AI 中文总结
本文证明一类具有可测漂移的 Wright--Fisher 随机热方程解的弱存在性与唯一性，通过随机对偶方法推广了噪声正则化结果。
AI 中文摘要
我们证明了一类由 Wright--Fisher 噪声驱动的、具有可测漂移 $b: [0,1] \to \mathbb{R}$ 的标量随机热方程的 $[0,1]$-值解的弱存在性和分布唯一性。我们对漂移仅有的假设是 Borel 可测性，以及单向不等式 $-Cu\leq b(u)\leq C(1-u)$ 对某个 $C<\infty$ 成立，这与必要条件 $b(1)\leq 0\leq b(0)$ 相容。这些条件远比先前可用的条件更一般。我们的证明依赖于随机热方程的解与一个沿着分支-合并粒子系统的图运行的投票方案之间的随机对偶，推广了存在对偶过程的漂移类。我们的结果及其证明将 Barnes、Mytnik 和 Sun 的噪声正则化结果及其解释置于更一般的基础上。
英文摘要
We prove weak existence and uniqueness in law of $[0,1]$-valued solutions for a class of scalar stochastic heat equations with measurable drifts $b : [0,1] \to \mathbb{R}$, driven by Wright--Fisher noise. The only assumptions we require on the drift are Borel-measurability, and the one-way inequalities $-Cu\leq b(u)\leq C(1-u)$ for some $C<\infty$, compatible with the necessary condition $b(1)\leq 0\leq b(0)$. These are vastly more general conditions than were previously available. Our proof relies on a stochastic duality between the solution of the stochastic heat equation, and a voting scheme running along the graph of a branching-coalescing particle system, generalising the class of drifts for which a dual process is available. Our results and their proof put the regularisation-by-noise result of Barnes, Mytnik and Sun, and its explanation, on a much more general footing.
Comments21 pages
