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Brownian motion on a pseudo sphere in Minkowski space \(\mathbb {R}^l_v\). (English) Zbl 1355.60110
Summary: For a Brownian motion moving on a pseudo sphere in Minkowski space \(\mathbb {R}^l_v\) of radius \(r\) starting from point \(X\), we obtain the distribution of hitting a fixed point on this pseudo sphere with \(l\geq 3\) by solving Dirichlet problems. The proof is based on the method of separation of variables and the orthogonality of trigonometric functions and Gegenbauer polynomials.
MSC:
60J65 Brownian motion
58J65 Diffusion processes and stochastic analysis on manifolds
83A05 Special relativity
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[1] Cohen de Lara, M, On drift, diffusion and geometry, J. Geom. Phys., 56, 1215-1234, (2006) · Zbl 1096.60030
[2] Cammarota, V; Gregorio, A; Macci, C, On the asymptotic behavior of the hyperbolic Brownian, J. Stat. Phys., 154, 1550-1568, (2013) · Zbl 1296.60066
[3] Cammarota, V; Orsingher, E, Hitting spheres on hyperbolic spaces, Theory Probab. Appl., 57, 419-443, (2013) · Zbl 1292.60080
[4] Dunkel, J; Hänggi, P, Relativistic Brownian motion, Phys. Rep., 471, 1-73, (2009)
[5] Dembo, A; Peres, Y; Rosen, J, Brownian motion on compact manifolds: cover time and late points, Electron. J. Probab., 8, 1-14, (2003) · Zbl 1063.58021
[6] Dembo, A; Peres, Y; Rosen, J; Zeitouni, O, Cover times for Brownian motion and random walks in two dimensions, Ann. Math., 160, 433-464, (2004) · Zbl 1068.60018
[7] Grigor’yan, A, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Am. Math. Soc., 36, 135-249, (1999) · Zbl 0927.58019
[8] Garbaczewski, P, Rotational diffusions as seen by relativistic observers, J. Math. Phys., 33, 3393-3401, (1992) · Zbl 0761.60077
[9] Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 4th edn. Academic Press, New York (1981) · Zbl 0918.65002
[10] Itô, K, On stochastic differential equations on a differential manifold I, Nagoya Math. J., 1, 35-47, (1950) · Zbl 0039.35103
[11] Lablée, O.: Spectrual Theory in Riemannian Geometry. European Mathematical Society (EMS), Zurich (2015) · Zbl 1328.53001
[12] Morava, J, Conformal invariants of Minkowski space, Proc. Am. Math. Soc., 95, 565-570, (1985) · Zbl 0625.53054
[13] Mckean, H.P.: Stochastic Integrals. Academic Press, New York (1969) · Zbl 0191.46603
[14] Mizrahi, S; Daboul, J, Squeezed states, generalized hermitz polynomials and pseudo-diffusion equation, Phys. A, 189, 635-650, (1992)
[15] Øksendal, B.: Stochastic Differential Equations, an Introduction with Applications, 6th edn. Springer, New York (2003) · Zbl 1025.60026
[16] O’Hara, P; Rondoni, L, Brownian motion in Minkowski space, Entropy, 17, 3581-3594, (2015) · Zbl 1338.60204
[17] O’neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983) · Zbl 0531.53051
[18] Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solution for Ordinary Differential Equations. Chapman & Hall, Boca Raton (2003) · Zbl 1015.34001
[19] Szegö, G.: Orthogonal Polynomials, 4th edn. AMS, Providence (1975) · Zbl 0305.42011
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