Quasi-invariance of the Gaussian measure for the two-dimensional stochastic cubic nonlinear wave equation

We consider the stochastic damped nonlinear wave equation ${\mathrm{\partial }}_t^{2}u+{\mathrm{\partial }}_{t}u+u-\mathrm{\Delta }u+u^3=\sqrt{2}⟨\mathrm{\nabla }{⟩}^{-s}\xi$ on the two-dimensional torus ${\mathbb{T}}^2$, where $\xi$ denotes a space-time white noise and $s>0$. We show that the measure ${\vec{\mu }}_s$ corresponding to the unique invariant measure for the flow of the associated linear equation is quasi-invariant under the nonlinear stochastic flow.