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Convex geometry and nonconfluent \(\Gamma\)-martingales. II: Well-posedness and \(\Gamma\)-martingale convergence. (English) Zbl 0758.58036

Summary: In the terminology of the first paper of this series [Stochastic analysis, Proc.Symp., Durham/UK 1990, Lond. Math. Soc. Lect. Note Ser. 167, 163-178 (1991; Zbl 0747.58051)], a closed domain \(\mathcal B\) of a manifold furnished with a connection \(\Gamma\) is said to have convex geometry, or property (A), if there is a bounded nonnegative \(\Gamma\)- convex function \(\mathcal Q\) defined on \({\mathcal B}\times{\mathcal B}\) vanishing only on the diagonal. It is said to have property (B) if solutions to its Dirichlet problem for \(\Gamma\)-martingales are unique and well-posed (depend continuously on their limiting values at time \(\infty\)) when they exist. In this paper it is shown that (A) and (B) are equivalent if \(\mathcal B\) is compact, strengthening Theorem 3.2 of the first paper of this series. In the course of the proof a result of independent interest is established: if the limits at time \(\infty\) of nontrivial \(\Gamma\)- martingales in \(\mathcal B\) are never nonrandom, and if the associated \(\Gamma\)-martingale Liouville property is well-posed, then all \(\Gamma\)- martingales in \(\mathcal B\) converge as time tends to infinity.

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
60G48 Generalizations of martingales
26B25 Convexity of real functions of several variables, generalizations

Citations:

Zbl 0747.58051
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