## Seminar on Stochastic Processes - 1999

March 18 - 20, 1999

### ABSTRACTS

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''.