Gaussian reciprocal diffusions and positive definite sturm-liouville operators

Abstract

This paper examines the connection between Gaussian reciprocal diffusions (GRDs) on a finite interval [0, T] and positive definite Sturm-Liouville boundary values problems (BVPs) on the same interval. We show that there exists a bijection between the set of GRDs on [0, T] and the set of positive definite Sturm-Liouville BVPs on [0, T]. The bijection occurs through the identification of GRD covariances with Sturm-Liouville Green's functions. Furthermore, every GRD x(t) can be formulated as a weak solution of its corresponding Sturm-Liouville BVP, where the interior forcing terms and the boundary forcing terms form a stochastic object whose covariance structure is determined by that of x(t). This formulation differs from the one presented in [20] in that it represents x(t) as the solution of its matching positive definite Sturm-Liouville BVP, as opposed to a Dirichlet BVP. Finally, for GRDs with a negative stress tensor [11], it is shown that the Sturm-Liouville BVP they satisfy can be reformulated as a fir...