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 \(t\).


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