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Bellman function in stochastic control and harmonic analysis

by F. Nazarov, S. Treil and A. Volberg


 Abstract. he stochastic optimal control uses the differential equation of
Bellman and its solution---the Bellman function. We show how the homonym
function in harmonic analysis is (and how it is not) the same stochastic
optimal control Bellman function. Then we present several creatures from
Bellman's Zoo: a function that proves the inverse Holder inequality, as well as several
other harmonic analysis Bellman functions and their corresponding Bellman
PDE's. Finally we translate the approach of Burkholder to the language of
``our" Bellman function.

The goal of this paper is almost entirely methodological: we relate the
ideas to each other, rather than presenting the new ones.

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