June  8, 2023

Seminar on Stochastic Processes - 1999

March 18 - 20, 1999


David Aldous (UC Berkeley)
From Random Walks on Discrete Tree Space to Diffusions on Continuous Tree Space

As a gross simplification of practical problems of reconstructing phylogenetic trees from DNA data via Markov Chain Monte Carlo, we consider two models of reversible Markov chains on the (finite) state space of all $n$-vertex trees. How does the mixing time ($1/$spectral gap) scale with $n$? In work-in-progress we try the weak convergence methodology. The $n \to \infty$ limit of spatially-rescaled random trees is the continuum random tree. So presumably each Markov chain, with suitable time-rescaling, converges weakly to some diffusion of the space of continuum trees. Understanding what these diffusions are is a challenging problem. For instance, a functional of one is the $k$-allele Wright-Fisher diffusion with negative [sic] mutation rates.

Bruce K. Driver (UC San Diego)
On path integral formulas on manifolds

This talk will discuss joint work with Lars Andersson on certain natural geometric approximation schemes for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral expression, $$ ``d\nu (\sigma )=Z^{-1}\exp \left( -\frac{1}{2}\int_{0}^{1}| \sigma ^{\prime}(s)|^{2}ds\right) \mathcal{D}\sigma ,'' $$ used in the physics literature for representing the heat semi-group on Riemannian manifolds.

Hans Föllmer (Humboldt Universität )
Some probabilistic aspects of insider trading

From a mathematical point of view, insider trading involves an enlargement of filtration. From an economic point of view, the question arises whether insider strategies based on the larger filtration can be detected by analyzing the resulting price fluctuation. In a simple model based on Brownian motion, Kyle and Back have shown that there exist insider strategies which maximize expected gain and yet remain inconspicuous, i.e., they leave invariant the underlying Wiener measure. We report on recent joint work with C.T. Wu and M. Yor on transformations of Brownian motion and on ``weak Brownian motions of order $k$'', which was motivated by such questions.

Leonid Mytnik (Technion)
A duality approach to proving uniqueness

A duality technique is useful in proving weak uniqueness of solutions to some martingale problems. We use the duality approach to establish the uniqueness in law for the heat equation with noise \[ \frac{\partial X_{t}}{\partial t} =\frac{1}{2}\Delta X_{t}+X^{\gamma}_{t}\dot{W} \] for $1/2<\gamma<1$; the proof requires the construction of an approximating sequence of dual processes. We also show how duality helped to prove the uniqueness in law for a system of SPDEs \[ \lfi\begin{array}{l} \frac{\partial U^{1}_{t}}{\partial t}= \frac{1}{2}\Delta U^{1}_{t} +\sqrt{U^{1}_{t}U^{2}_{t}}\,\dot{W}_{1}\\ \\ \frac{\partial U^{2}_{t}}{\partial t}= \frac{1}{2}\Delta U^{2}_{t} + \sqrt{U^{1}_{t}U^{2}_{t}}\,\dot{W}_{2}\\ \end{array} \right. \] which describes a mutually catalytic branching model.

Wendelin Werner (Univ. Paris Sud, Orsay)
Intersection exponents

This talk is based on joint work with Greg Lawler. We will discuss recent developments concerning critical exponents for two-dimensional conformal invariant systems. We will mainly focus on the so-called intersection exponents between planar Brownian paths. These exponents describe the decay when time runs to infinity of the probability that several planar Brownian paths do not intersect.

Just as for many two-dimensional statistical physics systems for which conformal invariance is predicted in the scaling limit (for instance self-avoiding walks, critical percolation etc), physicists conjectured that many of these Brownian intersection exponents take rational values. We will first show how to define generalized intersection exponents that loosely speaking descibe non-intersection iprobabilities between non-integer numbers of Brownian paths. In other words, the discrete sequence of intersection exponents can be replaced in a natural way by a continuous function. We will then show that these generalized exponents (and therefore also the usual ones) must satisfy certain functional relations that lead to general conjectures concerning their exact values.
In particular, this seems to indicate that some of the Brownian intersection exponents are not rational numbers. Finally, we will point out that these generalized Brownian intersection exponents are universal in the following sense: any set defined under a measure with a certain conformal invariance property that we shall discuss, behaves (as what all intersection exponents are concerned) exactly like a ``generalized union of planar Brownian paths''.