论文标题
归化的kolmogorov方程和具有对数电势的随机allen-cahn方程
Degenerate Kolmogorov equations and ergodicity for the stochastic Allen-Cahn equation with logarithmic potential
论文作者
论文摘要
证明了与随机的allen-cahn方程相关的一类简并kolmogorov方程证明，弗里德里奇的适合度已被证明。该模型的热力学一致性要求具有奇异的电势，并且在各自的潜在屏障上消失的乘法噪声系数，从而使相应的kolmogorov方程在空间中并不均匀地椭圆形。首先，讨论了不变的措施和奇特性的存在和独特性。然后，明确构建了一些正规化的kolmogorov方程的经典解决方案。最终，对原始的kolmogorov方程的特定缩放率存在 - 弗里德里奇（Friedrichs）的特定缩放收益率的存在对限制的爆炸率进行了尖锐的分析。
Well-posedness à la Friedrichs is proved for a class of degenerate Kolmogorov equations associated to stochastic Allen-Cahn equations with logarithmic potential. The thermodynamical consistency of the model requires the potential to be singular and the multiplicative noise coefficient to vanish at the respective potential barriers, making thus the corresponding Kolmogorov equation not uniformly elliptic in space. First, existence and uniqueness of invariant measures and ergodicity are discussed. Then, classical solutions to some regularised Kolmogorov equations are explicitly constructed. Eventually, a sharp analysis of the blow-up rates of the regularised solutions and a passage to the limit with a specific scaling yield existence à la Friedrichs for the original Kolmogorov equation.
