Logarithmic derivatives of variational and singular stochastic partial differential equations

Abstract

For a stochastic partial differential equation posed on a Gelfand triple and satisfying the fully local monotone conditions of Röckner, Shang and Zhang, we compute the logarithmic derivative of the law of the solution at a fixed time along a prescribed direction of the state space. The formula is intrinsic, being expressed through the Hilbert-Schmidt Malliavin derivative ${\Phi }_r={\mathcal{D}}_{r}X(t)$ and the covariance ${\gamma }_t={\int }_0^t{\Phi }_r{\Phi }_r^{\ast }\mathrm{d}r$ alone, so that neither the inversion of the first variation used in finite dimensions nor uniform Malliavin-Sobolev bounds on the Tikhonov family are called upon. It is obtained from an integration-by-parts identity on an abstract Hilbert space, a Moore-Penrose construction of a covering field on Wiener space, and a trace formula for the Tikhonov limit, specialised to the equation through the representation ${\Phi }_r=Y(t,r)\mathcal{B}(r,X(r))$ of the Malliavin derivative by the first variation; the stochastic $p$-Laplacian and the two-dimensional Navier-Stokes equation are treated in detail. Beyond the variational class, a scalar reduction gives an integration-by-parts identity for the law of a pairing $⟨u(t),\phi ⟩$ which passes to the renormalised limit for the singular equations of Bruned, Chandra, Chevyrev and Hairer, and which is represented by a logarithmic derivative under second-order Malliavin smoothness and negative-moment hypotheses.