On Intersections of Independent Nondegenerate Diffusion Processes

Let X(1)={X(1)(s),s∈ℝ+}andX(2)={X(2)(t),t∈ℝ+}X(1)={X(1)(s),s∈ℝ+}andX(2)={X(2)(t),t∈ℝ+} be two independent nondegenerate diffusion processes with values in ℝd. The existence and fractal dimension of intersections of the sample paths of X(1) and X(2) are studied. More generally, let E1,E2⊆(0,∞)andF⊂ℝdE1,E2⊆(0,∞)andF⊂ℝd be Borel sets. A necessary condition and a sufficient condition for P{X(1)(E1)∩X(2)(E2)∩F≠∅}>0P{X(1)(E1)∩X(2)(E2)∩F≠∅}>0 are proved in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 × E2 × F in the metric space (ℝ+×ℝ+×ℝd,pˆ),wherepˆ is an unsymmetric metric defined in ℝ+×ℝ+×ℝdℝ+×ℝ+×ℝd. Under reasonable conditions, results resembling those of Browian motion are obtained.