Abstract: In this talk, we study a stochastic process that is repelled by its own local time. Due to the singular nature of local time, there is no classical way to define such an object. In dimension d=1, we construct this process using the theory of martingale problems for SPDEs. Predictions from the physics literature suggest that this approach will fail in the critical dimension d=2 due to a logarithmic blowup, the precise rate of which is not known. A natural response is to take a weak-coupling limit, and we show that this produces a Brownian motion with an explicit diffusivity.
This is joint work with Giuseppe Cannizzaro and Lukas Gräfner.