A Stochastic Flow for the Stochastic Allen-Cahn Equation with Multiplicative Noise

We establish the existence of a stochastic flow on $L^{\mathrm{\infty }}(\mathbb{T})$ for the stochastic Allen-Cahn equation with multiplicative noise $$({\mathrm{\partial }}_t-{\mathrm{\partial }}_x^2)u=u-u^3+\sigma (u)\xi \text{on}{\mathbb{R}}_{+}\times \mathbb{T},$$ where $\xi$ is space-time white noise and $\sigma :\mathbb{R}\to \mathbb{R}$ is sufficiently smooth, bounded, and has bounded derivatives. Our strategy is to obtain pathwise a priori estimates via regularity structures. In fact, we consider a general singular multiplicative equation with superlinear damping, driven by noises of parabolic regularity $\alpha -2$, for $\alpha \in (0,1)$, which can be lifted to a weakly admissible model. We show that the required estimates hold whenever $$m>\frac{2-\alpha }{\alpha }{\epsilon }_{\alpha },\text{where}{\epsilon }_{\alpha }=1-\alpha (1-\frac{2}{3-\alpha })\in (0,1)\mathrm{.}$$ Thus the strength of the damping needs to be chosen only as a function of the regularity of the driving noise. Under an additional smoothness assumption on $\sigma$, we show that the stochastic flow is differentiable with respect to its initial condition.