Asymptotic behavior of Brownian polymers

Summary We consider a system that models the shape of a growing polymer. Our basic problem concerns the asymptotic behavior ofX_t, the location of the end of the polymer at timet. We obtain bounds onX_t in the (physically uninteresting) case thatd=1 and the interaction functionf(x)0. If, in addition,f(x) behaves for largex likeCx^– with <1 we obtain a strong law that gives the exact growth rate.Partially supported by the National Science Foundation and the Army Research Office through the Mathematical Sciences Institute (MSI) at Cornell UniversityPartially supported by SERC grant GR/G02307 and MSI, this work was done while the second author was visiting Cornell, April–July, 1990