Geometric stochastic heat equations

Abstract

We consider a natural class of $\mathbf{R}^d$-valued one-dimensional stochastic PDEs driven by space-time white noise that is formally invariant under the action of the diffeomorphism group on $\mathbf{R}^d$. This class contains in particular the KPZ equation, the multiplicative stochastic heat equation, the additive stochastic heat equation, and rough Burgers-type equations. We exhibit a one-parameter family of solution theories with the following properties: - For all SPDEs in our class for which a solution was previously available, every solution in our family coincides with the previously constructed solution, whether that was obtained using It\^o calculus (additive and multiplicative stochastic heat equation), rough path theory (rough Burgers-type equations), or the Hopf-Cole transform (KPZ equation). - Every solution theory is equivariant under the action of the diffeomorphism group, i.e. identities obtained by formal calculations treating the noise as a smooth function are valid. - Every solution theory satisfies an analogue of It\^o's isometry. - The counterterms leading to our solution theories vanish at points where the equation agrees to leading order with the additive stochastic heat equation. In particular, points 2 and 3 show that, surprisingly, our solution theories enjoy properties analogous to those holding for both the Stratonovich and It\^o interpretations of SDEs simultaneously. For the natural noisy perturbation of the harmonic map flow with values in an arbitrary Riemannian manifold, we show that all these solution theories coincide. In particular, this allows us to conjecturally identify the process associated to the Markov extension of the Dirichlet form corresponding to the $L^2$-gradient flow for the Brownian loop measure.