×

First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation \(\frac{\partial}{\partial t}=\pm \frac {\partial ^{N}}{\partial x^N}\). (English) Zbl 1128.60028

Summary: Consider the high-order heat-type equation \(\partial u/\partial t=\pm\partial^Nu/\partial x^N\) for an integer \(N>2\) and introduce the related Markov pseudo-process \((X(t))_{t\geq0}\). In this paper, we study several functionals related to \((X(t))_{t\geq0}\): the maximum \(M(t)\) and minimum \(m(t)\) up to time \(t\); the hitting times \(\tau_a^+\) and \(\tau_a^-\) of the half lines \((a,+\infty)\) and \((-\infty,a)\), respectively. We provide explicit expressions for the distributions of the vectors \((X(t),M(t))\) and \((X(t),m(t))\), as well as those of the vectors \((\tau_a^+,X(\tau_a^+))\) and \((\tau_a^-,X(\tau_a^-))\).

MSC:

60G20 Generalized stochastic processes
60J25 Continuous-time Markov processes on general state spaces
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J05 Discrete-time Markov processes on general state spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML