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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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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  • Beghin, Luisa; Hochberg, Kenneth J.; Orsingher, Enzo. Conditional maximal distributions of processes related to higher-order heat-type equations. Stochastic Process. Appl. 85 (2000), no. 2, 209–223. MR1731022
  • Beghin, L.; Orsingher, E. The distribution of the local time for "pseudoprocesses" and its connection with fractional diffusion equations. Stochastic Process. Appl. 115 (2005), no. 6, 1017–1040. MR2138812
  • Beghin, L.; Orsingher, E.; Ragozina, T. Joint distributions of the maximum and the process for higher-order diffusions. Stochastic Process. Appl. 94 (2001), no. 1, 71–93. MR1835846
  • Benachour, S.; Roynette, B.; Vallois, P. Explicit solutions of some fourth order partial differential equations via iterated Brownian motion. Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1996), 39–61, Progr. Probab., 45, Birkhäuser, Basel, 1999. MR1712233
  • H. Bohr. Über die gleichmässige Konvergenz Dirichletscher Reihen, J. für Math. 143 (1913), 203–211.
  • Funaki, Tadahisa. Probabilistic construction of the solution of some higher order parabolic differential equation. Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 5, 176–179. MR0533542
  • Hochberg, Kenneth J. A signed measure on path space related to Wiener measure. Ann. Probab. 6 (1978), no. 3, 433–458. MR0490812
  • Hochberg, Kenneth J.; Orsingher, Enzo. The arc-sine law and its analogs for processes governed by signed and complex measures. Stochastic Process. Appl. 52 (1994), no. 2, 273-292. MR1290699
  • Hochberg, Kenneth J.; Orsingher, Enzo. Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations. J. Theoret. Probab. 9 (1996), no. 2, 511–532. MR1385409
  • Krylov, V. Ju. Some properties of the distribution corresponding to the equation $\partial u/\partial t=(-1)^{q+1} \partial ^{2q}u/\partial x^{2q}$. Dokl. Akad. Nauk SSSR 132 1254–1257 (Russian); translated as Soviet Math. Dokl. 1 1960 760–763. MR0118953
  • Lachal, Aimé. Distributions of sojourn time, maximum and minimum for pseudo-processes governed by higher-order heat-type equations. Electron. J. Probab. 8 (2003), no. 20, 53 pp. (electronic). MR2041821
  • Lachal, Aimé. Lois conjointes du processus et de son maximum, des premier instant et position d'atteinte d'une demi-droite pour le pseudo-processus régi par l'équation $\frac{\partial}{\partial t}=\pm\frac{\partial^ N}{\partial x^ N}$. (French) [Joint law of the process and its maximum, first hitting time and place of a half-line for the pseudo-process driven by the equation $\frac{\partial}{\partial t}=\pm\frac{\partial^ N}{\partial x^ N}$] C. R. Math. Acad. Sci. Paris 343 (2006), no. 8, 525–530. MR2267588
  • Motoo, Minoru. An analogue to the stochastic integral for $\partial/\partial t=-\Delta^ 2$. Stochastic analysis and applications, 323–338, Adv. Probab. Related Topics, 7, Dekker, New York, 1984. MR0776986
  • Nakajima, Tadashi; Sato, Sadao. On the joint distribution of the first hitting time and the first hitting place to the space-time wedge domain of a biharmonic pseudo process. Tokyo J. Math. 22 (1999), no. 2, 399–413. MR1727883
  • Nikitin, Y.; Orsingher, E. On sojourn distributions of processes related to some higher-order heat-type equations. J. Theoret. Probab. 13 (2000), no. 4, 997–1012. MR1820499
  • Nishioka, Kunio. Monopoles and dipoles in biharmonic pseudo-process. Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 3, 47–50. MR1391894
  • Nishioka, Kunio. The first hitting time and place of a half-line by a biharmonic pseudo process. Japan. J. Math. (N.S.) 23 (1997), no. 2, 235–280. MR1486514
  • Nishioka, Kunio. Boundary conditions for one-dimensional biharmonic pseudo process. Electron. J. Probab. 6 (2001), no. 13, 27 pp. (electronic). MR1844510
  • Orsingher, Enzo. Processes governed by signed measures connected with third-order "heat-type” equations. Litovsk. Mat. Sb. 31 (1991), no. 2, 323–336; translation in Lithuanian Math. J. 31 (1991), no. 2, 220–231 (1992) MR1161372