Certain positive-definite kernels

Abstract

In one way or another, the extension of the standard Brownian motion process { B t : t ∈ [ 0 , ∞ ) } \{ {B_t}:t \in [0,\infty )\} to a (Gaussian) random field { B t : t ∈ R + d } \{ {B_t}:{\text {t}} \in {\mathbf {R}}_ + ^d\} involves a proof of the positive semi-definiteness of the kernel used to generalize ρ ( s , t ) = cov( B s , B t ) = s ∧ t \rho (s,t) = {\text {cov(}}{{\text {B}}_s}{\text {,}}{{\text {B}}_t}{\text {) = s}} \wedge t to multidimensional time. Simple direct analytical proofs are provided here for the cases of (i) the Lévy multiparameter Brownian motion, (ii) the Chentsov Brownian sheet, and (iii) the multiparameter fractional Brownian field.