## Distributions of sojourn time, maximum and minimum for pseudo-processes governed by higher-order heat-type equations.(English)Zbl 1064.60078

Summary: The higher-order heat-type equation $$\partial u/\partial t=\pm \partial^n u/\partial x^n$$ has been investigated by many authors. With this equation a pseudo-process $$(X_t)_{t\geq 0}$$ is associated which is governed by a signed measure. In the even-order case, V. Yu. Krylov [Sov. Math., Dokl. 1, 760–763 (1960); translation from Dokl. Akad. Nauk SSSR 132, 1254–1257 (1960; Zbl 0095.32703)] proved that the classical arc-sine law of Paul Lévy for standard Brownian motion holds for the pseudo-process $$(X_t)_{t\geq 0}$$, that is, if $$T_t$$ is the sojourn time of $$(X_t)_{t\geq 0}$$ in the half line $$(0,+ \infty)$$ up to time $$t$$, then $$\mathbb{P}(T_t\in ds)=\frac{ds}{\pi\sqrt{s(t-s)}}$$, $$0<s<t$$. E. Orsingher [Lith. Math. J. 31, 220–231 (1991) and Lit. Mat. Sb. 31, 323–336 (1991; Zbl 0729.60072)] and next K. J. Hochberg and E. Orsingher [Stochastic Processes Appl. 52, 273–292 (1994; Zbl 0811.60028)] obtained a counterpart to that law in the odd cases $$n=3, 5,7$$. Actually Hochberg and Orsingher proposed a more or less explicit expression for that new law in the odd-order general case and conjectured a quite simple formula for it. The distribution of $$T_t$$ subject to some conditioning has also been studied by Y. Nikitin and E. Orsingher [J. Theor. Probab. 13, 997–1012 (2000; Zbl 0993.60042)] in the cases $$n=3,4$$.
In this paper, we prove that the conjecture of Hochberg and Orsingher is true and we extend the results of Nikitin and Orsingher for any integer $$n$$. We also investigate the distributions of maximal and minimal functionals of $$(X_t)_{t\geq 0}$$, as well as the distribution of the last time before becoming definitively negative up to time $$t$$.

### MSC:

 60G20 Generalized stochastic processes 60J25 Continuous-time Markov processes on general state spaces

### Keywords:

Arc-sine law; Vandermonde systems and determinants
Full Text: