## 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

#### 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

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

Rights

#### 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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