“Polymers appear pervasively in our lives: from everyday materials such as paints, lubricants, car parts, etc. to the very substance of life, DNA. Their wide range of applications means that polymers have been studied extensively in various fields. From the probability theory standpoint, the physical properties of polymers have inspired a whole mathematical theory for the object. We are interested in studying the recurrence property of a polymer, that is, whether it returns to a given set infinitely often with positive probability. Our talk concerns a certain \(d\)-dimensional model, namely a solution \((U_t(x)) = (U_t(x))_{t \ge 0, x \in \mathbb{R}}\) of the stochastic heat equation, where \(t\) is the time variable, \(x\) is the one-dimensional space variable, and \(U_t(x)\) is in \(\mathbb{R}^d\).

We introduce different modes of recurrence/ transience as follows. Roughly speaking, \((U_t(x))\) is point recurrent if for every point in \(\mathbb{R}^d\), it returns to that point infinitely often at randomly large times almost surely (a.s.). Neighborhood recurrence is more interesting: for every point in \(\mathbb{R}^d\), if we fix an arbitrarily small \(d\)-dimensional ball with radius \(\epsilon > 0\) centered at that point, then \((U_t(x))\) will return to that ball infinitely often at randomly large times a.s. Finally, it is transient if, for every point in \(\mathbb{R}^d\) and any fixed ball centered at that point, it gets away from that ball after some (random) time a.s. Now, if we let this fixed ball grow/ shrink, then how does this model go from being neighborhood recurrence to being transience, and vice versa? We study the question of recurrence and transience when the growth rate of the ball is a power of time, i.e. of the form \(f(t) = t^{\alpha}\), where \(\alpha \in \mathbb{R}\).”