XIII: 33, 385-399, LNM 721 (1979)
LE JAN, Yves
Martingales et changement de temps (
Martingale theory,
Markov processes)
The first part of the paper concerns changes of time by a continuous (not strictly increasing) process, with a detailed computation, for instance, of the continuous martingale part of a time-changed martingale. This is a useful addition to
1108 and
1109. The second part is an application to classical potential theory: the martingale is a harmonic function along Brownian motion in a domain, stopped at the boundary; the change of time is defined by a boundary local time. Then the time-changed Brownian motion is a Markov process on the boundary, the time-changed martingale is purely discontinuous, and the computation of its quadratic norm leads to the Douglas formula, which expresses the Dirichlet integral of the harmonic function by a quadratic double integral of its restriction to the boundary
Keywords: Changes of time,
Energy,
Douglas formulaNature: Original
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