Itô-Stratonovich Conversion in Infinite Dimensions for Unbounded, Time-Dependent, Nonlinear Operators

We prove that a solution, in a variational framework, to the Stratonovich stochastic partial differential equation with noise $G(t,{\mathrm{\Psi }}_t)\circ d{W}_t$ is given by a solution to the Itô equation with Itô-Stratonovich corrector $\frac{1}{2}{\sum }_{i=1}^{\mathrm{\infty }}{D}_u{G}_i(t,{\mathrm{\Psi }}_t)[G_i(t,{\mathrm{\Psi }}_t)]dt$. Here $G_i$ denotes the action of $G$ on the $i^{th}$ component of the cylindrical noise, and $D_u{G}_i$ its Fréchet partial derivative in the Hilbert space for which the Itô form is satisfied. The noise operator $G$ may be time-dependent, nonlinear, and unbounded in the sense of differential operators; in the latter case, one must pass to a larger space in order to solve the Stratonovich equation. Our proof relies on martingale techniques, and the results apply to fluid equations with time-dependent and nonlinear transport noise.