A symmetric random evolution X(t) = (X 1 (t), …, X m (t)) controlled by a homogeneous Poisson process with parameter λ > 0 is considered in the Euclidean space ℝm, m ≥ 2. We obtain an asymptotic relation for the transition density p(x, t), t > 0, of the process X(t) as λ → 0 and describe the behavior of p(x, t) near the boundary of the diffusion domain in spaces of different dimensions.
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References
E. V. Tolubinskii, Theory of Transport Processes [in Russian], Naukova Dumka, Kiev (1969).
W. Stadje, “The exact probability distribution of a two-dimensional random walk,” J. Statist. Phys., 46, 207–216 (1987).
J. Masoliver, J. M. Porrá, and G. H. Weiss, “Some two- and three-dimensional persistent random walks,” Physica A, 193, 469–482 (1993).
A. D. Kolesnik and E. Orisingher, “A planar random motion with an infinite number of directions controlled by the damped wave equation,” J. Appl. Probab., 42, 1168–1182 (2005).
A. D. Kolesnik, “A note on planar random motion at finite speed,” J. Appl. Probab., 44, 838–842 (2007).
W. Stadje, “Exact probability distributions for non-correlated random walk models,” J. Statist. Phys., 56, 415–435 (1989).
A. D. Kolesnik, “A four-dimensional random motion at finite speed,” J. Appl. Probab., 43, 1107–1118 (2006).
A. D. Kolesnik, “A limit theorem for symmetric Markovian random evolution in ℝm,” Theory Stochast. Process., 14, No. 1, 69–75 (2008).
M. Pinsky, “Isotropic transport process on a Riemann manifold,” Trans. Amer. Math. Soc., 218, 353–360 (1976).
A. F. Turbin, “One-dimensional processes of Brownian motion as an alternative to the Einstein–Wiener–Lévy model,” in : Fractal Analysis and Related Problems [in Russian], Vol. 2, (1998), pp. 47–60.
V. S. Korolyuk and A. V. Svishchuk, Semi-Markov Random Evolutions [in Russian], Naukova Dumka, Kiev (1992).
G. Papanicolaou, “Asymptotic analysis of transport processes,” Bull. Amer. Math. Soc., 81, 330–392 (1975).
M. Pinsky, Lectures on Random Evolution, World Scientific, River Edge (1991).
A. D. Kolesnik, “Random motions at finite speed in higher dimensions,” J. Statist. Phys., 131, 1039–1065 (2008).
V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1981).
I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Nauka, Moscow (1971).
A. D. Kolesnik, “Discontinuous term of the distribution for Markovian random evolution in ℝ3,” Bull. Acad. Sci. Moldova, Ser. Math., 2 (51), 62–68 (2006).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1631 – 1641, December, 2008.
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Kolesnik, A.D. Asymptotic relation for the density of a multidimensional random evolution with rare poisson switchings. Ukr Math J 60, 1915–1926 (2008). https://doi.org/10.1007/s11253-009-0180-6
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DOI: https://doi.org/10.1007/s11253-009-0180-6