Brownian Particle in the Curl of 2-D Stochastic Heat Equations

We study the long time behaviour of a Brownian particle evolving in a dynamic random environment. Recently, Cannizzaro et al. (Ann Probab 50(6):2475–2498, 2022) proved sharp \(\sqrt{\log }\)-super diffusive bounds for a Brownian particle in the curl of (a regularisation of) the 2-D Gaussian Free Field (GFF) \(\underline{\omega }\). We consider a one parameter family of Markovian and Gaussian dynamic environments which are reversible with respect to the law of \(\underline{\omega }\). Adapting their method, we show that if \(s\ge 1\), with \(s=1\) corresponding to the standard stochastic heat equation, then the particle stays \(\sqrt{\log }\)-super diffusive, whereas if \(s<1\), corresponding to a fractional heat equation, then the particle becomes diffusive. In fact, for \(s<1\), we show that this is a particular case of Komorowski and Olla (J Funct Anal 197(1):179–211, 2003), which yields an invariance principle through a Sector Condition result. Our main results agree with the Alder–Wainwright scaling argument (see Alder and Wainwright in Phys Rev Lett 18:988–990, 1967; Alder and Wainwright in Phys Rev A 1:18–21, 1970; Alder et al. in Phys Rev A 4:233–237, 1971; Forster et al. in Phys Rev A 16:732–749, 1977) used originally in Tóth and Valkó (J Stat Phys 147(1):113–131, 2012) to predict the \(\log \)-corrections to diffusivity. We also provide examples which display \(\log ^a\)-super diffusive behaviour for \(a\in (0,1/2]\).