Convex geometry and nonconfluent γ-martingales II: well-posedness and γ-martingale convergence
Wilfrid S. Kendall · Stochastics and stochastics reports · 1992
In the terminology of the first paper of this series, a closed domain of a manifold furnished with a connection γ is said to have convex geometry, or property (A), if there is a bounded nonnegative γ-convex function defined on vanishing only on the diagonal. It is said to have property (B) if solutions to its Dirichlet problem for γ-martingales are unique and well-posed (depend continuously on their limiting values at time ∞) when they exist. In this paper it is shown that (A) and (B) are equivalent if 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 ∞ of nontrivial γ-martingales in are never nonrandom, and if the associated γ-martingale Liouville property is well-posed, then all γ-martingales in converge as time tends to infinity