Electronic Journal of Probability

First Hitting Time and Place, Monopoles and Multipoles for Pseudo-Processes Driven by the Equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$

Aimé Lachal

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Abstract

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 $(\tau_a^-,X(\tau_a^-))$.

Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 11, 300-353.

Dates
Accepted: 28 March 2007
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464818482

Digital Object Identifier
doi:10.1214/EJP.v12-399

Mathematical Reviews number (MathSciNet)
MR2299920

Zentralblatt MATH identifier
1128.60028

Subjects
Primary: 60G20: Generalized stochastic processes
Secondary: 60J25: Continuous-time Markov processes on general state spaces

Keywords
pseudo-process joint distribution of the process and its maximum/minimum first hitting time and place Multipoles Spitzer's identity

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Lachal, Aimé. First Hitting Time and Place, Monopoles and Multipoles for Pseudo-Processes Driven by the Equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$. Electron. J. Probab. 12 (2007), paper no. 11, 300--353. doi:10.1214/EJP.v12-399. https://projecteuclid.org/euclid.ejp/1464818482


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