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Joint distribution of the process and its sojourn time on the positive half-line for pseudo-processes governed by high-order heat equation. (English) Zbl 1231.60032

Summary: Consider the high-order heat-type equation \(\partial u/\partial t=\pm \partial ^{N} u/\partial x^{N}\) for an integer \(N>2\) and introduce the related Markov pseudo-process \((X(t))_{t\geq 0}\). In this paper, we study the sojourn time \(T(t)\) in the interval \([0,+\infty \)) up to a fixed time \(t\) for this pseudo-process. We provide explicit expressions for the joint distribution of the couple \((T(t),X(t))\).

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
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