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Explicit formula for evolution semigroup for diffusion in Hilbert space

机译：Evolution Semigroup在希尔伯特空间扩散的明确公式

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摘要

A parabolic partial differential equation u(t)'(t, x) = Lu(t, x) is considered, where L is linear second-order differential operator with time-independent (but dependent on x) coefficients. We assume that the spatial coordinate x belongs to a finite- or infinite-dimensional real separable Hilbert space H. The aim of the paper is to prove a formula that expresses the solution of the Cauchy problem for this equation in terms of initial condition and coefficients of the operator L. Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup using a Feynman formula (i.e. we write it in the form of a limit of a multiple integral over H as the multiplicity of the integral tends to infinity), Which gives us a unique solution to the Cauchy problerri in the uniform closure of the set of smooth cylindrical functions on H. This solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in L is zero, we prove that the strongly continuous resolving semigroup indeed exists (which implies the existence of a unique solution to the Cauchy problem in the class mentioned above), and that the solution to the Cauchy problem depends continuously on the coefficients of the equation.
机译：考虑一个抛物线部分微分方程U（t）'（t，x）= lu（t，x），其中L是线性二阶差分运算符，其与时间无关（但依赖于x）系数。我们假设空间坐标X属于有限或无限的维度可分离的Hilbert空间H.以初始条件和系数在初始条件和系数方面证明了该等式表达了Cauchy问题解决方案的公式在操作员L.假设存在强烈连续的解析半群的这种方程式，我们使用Feynman公式构建该半群的表示（即，我们以多个整数的形式写入Huply以上的多个积分倾向于无穷大），这使我们为Cauchy Probleri提供了一种独特的Cauchy Probleri在H上的平滑圆柱功能的均匀封闭中。该解决方案在初始条件下连续取决于初始条件。在L中的第一衍生术语的系数为零的情况下，我们证明了强烈连续的分辨半群确实存在（这意味着在上面提到的班级中的Cauchy问题的独特解决方案），而且Cauchy问题的解决方案连续地取决于等式的系数。

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