Past Stochastic Analysis Seminar

Based on ideas from rough path analysis and operator splitting, the Kusuoka-Lyons-Victoir scheme provides a family of higher order methods for the weak approximation of stochastic differential equations. Out of this family, the Ninomiya-Victoir method is especially simple to implement and to adjust to various different models. We give some examples of models used in financial engineering and comment on the performance of the Ninomiya-Victoir scheme and some modifications when applied to these models.

Pathwise Holder convergence with optimal rates is proved for the implicit Euler scheme associated with semilinear stochastic evolution equations with multiplicative noise. The results are applied to a class of second order parabolic SPDEs driven by space-time white noise. This is joint work with Sonja Cox.

"The model of Vertex Reinforced Random Walk (VRRW) on Z goes back to Pemantle & Volkov, '99, who proved a result of localization on 5 sites with positive probability. They also conjectured that this was the a.s. behavior of the walk. In 2004, Tarrès managed to prove this conjecture. Then in 2006, inspired by Davis'paper '90 on the edge reinforced version of the model, Volkov studied VRRW with weight on Z. 

He proved that in the strongly reinforced case, i.e. when the weight sequence is reciprocally summable, the walk localizes a.s. on 2 sites, as expected. He also proved that localization is a.s. not possible for weights growing sublinearly, but like a power of n. However, the question of localization remained open for other weights, like n*log n or n/log n, for instance. In the talk I will first review these results and formulate more precisely the open questions. Then I will present some recent results giving partial answers. This is based on joint (partly still on-going) work with Anne-Laure Basdevant and Arvind Singh."

We find the order of growth of the typical number of components of zero sets of smooth random functions of several real variables. This might be thought as a statistical version of the (first half of) 16th Hilbert problem. The primary examples are various ensembles of Gaussian real-valued polynomials (algebraic or trigonometric) of large degree, and smooth Gaussian functions on the Euclidean space with translation-invariant distribution.

Joint work with Fedor Nazarov.

It is well known that electrical resistance arguments provide (usually) the best method for determining whether a graph is transient or recurrent. In this talk I will discuss a similar characterization of 'sub-diffusive behaviour' -- this occurs in spaces with many obstacles or traps.

The characterization is in terms of the energy of functions in annuli.

We consider the approximation of the marginal distribution of solutions of stochastic partial differential equations by splitting schemes. We introduce a functional analytic framework based on weighted spaces where the Feller condition generalises. This allows us to apply the theory of strongly continuous semigroups. The possibility of achieving higher orders of convergence through cubature approximations is discussed.

Applications of these results to problems from mathematical finance (the Heath-Jarrow-Morton equation of interest rate theory) and fluid dynamics (the stochastic Navier-Stokes equations) are considered. Numerical experiments using Quasi-Monte Carlo simulation confirm the practicality of our algorithms.

Parts of this work are joint with J. Teichmann and D. Veluscek.

I will discuss properties of stochastic differential equations and numerical algorithms for sampling Gibbs (i.e smooth) measures. Methods such as Langevin dynamics are reliable and well-studied performers for molecular sampling.   I will show that, when the objective of simulation is sampling of the configurational distribution, it is possible to obtain a superconvergence result (an unexpected increase in order of accuracy) for the invariant distribution.   I will also describe an application of thermostats to the Hamiltonian vortex method in which the energetic interactions with a bath of weak vortices are treated as thermal fluctuations

This is joint work with P. Caputo and D. Chafai. In this talk, we
will consider various probability distributions on the set of stochastic
 matrices with n states and on the set of Laplacian/Kirchhoff
matrices on n states. They will arise naturally from the conductance model on
n states with i.i.d conductances. With the help of random matrix
theory, we will study the spectrum of these processes.

To study turbulence,B. Mandelbrot introduced a random fractal which is called

now Mandelbrot percolation or fractal percolation. The construction is as follows:

given an integer M _ 2 and a probability 0

This talk is devoted to Talagrand's transport-entropy inequality and its deep connections to the concentration of measure phenomenon, large deviation theory and logarithmic Sobolev inequalities. After an introductive part on the field, I will present recent results obtained with P-M Samson and C. Roberto establishing the equivalence of Talagrand's inequality to a restricted version of the Log-Sobolev inequality. If time enables, I will also present some works in progress about transport inequalities in a discrete setting.

Abstract: First we provide a survey on the long-time behaviour of stochastic delay equations with bounded memory, addressing existence and uniqueness of invariant measures, Lyapunov spectra, and exponential growth rates.

Then, we study the very simple one-dimensional equation $dX(t)=X(t-1)dW(t)$ in more detail and establish the existence of a deterministic exponential growth rate of a suitable norm of the solution via a Furstenberg-Hasminskii-type formula.

Parts of the talk are based on joint work with Martin Hairer and Jonathan Mattingly. 

 Abstract:  In this talk we will introduce a new particle approximation scheme to solve the stochastic filtering problem. This new scheme makes use of the Kusuoka-Lyons-Victoir (KLV) method to approximate the dynamics of the signal. In order to control the computational cost, a partial sampling procedure based on the tree based branching algorithm (TBBA) is performed. The novelty of the method lies in the fact that the weights used in the TBBA are computed combining the cubature weights and the filtering weights. In this way, we can avoid the sample degeneracy problem inherent to particle filters. We will also present some simulations showing the performance of the method.

It is a general belief that the heat kernels on fractals should exhibit highly oscillatory behaviors as opposed to the classical case of Riemannian manifolds.

For example, on a class of finitely ramified fractals, called (affine) nested fractals, a canonical ``Brownian motion" has been constructed and its transition density (heat kernel) $p_{t}(x,y)$ satisfies $c_{1} \leq t^{d_{s}/2} p_{t}(x,x) \leq c_{2}$ for $t \leq 1$ for any point $x$ of the fractal; here $d_{s}$ is the so-called spectral dimension. Then it is natural to ask whether the limit of this quantity as $t$ goes to 0 exists or not, and it has been conjectured NOT to exist by many people.

In this talk, I will present partial affirmative answers to this conjecture. First, for a general (affine) nested fractal, the non-existence of the limit is shown to be true for a ``generic" (in particular, almost every) point. Secondly, the same is shown to be valid for ANY point of the fractal in the particular cases of the $d$-dimensional standard Sierpinski gasket with $d\geq 2$ and of the $N$-polygasket with $N\geq 3$ odd, e.g. the pentagasket ($N=5$) and the heptagasket ($N=7$).

Abstract : The question adressed in this talk is the following one : how are the extreme eigenvalues of a matrix X moved by a small rank perturbation P of X ?
We shall consider this question in its generic apporach, i.e. when the matrices X and P are chosen at random independently and in isotropic ways.
We shall give a general answer, uncovering a remarkable phase transition phenomenon: the limit of the extreme eigenvalues of the perturbed matrix differs from the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. We also examine the consequences of this eigenvalue phase transition on the associated eigenvectors and generalize our results to examine the case of multiplicative perturbations or of additive perturbations for the singular values of rectangular matrices.

Abstract: We will discuss systems of diffusion processes on the real line, in which the dynamics of every single process is determined by its rank in the entire particle system. Such systems arise in mathematical finance and statistical physics, and are related to heavy-traffic approximations of queueing networks. Motivated by the applications, we address questions about invariant distributions, convergence to equilibrium and concentration of measure for certain statistics, as well as hydrodynamic limits and large deviations for these particle systems. Parts of the talk are joint works with Amir Dembo, Tomoyuki Ichiba, Ioannis Karatzas, Soumik Pal and Ofer Zeitouni

Abstract: We will discuss systems of diffusion processes on the real line, in which the dynamics of every single process is determined by its rank in the entire particle system. Such systems arise in mathematical finance and statistical physics, and are related to heavy-traffic approximations of queueing networks. Motivated by the applications, we address questions about invariant distributions, convergence to equilibrium and concentration of measure for certain statistics, as well as hydrodynamic limits and large deviations for these particle systems. Parts of the talk are joint works with Amir Dembo, Tomoyuki Ichiba, Ioannis Karatzas, Soumik Pal and Ofer Zeitouni

The notion quantization originates from information theory, where it refers to the approximation of a continuous signal on a discrete set. Our research on quantization is mainly motivated by applications in quadrature problems. In that context, one aims at finding for a given probability measure $\mu$ on a metric space a discrete approximation that is supported on a finite number of points, say $N$, and is close to $\mu$ in a Wasserstein metric.

In general it is a hard problem to find close to optimal quantizations, if  $N$ is large and/or  $\mu$ is given implicitly, e.g. being the marginal distribution of a stochastic differential equation. In this talk we analyse the efficiency of empirical measures in the constructive quantization problem. That means the random approximating measure is the uniform distribution on $N$ independent $\mu$-distributed elements.

We show that this approach is order order optimal in many cases. Further, we give fine asymptotic estimates for the quantization error that involve moments of the density of the absolutely continuous part of $\mu$, so called high resolution formulas. The talk ends with an outlook on possible applications and open problems.

The talk is based on joint work with Michael Scheutzow (TU Berlin) and Reik Schottstedt (U Marburg).

We present some recent results on the metastability of continuous time Markov chains on finite sets using potential theory. This approach is applied to the case of supercritical zero range processes.

I will report joint work with Wolfhard Hansen, Tomasz Jakubowski, Sebastian Sydor and Karol Szczypkowski on perturbations of semigroups and integral kernels, ones which produce comparable semigroups and integral kernels.

Several stochastic simulation algorithms (SSAs) have been recently

proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this talk, two commonly used SSAs will  be studied. The first SSA is an on-lattice model described by the  reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual  molecules and their reactive collisions. The connections between SSAs  and the deterministic models (based on reaction- diffusion PDEs) will  be presented. I will consider chemical reactions both at a surface  and in the bulk. I will show how the "microscopic" parameters should  be chosen to achieve the correct "macroscopic" reaction rate. This  choice is found to depend on which SSA is used. I will also present  multiscale algorithms which use models with a different level of  detail in different parts of the computational domain

We treat the first n terms of general orthogonal series evolving with n as the partial sum process, and proved that under Menshov-Rademacher condition, the partial sum process can be enhanced into a geometric 2-rough process. For Fourier series, the condition can be improved, with an equivalent condition on limit function identified.

We consider the forest fire process on Z: on each site, seeds and matches fall at random, according to some independent Poisson processes. When a seed falls on a vacant site, a tree immediately grows. When a match falls on an occupied site, a fire destroys immediately the corresponding occupied connected component. We are interested in the asymptotics of rare fires. We prove that, under space/time re-scaling, the process converges (as matches become rarer and rarer) to a limit forest fire process.
Next, we consider the more general case where seeds and matches fall according to some independent stationary renewal processes (not necessarily Poisson). According to the tail distribution of the law of the delay between two seeds (on a given site), there are 4 possible scaling limits.
We finally introduce some related coagulation-fragmentation equations, of which the stationary distribution can be more or less explicitely computed and of which we study the scaling limit.

We will consider a branching Brownian motion where particles have a drift $-\rho$, binary branch at rate $\beta$ and are killed if they hit the origin. This process is supercritical  if $\beta>\rho^2/2$ and we will discuss the survival probability in the regime as criticality is approached. (Joint work with Elie Aidekon)

We describe a construction of the Brownian measure on Jordan curves with respect to the Weil-Petersson metric. The step from Brownian motion on the diffeomorphism group of the circle to Brownian motion on Jordan curves in the complex plane requires probabilistic arguments well beyond the classical theory of conformal welding, due to the lacking quasi-symmetry of canonical Brownian motion on Diff(S1). A new key step in our construction is the systematic use of a Kählerian diffusion on the space of Jordan curves for which the welding functional gives rise to conformal martingales.

To identify the Martin boundary for a transient Markov chain with Green's function G(x,y), one has to identify all possible limits Lim G(x,y_n)/G(0,y_n) with y_n "tending to infinity". For homogeneous random walks, these limits are usually obtained from the exact asymptotics of Green's function G(x,y_n). For non-homogeneous random walks, the exact asymptotics af Green's function is an extremely difficult problem. We discuss several examples where Martin boundary can beidentified by using large deviation technique. The minimal Martin boundary is in general not homeomorphic to the "radial"  compactification obtained by Ney and Spitzer for homogeneous random walks in Z^d : convergence of a sequence of points y_n toa point on the Martin boundary does not imply convergence of the sequence y_n/|y_n| on the unit sphere. Such a phenomenon is a consequence of non-linear optimal large deviation trajectories.