Les processus de Dirichlet en tant qu’espace de Banach. (Dirichlet processes as Banach spaces). (French) Zbl 0602.60069

What is called a Dirichlet process is any sum of a square integrable martingale and a process of vanishing quadratic variation (plus some natural assumptions of adaptivity and continuity), but when the last is defined the partitions of the time interval [0,1] are supposed to be defined by stopping times rather than fixed times. The class \({\mathcal D}\) of all such processes is proved to be a Banach space in which semimartingales are dense. When the process \(X_ t\), \(t\in [0,1]\), and the function F obey certain conditions it is proved that \(F(X_ t)- F(X_ 0)\) is a Dirichlet process and its martingale part is explicitly given. A necessary condition and a sufficient condition is found for a function f so that \(f(B_ t)\) is a Dirichlet process \((B_ t\) is the Brownian motion).
Reviewer: O.Enchev


60J60 Diffusion processes
60J65 Brownian motion
60G44 Martingales with continuous parameter
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