论文标题
重新排列的随机热方程
Rearranged Stochastic Heat Equation
论文作者
论文摘要
这项工作的目的是在实际线路上的概率度量范围内进行明确构造强大的feller semigroup，该概率措施的范围额外映射了可测量的函数到Lipschitz连续功能中，Lipschitz常数以少量的时间以可集成的方式吹出。我们的构造依赖于由彩色噪声驱动的圆上的随机热方程的重新安排。正式地，这个随机方程将无限维度的反射方程式写入。在重排的作用下，该解决方案被迫生活在对实际线上概率测量空间的分位数函数的空间中。我们证明了通过EULER方案来求解的方程式，在该方案中，我们将在圆圈上随机变量的空间中交替使用重新排列操作，该动力学将随机变量投射到位点函数的子集上。第一个挑战是证明该方案很紧。第二个是为限制反射方程式提供一致的理论，尤其是以相关的方式解释反思术语。我们工作的最后一步是通过调整Bismut-Elworthy-Li公式的早期想法来确定上述Semigroup的Lipschitz财产。
The purpose of this work is to provide an explicit construction of a strong Feller semigroup on the space of probability measures over the real line that additionally maps bounded measurable functions into Lipschitz continuous functions, with a Lipschitz constant that blows up in an integrable manner in small time. Our construction relies on a rearranged version of the stochastic heat equation on the circle driven by a coloured noise. Formally, this stochastic equation writes as a reflected equation in infinite dimension. Under the action of the rearrangement, the solution is forced to live in a space of quantile functions that is isometric to the space of probability measures on the real line. We prove the equation to be solvable by means of an Euler scheme in which we alternate flat dynamics in the space of random variables on the circle with a rearrangement operation that projects back the random variables onto the subset of quantile functions. A first challenge is to prove that this scheme is tight. A second one is to provide a consistent theory for the limiting reflected equation and in particular to interpret in a relevant manner the reflection term. The last step in our work is to establish the aforementioned Lipschitz property of the semigroup by adapting earlier ideas from the Bismut-Elworthy-Li formula.
