A Support Theorem for Singular Stochastic PDEs

We present a support theorem for subcritical parabolic stochastic partial differential equations (SPDEs) driven by Gaussian noises. In the spirit of the classical theorem by Stroock and Varadhan for ordinary stochastic differential equations, we identify the support of the solution to singular SPDEs with the closure of the union of the support of solutions to approximate and renormalized equations. We implement our approach in the setting of regularity structures and obtain a general result covering a range of singular SPDEs (including $\Phi^4_3$, $\Phi^d_2$, KPZ, PAM (2D+3D), SHE, ...). As a Corollary to our result we obtain the uniqueness of invariant measures for various interesting SPDEs. This is a joint work with Martin Hairer.