## Electronic Journal of Probability

### Joint Distribution of the Process and its Sojourn Time on the Positive Half-Line for Pseudo-Processes Governed by High-Order Heat Equation

#### Abstract

Consider the high-order heat-type equation $\partial_t u=\pm \partial^n_x u$ for an integer $n>2$ and introduce the related Markov pseudo-process $(X(t))_{t\geq0}$. 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 $(T(t),X(t))$.

#### Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 28, 895-931.

Dates
Accepted: 17 June 2010
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819813

Digital Object Identifier
doi:10.1214/EJP.v15-782

Mathematical Reviews number (MathSciNet)
MR2653948

Zentralblatt MATH identifier
1231.60032

Rights

#### Citation

Lachal, Aimé; Cammarota, Valentina. Joint Distribution of the Process and its Sojourn Time on the Positive Half-Line for Pseudo-Processes Governed by High-Order Heat Equation. Electron. J. Probab. 15 (2010), paper no. 28, 895--931. doi:10.1214/EJP.v15-782. https://projecteuclid.org/euclid.ejp/1464819813

#### References

• M. Abramowitz and I.-A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, 1992.
• L. Beghin, K.-J. Hochberg and E. Orsingher. Conditional maximal distributions of processes related to higher-order heat-type equations. Stochastic Process. Appl. 85 (2000), no. 2, 209-223.
• L. Beghin and E. Orsingher. The distribution of the local time for "pseudoprocesses" and its connection with fractional diffusion equations. Stochastic Process. Appl. 115 (2005), 1017-1040.
• L. Beghin, E. Orsingher and T. Ragozina. Joint distributions of the maximum and the process for higher-order diffusions. Stochastic Process. Appl. 94 (2001), no. 1, 71-93.
• H. Bohr. Über die gleichmässige Konvergenz Dirichletscher Reihen. J. für Math. 143 (1913), 203-211.
• V. Cammarota and A. Lachal. arXiv:1001.4201 [math.PR].
• A. Erdélyi, Editor. Higher transcendental functions, vol. III, Robert Krieger Publishing Company, Malabar, Florida, 1981.
• K.-J. Hochberg. A signed measure on path space related to Wiener measure. Ann. Probab. 6 (1978), no. 3, 433-458.
• K.-J. Hochberg and E. Orsingher. The arc-sine law and its analogs for processes governed by signed and complex measures. Stochastic Process. Appl. 52 (1994), no. 2, 273-292.
• V. Yu. Krylov. Some properties of the distribution corresponding to the equation $frac{partial u}{partial t}=(-1)^{q+1}frac{partial^{2q} u}{partial^{2q} x}$. Soviet Math. Dokl. 1 (1960), 760-763.
• A. Lachal. Distribution of sojourn time, maximum and minimum for pseudo-processes governed by higer-order heat-typer equations. Electron. J. Probab. 8 (2003), paper no. 20, 1-53.
• A. Lachal. 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 $frac{partial}{partial t}=pmfrac{partial^N}{partial x^N}$. C. R. Acad. Sci. Paris, SÃƒÂˆr. I 343 (2006), no. 8, 525-530.
• A. Lachal. First hitting time and place, monopoles and multipoles for pseudo-procsses driven by the equation $frac{partial}{partial t}=pm frac{partial^N}{partial x^N}$. Electron. J. Probab. 12 (2007), 300-353.
• A. Lachal. First hitting time and place for the pseudo-process driven by the equation $frac{partial}{partial t}=pmfrac{partial^n}{partial x^n}$ subject to a linear drift. Stoch. Proc. Appl. 118 (2008), 1-27.
• T. Nakajima and S. Sato. 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.
• Ya. Yu. Nikitin and E. Orsingher. On sojourn distributions of processes related to some higher-order heat-type equations. J. Theoret. Probab. 13 (2000), no. 4, 997-1012.
• K. Nishioka. Monopole and dipole of a biharmonic pseudo process. Proc. Japan Acad. Ser. A 72 (1996), 47-50.
• K. Nishioka. The first hitting time and place of a half-line by a biharmonic pseudo process. Japan J. Math. 23 (1997), 235-280.
• K. Nishioka. Boundary conditions for one-dimensional biharmonic pseudo process. Electronic Journal of Probability 6 (2001), paper no. 13, 1-27.
• E. Orsingher. Processes governed by signed measures connected with third-order "heat-type" equations. Lithuanian Math. J. 31 (1991), no. 2, 220-231.
• F. Spitzer. A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82 (1956), 323-339.