Some Sample Function Properties of the Two-parameter Gaussian Process

Abstract

Let $\{X(s, t; \omega): (s, t) \in \lbrack 0, \infty) \times \lbrack 0, \infty)\}$ be a two parameter Gaussian process with mean function zero and covariance function $R(s_1, t_1; s_2, t_2) = \min (s_1, s_2) \min (t_1, t_2)$. This paper derives a multiparameter law of the iterated logarithm and modulus of continuity for the process $X(s, t; \omega)$. Estimates are also given which enable the author to define an Ito type integral for a suitable class of functions and to solve a diffusion equation involving the process.