Introduction

In Kendall (1990) it is explained how three nonlinear Dirichlet problems are closely connected to a problem about the existence of a certain convex surrogate distance function. Here we consider an aspect of these relationships in a setting more general than the Riemannian case of Kendall (1990). The problems are as follows. Suppose M is a smooth manifold equipped with a connection Λ and separately with a reference Riemannian structure (the connection need not be compatible with the metric!). Consider B a closed region in M. (In the sequel B is generally compact, but we prefer to state the following properties as applicable to a general region.)

(A): Does Β have Λ-convex geometry? That is to say, does there exist a (product-connection) convex function Q : Β × Β → [0, 1] vanishing only on the diagonal Δ = {(x, x) : x ∈ Β}? Here the “Λ” in “Λ-convex” refers to the use of the connection Λ to build the product-connection, instead of the Levi–Civita connection supplied by the reference Riemannian metric. (In the rest of the paper the prefix “Λ” is omitted; by “convex” we mean “Λ-convex” unless indicated otherwise.)

(B): Dirichlet problem for Λ-martingales lying in Β. This problem requires one to find Λ-martingales X (under a given filtration) attaining a given terminal value X(∞). In the following the heading (B) refers specifically to whether the Dirichlet problem is well-posed and has unique solutions.