On Sojourn Distributions of Processes Related to Some Higher-Order Heat-Type Equations

It is well known that the sojourn time of Brownian motion B(t), t>0 on the positive half-line, during the interval [0, t] and under the condition B(t)=0, is uniformly distributed, while it has the form of the "corrected arc-sine law" when the condition B(t)>0 is assumed. We find the analogues of these laws for "processes" X(t), t>0 governed by signed measures whose densities are the fundamental solutions of third and fourth-order heat-type equations. Surprisingly, both laws hold for the fourth-order "process." The uniform law is still valid for the thirdorder "process" but a different law emerges when the condition X(t)>0 is considered.