Results on Local Times of a Class of Multiparameter Gaussian Processes

Abstract

In this paper, we introduce a class of Gaussian processes $Y = {\left\{ {Y{\left( t \right)}:t \in R^{N}_{ + } } \right\}}$, the so called bifractional Brownian motion with the indexes H = (H₁, • • •,H _N ) and α. We consider the (N, d,H,α) Gaussian random field

$$ X{\left( t \right)} = {\left( {X_{1} {\left( t \right)},\raise0.145em\hbox{${\scriptscriptstyle \bullet}$}\raise0.145em\hbox{${\scriptscriptstyle \bullet}$}\raise0.145em\hbox{${\scriptscriptstyle \bullet}$},X_{d} {\left( t \right)}} \right)}, $$

where X₁(t), • • •,X _d (t) are independent copies of Y (t). At first we show the existence and join continuity of the local times of $X = {\left\{ {X{\left( t \right)},t \in R^{N}_{ + } } \right\}}$, then we consider the Hölder conditions for the local times.