Non-metric connections for non-Markovian dynamics

Investigate whether non-metric affine connections can systematically classify and characterize non-Markovian dynamics and generalized diffusion processes obtained by allowing time-dependent noise amplitudes or incorporating memory effects into stochastic dynamics.

Background

The paper establishes a correspondence between alpha-geodesics on the Gaussian statistical manifold and continuous stochastic processes driven by linear stochastic differential equations with time-independent additive noise. The authors note that a curve of one-time marginal distributions on the statistical manifold can also be lifted to other stochastic dynamics, including processes with time-dependent noise amplitudes and non-Markovian stochastic differential equations.

The unresolved problem is to determine whether the non-metric affine-connection framework can be extended beyond the linear Gauss-Markov setting so that it systematically classifies and characterizes these broader classes of stochastic dynamics and generalized diffusion processes.