Longitudinal Brownian field with transverse covariance ($\mathbb{E}[d{\mathcal{B}}_x(z)d{\mathcal{B}}_x(z^{\prime })]=B(z-z^{\prime })dx$)

A longitudinal Brownian field ${\mathcal{B}}_x(z)$ is a centered Gaussian random field withwhere $B$ is a valid transverse covariance kernel. For each fixed $z$ it is a scaled Brownian motion, and increments over disjoint longitudinal intervals are independent. Such a field drives a Markov model of wave propagation in a random medium. Its formal longitudinal derivative is Gaussian white noise in $x$, with transverse correlations described by $B$; it is not in general a Brownian sheet.