Dimension Walks on Generalized Spaces

There has been an increasing interest in stochastic processes that are defined over product spaces in many branches of applied sciences, including climate modeling, atmospheric sciences, geophysical science, and even finance. Our work provides the mathematical foundations for certain classes of stochastic processes that are defined over special classes of product spaces. Specifically, let d, k be positive integers. We call generalized spaces the Cartesian product of the d-dimensional sphere, 𝕊d, with the k-dimensional Euclidean space, ℝk. We consider the class 𝒫 (𝕊d × ℝk) of continuous functions φ : [−1, 1] × [0, ∞) → ℝ such that the mapping C:(Sd×ℝk)2→ℝ, defined as C ((x, y), (x′, y′)) = φ (cos θ (x, x′), ‖y − y′‖), (x, y), (x′, y′) ∈ 𝕊d × ℝk, is positive definite. We propose linear operators that allow for walks through dimensions within generalized spaces while preserving positive definiteness.