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The calculus of boundary processes. (English) Zbl 0561.60081
A diffusion process \((X_ t,Z_ t)\), \(t\geq 0\), is considered taking values in the half space \(R^ d\times R^+\) with the reflexion and the drift on the boundary and the local time \(L_ t\), \(t\geq 0\), on it. By means of probabilistic methods the properties of smoothness of the transition probability function of the strong Markov process \((A_ t\), \(X_{A_ t})\), \(t\geq 0\), are investigated, where \(A_ t\), \(t\geq 0\), is the right-continuous inverse of \(L_ t\), \(t\geq 0.\)
The techniques of stochastic flows, the calculus of variations on diffusions and jump processes, the Ito’s theory of excursions and the stochastic calculus of semimartingales are developed and adapted. The analogous problem in the case of a diffusion process \((X_ t,Z_ t)\), \(t\geq 0\), taking values in \(R^ d\times R\) with the reflexion and the drift on the two-sided boundary \(R^ d\times \{0\}\) is also discussed.
Reviewer: B.Grigelionis

MSC:
60J60 Diffusion processes
60J50 Boundary theory for Markov processes
60J55 Local time and additive functionals
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