Geometric stochastic heat equations

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