The Effect of Transport Noise on Interface Formulation

The paper discusses a mass conserving Allen–Cahn equation with a small parameter \(\epsilon >0\) $ϵ>0$ perturbed by transport type noise involving the smeared noise \({\dot{w}}^\epsilon _i\) ${\dot{w}}_i^{ϵ}$ (formally converging to white noise \({\dot{w}}_i\) ${\dot{w}}_i$ ), and investigate its sharp interface limit as \(\epsilon \rightarrow 0\) $ϵ\to 0$ . We restrict our discussion over the two-dimensional domain to construct the hypersurface driven by the stochastic term appearing due to the contribution from the transport term when \(\epsilon \rightarrow 0\) $ϵ\to 0$ . We make use of asymptotic expansion method to discuss the sharp interface limit.