Higher-order derivative of local times for space–time anisotropic Gaussian random fields

Let $X=\{X\left(t\right),t\in R^N\}$ be a centered space–time anisotropic Gaussian random field values in $R^d$. Under some general conditions, the existence and smoothness (in the sense of Meyer-Watanabe) of the higher-order derivative of the local times of $X\left(t\right)$ are studied. Moreover, we show that the derivatives of the local time of $X\left(t\right)$ is jointly continuous on $R^d\times {[0,1]}^N$. The existing results on local times of fractional Brownian motion and other Gaussian random fields are extended to higher-order derivative of local times of more general space–time anisotropic Gaussian random fields.