Quadrature Domains and Brownian Motion (A Heuristic Approach)

  • Henrik Shahgholian
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 156)


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 .


Brownian motion quadrature domains variational inequalities 


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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Henrik Shahgholian
    • 1
  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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