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 $(\max_{0\leqs\leqt}X(s),X(t))$ 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