Joint distributions of the maximum and the process for higher-order diffusions

For processes X(t),t>0 governed by signed measures whose density is the fundamental solution of third and fourth-order heat-type equations (higher-order diffusions) the explicit form of the joint distribution of (max0⩽s⩽t X(s),X(t)) is derived. The expressions presented include all results obtained so far and, for the third-order case, prove to be genuine probability distributions. The case of more general fourth-order equations is also investigated and the distribution of the maximum is derived.