Quadrature Domains and Brownian Motion (A Heuristic Approach)

Abstract

In this note we will make an attempt to link the theory of the so-called quadrature domains (QD) to stochastic analysis. We show that a QD, with the underlying measure μ, can be represented as the set of points x, for which the expectation value (average reward)

$$E^x \left( { - \theta + \int_0^\theta {\mu \left( {X_t } \right)} } \right),$$

is positive for some (bounded) stopping time θ. Here X _t denotes the Brownian motion starting at the point x, and E^x denotes the expectation with respect to the underlying probability measure P^x.