Martingale measure associated with the critical $2d$ stochastic heat flow

Abstract

In [CSZ23], the authors proved the convergence of the finite dimensional time distribution of the rescaled random fields derived from the discrete stochastic heat equation of $2d$-directed polymers in random environment in the critical window. The scaling limit is called critical $2d$ stochastic heat flow (SHF). In this paper, we will show that the critical $2d$ SHF is a continuous semimartingale. Moreover, we will consider the martingale problem associated with the critical $2d$ SHF in a similar fashion to the super Brownian motion which is one of the well-known measure valued process. Also, we define the martingale measure associated with the critical $2d$ SHF in the sense of [Wal86, Chapter 2]. The quadratic variation of the martingale measure gives information of the regularity of the critical $2d$ SHF.