AI 中文总结
本文针对带乘性噪声的随机Allen-Cahn方程，通过正则结构建立其在L^∞(T)上的随机流存在性，推导了阻尼强度的选择条件，还证明该随机流关于初始条件可微。
AI 中文摘要
我们在L^∞(T)空间上建立带乘性噪声的随机Allen-Cahn方程的随机流的存在性，该方程为(∂_t - ∂_x^2)u = u - u^3 + σ(u)ξ，定义在R_+×T上，其中ξ是时空白噪声，σ:R→R足够光滑、有界且导数有界。我们的策略是通过正则结构获得路径先验估计。实际上，我们考虑一类带超线性阻尼的一般奇异乘性方程，由抛物正则性α-2的噪声驱动，α∈(0,1)，这类方程可提升为弱容许模型。我们证明，当m > [(2-α)/α]ε_α（其中ε_α=1-α(1-2/(3-α))∈(0,1)）时，所需估计成立，因此阻尼强度仅需根据驱动噪声的正则性选择。在σ满足额外光滑性假设的条件下，我们证明该随机流关于初始条件是可微的。
英文摘要
We establish the existence of a stochastic flow on $L^{\infty} (\mathbb{T})$ for the stochastic Allen-Cahn equation with multiplicative noise \[ (\partial_t - \partial_x^2) u = u - u^3 + σ(u) ξ\quad \text{on} \quad \mathbb{R}_+ \times \mathbb{T}, \] where $ξ$ is space-time white noise and $σ: \mathbb{R} \rightarrow \mathbb{R}$ 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 $α- 2$, for ${α\in (0, 1)}$, which can be lifted to a weakly admissible model. We show that the required estimates hold whenever \[ m > \frac{2 - α}α \varepsilon_α, \quad \text{where} \quad \varepsilon_α = 1 - α\left( 1 - \frac{2}{3 - α} \right) \in (0, 1) . \] 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 $σ$, we show that the stochastic flow is differentiable with respect to its initial condition.
Comments66 pages
