ON INTERSECTIONS OF INDEPENDENT NONDEGENERATE DIFFUSION PROCESSES

doi:10.1016/S0252-9602(13)60132-0

Abstract

Abstract:

Let X⁽¹⁾ = {X⁽¹⁾(s), s ∈ R₊} and X⁽²⁾ = {X⁽²⁾(t), t ∈ R₊} be two inde-pendent nondegenerate diffusion processes with values in R^d. The existence and fractal dimension of intersections of the sample paths of X⁽¹⁾ and X⁽²⁾ are studied. More gener-ally, let E₁, E₂ ⊆ (0, ∞) and F ⊂ R^d be Borel sets. A necessary condition and a sufficient condition for P{X⁽¹⁾(E₁) ∩X⁽²⁾(E₂) ∩F 6≠Φ} > 0 are proved in terms of the Bessel-Riesz type capacity and Hausdorff measure of E₁×E₂×F in the metric space (R₊×R₊×R^d, ρ), where ρ is an unsymmetric metric defined in R₊ ×R₊ ×R^d. Under reasonable conditions, results resembling those of Browian motion are obtained.

Key words: intersection, diffusion processes, hitting probability, polar set, Hausdorff dimension

CLC Number:

60G15

CHEN Zhen-Long. ON INTERSECTIONS OF INDEPENDENT NONDEGENERATE DIFFUSION PROCESSES[J].Acta mathematica scientia,Series B, 2014, 34(1): 141-161.

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References

[1] Dvoretzky A, Erd¨os P, Kakutani S. Double points of paths of Brownian motion in n-space. Act Sci Math, 1950, 12: 75–81

[2] Khoshnevisan D. Intersections of Brownian motions. Expos Math, 2003, 21: 97–114

[3] Taylor S J. The measure theory of random fractals. Math Proc Camb Philos Soc, 1986, 100: 383–406

[4] Xiao Y. Random fractals and Markov processes//Michel L Lapidus, Machiel van Frankenhuijsen, eds. A Jubilee of Benoit Mandelbrot on Fractal Geometry and Applications. American Mathematical Society, 2004: 261–338

[5] Chen Z, Xiao Y. On intersections of independent anisotropic Gaussian random fields. Science China Mathematics, 2012, 55: 2217–2232

[6] Bierm´e H, Lacaux C, Xiao Y. Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields. Bull London Math Soc, 2009, 41: 253–273

[7] Xiao Y. Sample path properties of anisotropic Gaussian random fields//Khoshnevisan D, Rassoul-Agha F, eds. A Minicourse on Stochastic Partial Differential Equations, Lecture Notes in Math 1962. New York: Springer, 2009: 145–212

[8] Rosen J. Self-intersections of random fields. Ann Probab, 1984, 12: 108–119

[9] Rosen J. The intersection local time of fractional Brownian motion in the plane. J Multivar Anal, 1987, 23: 37–46

[10] Hu Y, Nualart D. Renormalized self-intersection local time for fractional Brownian motion. Ann Probab, 2005, 33: 948–983

[11] Wu D, Xiao Y. Regularity of intersection local times of fractional Brownian motions. J Theoret Probab, 2010, 23: 972–1001

[12] Sheu S J. Some estimates of the transition density of a nondegenerate diffusion Markov process. Ann Probab, 1991, 19: 538–561

[13] Ikeda N, Watanable S. Stochastic Differential Equations and Diffusion Processes. New York: North-Holland Publishing Company, 1981

[14] Yang X. The intersection and the polar functions of nondegenerate diffusion processes. J Sys Sci and Math Scis, 1999, 19: 56–64

[15] Yang X. The necessary conditions of the polarity of nondegenerate diffusion processes. J Sys Sci and Math Scis, 2002, 22: 67–77

[16] Yang X. The polarity and the intersection of nondegenerate diffusion processes. Acta Math Appl Sinica, 2003, 26: 242–251

[17] Chen Z. The level sets and the polar sets for nondegenerate diffusion processes. J Sys Sci and Math Scis, 2006, 26: 245–256

[18] Kahane J P. Points multiples des processus de L´evy sym´etriques restreints `a un ensemble de valurs du temps. S´em Anal Harm, Orsay, 1983, 38: 74–105

[19] Kahane J P. Some Random Series of Functions. 2nd ed. London: Cambridge University Press, 1985

[20] Evans S N. Potential theory for a family of several Markov processes. Ann Inst H Poincar´e Probab Statist, 1987, 23: 499–530

[21] Tongring N. Which sets contain multiple points of Brownian motion? Math Proc Cambridge Philos Soc, 1988, 103: 181–187

[22] Fitzsimmons P J, Salisbury T S. Capacity and energy for multiparameter processes. Ann Inst H Poincar´e Probab Statist, 1989, 25: 325–350

[23] Peres Y. Probability on trees: an introductory climb//Lectures on Probability Theory and Statistics (Saint-Flour). Lecture Notes in Math, 1717. Berlin: Springer, 1999: 193–280

[24] Khoshnevisan D, Xiao Y. L´evy processes: capacity and Hausdorff dimension. Ann Probab, 2005, 33: 841–878

[25] Dalang R C, Khoshnevisan D, Nualart E, Wu D, Xiao Y. Critical Brownian sheet does not have double points. Ann Probab, 2012, 40: 1829–1859

[26] Falconer K J. Fractal Geometry-Mathematical Foundations and Applications. Chichester: John Wiley and Sons Ltd, 1990

[27] Rogers C A. Hausdorff Measures. London: Cambridge University Press, 1970

[28] Taylor S J, Tricot C. Packing measure and its evaluation for a Brownian path. Trans Amer Math Soc, 1985, 288: 679–699

[29] Garsia A M, Rodemich E, Rumsey H Jr. A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ Math J, 1970, 20: 565–578

[30] Cs´aki E, Cs¨org?o M. Inequalities for increments of stochastic processes and moduli of continuity. Ann Probab, 1992, 20: 1031–1052

[31] Khoshnevisan D. Multiparameter Processes. New York: Springer-Verlag, 2002

[32] Tricot C. Two definitions of fractional dimension. Math Proc Camb Philos Soc, 1982, 91: 57–74

[33] Chen Z. Intersections and polar functions of fractional Brownian sheets. Acta Mathematica Scientia, 2008, 28B: 779–796