Krzysztof Burdzy

Some Path Properties of Iterated Brownian Motion

Suppose that X1, X2 and Y are independent standard Brownian motions starting from 0 and let

Configurational transition in a Fleming - Viot-type model and probabilistic interpretation of Laplacian eigenfunctions

X\left( t \right) = \left\{ {\begin{array}{*{20}{c}} {{X^1}\left( t \right) if t \geqslant 0,} \\ {{X^2}\left( { - t} \right) if t < 0.} \end{array}} \right.

On Neumann eigenfunctions in lip domains

We will consider the process \( \left\{ {Z\left( t \right)\underline{\underline {df}} X\left( {Y\left( t \right)} \right),t \geqslant 0} \right\}\) which we will call “iterated Brownian motion” or simply IBM. Funaki (1979) proved that a similar process is related to “squared Laplacian.” Krylov (1960) and Hochberg (1978) considered finitely additive signed measures on the path space corresponding to squared Laplacian (there exists a genuine probabilistic approach, see, e.g., Mądrecki and Rybaczuk (1992). A paper of Vervaat (1985) contains a section on the composition of self-similar processes.

Lifetimes of conditioned diffusions

We study a coordination game with randomly changing payoffs and small frictions in changing actions. Using only backwards induction, we find that players must coordinate on the risk-dominant equilibrium. More precisely, a continuum of fully rational players are randomly matched to play a symmetric 2 x 2 game. The payoff matrix changes according to a random walk. Players observe these payoffs and the population distribution of actions as they evolve. The game has frictions: opportunities to change strategies arrive from independent random processes, so that the players are locked into their actions for some time. As the frictions disappear, each player ignores what the others are doing and switches at her first opportunity to the risk-dominant action. History dependence emerges in some cases when frictions remain positive.

The hot spots problem in planar domains with one hole

We analyse and simulate a two-dimensional Brownian multi-type particle system with death and branching (birth) depending on the position of particles of different types. The system is confined in a two-dimensional box, whose boundaries act as the sink of Brownian particles. The branching rate matches the death rate so that the total number of particles is kept constant. In the case of m types of particle in a rectangular box of size and elongated shape we observe that the stationary distribution of particles corresponds to the mth Laplacian eigenfunction. For smaller elongations a > b we find a configurational transition to a new limiting distribution. The ratio a/b for which the transition occurs is related to the value of the mth eigenvalue of the Laplacian with rectangular boundaries.

Hölder domains and the boundary Harnack principle

The boundary Harnack principle for the ratio of positive harmonic functions is shown to hold in twisted Holder domains of order a for a E (1/2, 1]. For each a E (0, 1/2), there exists a twisted H6lder domain of order a for which the boundary Harnack principle fails. Extensions are given to L-harmonic

A probabilistic proof of the boundary Harnack principle

Atars research partially supported by the fund for the promotion of research at the Technion. Burdzy gratefull acknowledges the hospitality and financial support of Technion (Israel) and Institut Mittag-Leffler (Sweden). This research was partially supported by NSF Grant DMS-0071486 and ISF Grant 12/98.

Non-intersection exponents for Brownian paths

SummaryWe investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and non-divergence form can be found. The critical value of the parameter is found for each of the following classes of domains:Lp-domains (p=n−1), uniformly regular twistedLp-domains (p=n−1), and twisted Hölder domains (α=1/3). A related parabolic boundary Harnack principle is proved.