= Longitudinal Brownian field with transverse covariance
{title2=$\mathbb E[d\mathcal B_x(z)d\mathcal B_x(z')]=B(z-z')dx$}
= Brownian field
{c}
{synonym}
A longitudinal Brownian field $\mathcal B_x(z)$ is a centered <Gaussian random field> with
$$
\mathbb E[\mathcal B_x(z)\mathcal B_{x'}(z')]=\min(x,x')B(z-z'),\qquad x,x'\geq0,
$$
where $B$ 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.
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