Survival probability of an immobile target surrounded by mobile traps

We study analytically, in one dimension, the survival probability P-s(t) up to time t of an immobile target surrounded by mutually noninteracting traps, each performing a continuous-time random walk (CTRW) in continuous space. We consider a general CTRW with symmetric and continuous (but otherwise arbitrary) jump length distribution f(eta) and arbitrary waiting time distribution psi(tau). The traps are initially distributed uniformly in space with density rho. We prove an exact relation, valid for all time t, between P-s(t) and the expected maximum E[M(t)] of the trap process up to time t, for rather general stochastic motion x(trap)(t) of each trap. When x(trap)(t) represents a general CTRW with arbitrary f(eta) and psi(tau), we are able to compute exactly the first two leading terms in the asymptotic behavior of E[M(t)] for large t. This allows us subsequently to compute the precise asymptotic behavior, P-s(t) similar to a exp[-bt(theta)], for large t, with exact expressions for the stretching exponent. and the constants a and b for arbitrary CTRW. By choosing appropriate f(eta) and psi(tau), we recover the previously known results for diffusive and subdiffusive traps. However, our result is more general and includes, in particular, superdiffusive traps as well as totally anomalous traps.