For both Brownian and enhanced random walks, we can distinguish between systems where the random walker returns to the origin with certainty, and where the random walker may escape from the trapping origin with a probability strictly larger than zero. Examples for the first case are Brownian walks in one and two dimensions, where it is known that the asymptotic probability f(t) to be trapped at time t-->infinity behaves differently from the asymptotic probability p(t) to pass through the origin in the same system without trap. We also find this result for Levy flights in one dimension with the exponent of the characteristic function greater than or equal to 1. On the other hand, we compute p(t) and f(t) for various systems with a nonzero escape probability. In particular, we consider an anisotropic walk, which behaves Brownian in one direction and executes Levy flights along the second direction. For these cases we are able to prove that f(t) and p(t) follow the same inverse power law asymptotically, the ratio given by the squared escape probability. [S1063-651X(98)01510-4]. [References: 24]
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