in Bouchard, Elie and Touzi (2008), our general result also allows to consider optimal control problems with moment constraints of the form $\Esp{g(Z^\nu(T))}\ge 0$ or $\Pro{g(Z^\nu(T))\ge 0}\ge p$.

It is well known that the description of the asymptotic behaviour of products of i.i.d random matrices can be derived from the properties of the Lyapunov exponents of these matrices. So far, the fact that the matrices in question are IDENTICALLY distributed, had been crucial for the existing theories. The goal of this work is to explain how and under what conditions one might be able to control products of NON-IDENTICALLY distributed matrices.

We present a new class of self interacting Markov chain models. In contrast to traditional Markov chains, their time evolution may depend on the occupation measure of the past values. We propose a theoretical basis based on measure valued processes and semigroup technics to analyze their asymptotic behaviour as the time parameter tends to infinity. We exhibit different types of decays to equilibrium depending on the level of interaction. In the end of the talk, we shall present a self interacting methodology to sample from a sequence of target probability measures of increasing complexity. We also analyze their fluctuations around the limiting target measures.

We consider transient random walks in random environment on Z with zero asymptotic speed. In a seminal paper, Kesten, Kozlov and Spitzer proved that the hitting time of the level "n" converges in law, after a proper normalization, towards a positive stable law, but the question of the description of its parameter was left open since that time. A new approach to this problem, based on a precise description of Sinai's potential, leads to a complete characterization of this stable law, making a tight link with Kesten's renewal series. The case of Dirichlet environment turns out to be remarkably explicit. Quenched results on this model will be presented if time permits.

We present numerical schemes for nonlinear stochastic differential equations whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action and subsequently to the corresponding Lie algebra.

We construct an approximation to the stochastic flow in the Lie algebra via closed operations and then push back to the manifold, thus ensuring our approximation lies in the manifold. We call such schemes stochastic Munthe-Kaas methods after their deterministic counterparts. We also present stochastic Lie group integration schemes based on Castell--Gaines methods. They become stochastic Lie group integrator schemes if we use Munthe-Kaas methods as the underlying ordinary differential integrator. Lastly, we demonstrate our methods by presenting some numerical examples

We will examine the typical structure of random polytopes by projecting the three fundamental regular polytopes: the simplex, cross-polytope, and hypercube. Along the way we will explore the implications of their structure for information acquisition and optimization. Examples of these implications include: that an N-vector with k non-zeros can be recovered computationally efficiently from only n random projections with n=2e k log(N/n), or that for a surprisingly large set of optimization problems the feasible set is actually a point. These implications are driving a new signal processing paradigm, Compressed Sensing, which has already lead to substantive improvements in various imaging modalities. This work is joint with David L. Donoho.

We will consider a (sub) critical Galton-Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We shall specify the law of this allelic partition in terms of the distribution of the number of clone-children and the number of mutant-children of a typical individual. Some limit theorems related to the distribution of the allelic partition will be also presented.

We modify the usual Erdos-Renyi random graph evolution by letting connected clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a Poisson flow of lightnings. In a range of the intensity of rate of lightnings, the system sticks to a permanent critical state (i.e. exhibits so-called self-organised critical behaviour). The talk will be based on joint work with Balint Toth.

The uniform spanning forest (USF) in a graph

is a random spanning forest obtained as the limit of uniformly chosen spanning

trees on finite subgraphs. The USF is known to have stochastic dimension 4 on

graphs that are "at least 4 dimensional" in a certain sense. In this

talk I will look at more detailed estimates on the geometry of a fixed

component of the USF in the special case of the d-dimensional integer lattice,

d > 4. This is motivated in part by the study of random walk restricted to a

fixed component of the USF.

The system

u_t = Delta u + buv - cu + u^{1/2} dW

v_t = - uv

models the evolution of a branching population and its usage of a non-renewable resource.

A phase diagram in the parameters (b,c) describes its long time evolution.

We describe this, including some results on asymptotics in the phase diagram for small and large values of the parameters.