Scaling Properties of a Moving Polymer

We set up an SPDE model for a moving, weakly self-avoiding polymer with intrinsic length $J$ taking values in $(0,\infty)$. Our main result states that the effective radius of the polymer is approximately $J^{5/3}$; 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 $\mathbf{R}^2$, we offer a conjecture that the effective radius is approximately $J^{5/4}$.