Logarithmic derivatives of variational and singular stochastic partial differential equations
For a stochastic partial differential equation posed on a Gelfand triple and satisfying the fully local monotone conditions of R\"ockner, 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 $\gamma_t = \int_0^t \Phi_r \Phi_r^{*} \,\mathrm{d}r$ 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 $\langle u(t),\varphi\rangle$ 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.