论文标题
kardar-parisi-zhang方程的奇迹性和同步
Ergodicity and synchronization of the Kardar-Parisi-Zhang equation
论文作者
论文摘要
实际线上的Kardar-Parisi-Zhang（KPZ）方程是众所周知的，以线性漂移为固定分布（Modulo添加剂常数）。我们表明，这些解决方案很有吸引力，被称为一种力 - 一种解决方案（1F1S）原理或同步：KPZ方程的解决方案始于遥远的过去，从初始条件开始，带有给定的斜率几乎可以肯定地汇聚到Brownian Motion，并以这种漂移的形式融合到Brownian Motion，这特别表明这些不景气的度量完全令人震惊。我们的证明构建了该方程的Busemann过程，从而提供了所有这些固定溶液的自然固定耦合。然后，除了最多可计数的随机渐近斜率外，同步（在单个完整概率事件上）同时保持（在一个完整的概率事件上）。同步失败的一组特殊斜率几乎是空的，要么几乎肯定是密集的。一路上，我们证明了一个形状定理，这意味着KPZ方程的随机均质化几乎可以，为此，Busemann过程给出了校正器的过程。我们还表明，向前和向后的点对点和点对点连续聚合物聚合到半无限连续性聚合物，其过渡是DOOB通过有限长度聚合物的跃迁的busemann函数。
The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to admit Brownian motion with a linear drift as a stationary distribution (modulo additive constants). We show that these solutions are attractive, a result known as a one force--one solution (1F1S) principle or synchronization: the solution to the KPZ equation started in the distant past from an initial condition with a given slope will converge almost surely to a Brownian motion with that drift, which shows in particular that these invariant measures are totally ergodic. Our proof constructs the Busemann process for the equation, which gives the natural jointly stationary coupling of all of these stationary solutions. Synchronization then holds simultaneously (on a single full probability event) for all but an at most countable random set of asymptotic slopes. This set of exceptional slopes of instability for which synchronization fails is either almost surely empty or almost surely dense. Along the way, we prove a shape theorem which implies almost sure stochastic homogenization of the KPZ equation, for which the Busemann process gives the process of correctors. We also show that the forward and backward point-to-point and point-to-line continuum polymers converge to semi-infinite continuum polymers whose transitions are Doob transforms via Busemann functions of the transitions of the finite length polymers.
