First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation $\partial/\partial t = \pm\partial^N/\partial x^N$

Details First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation $\partial/\partial t = \pm\partial^N/\partial x^N$ First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation $\partial/\partial t = \pm\partial^N/\partial x^N$

2007-02-19

arxiv.org/abs/math/0702541v1

Electronical Journal of Probability, vol. 12 (2007), pp. 300-353

Mathematics

Probability

51 pages

Scientific paper

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 $(X(t))_{t\ge 0}$. In this paper, we study several functionals related to $(X(t))_{t\ge 0}$: 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 $(\tau_a^-,X(\tau_a^-))$.

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