Trapping reactions with randomly moving traps: exact asymptotic results for compact exploration

In a recent paper, Bray and Blythe have shown that the survival probability P(A)(t) of an A particle diffusing with a diffusion coefficient D(A) in a one-dimensional system with diffusive traps B is independent of D(A) in the asymptotic limit t--> infinity and coincides with the survival probability of an immobile target in the presence of diffusive traps. Here, we show that this remarkable behavior has a more general range of validity and holds for systems of an arbitrary dimension d, integer or fractal, provided that the traps are "compactly exploring" the space, i.e., the "fractal" dimension d(w) of traps' trajectories is greater than d. For the marginal case when d(w)=d, as exemplified here by conventional diffusion in two-dimensional systems, the decay form is determined up to a numerical factor in the characteristic decay time.