We set up an SPDE model for a moving, weakly self-avoiding polymer with intrinsic length J taking values in (0, ∞). Our main result states that the effective radius of the polymer is approximately J5/3; evidently for large J the polymer undergoes stretching. This contrasts with the equilibrium situation without the time variable, where many earlier results show that the effective radius is approximately J.
For such a moving polymer taking values in R2, we offer a conjecture that the effective radius is approximately J5/4.