ON INTERSECTIONS OF INDEPENDENT NONDEGENERATE DIFFUSION PROCESSES

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

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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