Comparison Theorems for Sample Function Growth

The growth rate at 0 of a Levy process is compared with the growth rate at a local minimum, m, of the process. For the lim inf it is found that the growth rate at m is the same as that on the set of "ladder points" following 0, parameterized by inverse local time; this result gives a precise meaning to the notion that a Levy process leaves its minima "faster" than it leaves 0. A less precise result is obtained for the lim sup.