Cyclic planar random evolution with four directions

Alexander D. Kolesnik

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2004), pp. 27-32.

Résumé

A four-direction cyclic random motion with constant finite speed $v$ in the plane $R^2$ driven by a homogeneous Poisson process of rate $\lambda >0$ is studied. A fourth-order hyperbolic equation with constant coefficients governing the transition law of the motion is obtained. A general solution of the Fourier transform of this equation is given. A special non-linear automodel substitution is found reducing the governing partial differential equation to the generalized fourth-order ordinary Bessel differential equation, and the fundamental system of its solutions is explicitly given.

Bibliographie

[1] Brooks E. A., “Probabilistic methods for a linear reaction-hyperbolic system with constant coefficients”, Ann. Appl. Probab., 9 (1999), 719–731 | DOI | MR | Zbl

[2] Di Crescenzo A., “Exact transient analysis of a planar random motion with three directions”, Stochastics Stoc. Reports, 72 (2001), 175–189 | DOI | MR

[3] Foong S. C., “First-passage time, maximum displacement, and Kac's solution of the telegraph equation”, Phys. Rev., 46 (1992), 707–710 | DOI | MR

[4] Foong S. K., Kanno S., “Properties of the telegrapher's random process with or without a trap”, Stoc. Proc. Appl., 53 (1994), 147–173 | DOI | MR | Zbl

[5] Kolesnik A. D., “On a model of Markovian random evolution in a plane”, Analytic Methods of Investigation of Evolution of Stochastic Systems, Inst. Math. Ukrain. Acad Sci., Kiev, 1989, 55–61 (In Russian) | MR

[6] Kolesnik A. D., “Applied models of random evolutions”, Z. Angew. Math. Mech., 76(S3) (1996), 477–478 | DOI | Zbl

[7] Kolesnik A. D., “The equations of Markovian random evolution on the line”, J. Appl. Prob., 35 (1998), 27–35 | DOI | MR | Zbl

[8] Kolesnik A. D., Turbin A. F., “The equation of symmetric Markovian random evolution in a plane”, Stoc. Proc. Appl., 75 (1998), 67–87 | DOI | MR | Zbl

[9] Kolesnik A. D., Orsingher E., “Analysis of a finite-velocity planar random motion with reflection”, Theor. Probab. Appl., 46 (2001), 138–147 (In Russian) | MR | Zbl

[10] Orsingher E., “Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws”, Stoc. Proc. Appl., 34 (1990), 49–66 | DOI | MR | Zbl

[11] Orsingher E., “Exact joint distribution of a planar random motion”, Stochastics and Stoc. Reports, 69 (2000), 1–10 | MR | Zbl

[12] Orsingher E., San Martini A., “Planar random evolution with three directions”, Exploring Stochastic Laws, 1995, 357–366 | MR | Zbl

[13] Papanicolaou G., “Asymptotic analysis of transport processes”, Bull. Amer. Math. Soc., 81 (1975), 330–391 | DOI | MR

[14] Ratanov N. E., “Random walks in an inhomogeneous one-dimensional medium with reflecting and absorbing barriers”, Theoret. Math. Physics, 112 (1997), 857–865 | DOI | MR | Zbl

[15] Tolubinsky E. V., The Theory of Transfer Processes, Naukova Dumka, Kiev, 1969 (In Russian)

[16] Turbin A. F., Plotkin D. Ya., “Bessel equation and functions of higher order”, Asymptotic Methods in the Problems of Theory of Random Evolutions, Inst. Math. Ukrain. Acad. Sci., Kiev, 1991, 112–121 (In Russian) | MR