A parabolic partial differential equation [Formula: see text] is considered, where [Formula: see text] is a linear second-order differential operator with time-independent (but dependent on [Formula: see text]) coefficients. We assume that the spatial coordinate [Formula: see text] belongs to a finite- or infinite-dimensional real separable Hilbert space [Formula: see text]. 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 [Formula: see text]. 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 [Formula: see text] as the multiplicity of the integral tends to infinity), which gives us a unique solution to the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on [Formula: see text]. This solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in [Formula: see text] 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.

中文翻译：

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希尔伯特空间中扩散演化半群的显式公式

考虑抛物线偏微分方程 [公式：参见文本]，其中 [公式：参见文本] 是线性二阶微分算子，具有与时间无关的（但取决于 [公式：参见文本]）系数。我们假设空间坐标[公式：见正文]属于有限维或无限维实可分希尔伯特空间[公式：见正文]。本文的目的是证明一个公式，该公式根据初始条件和算子的系数表达该方程的柯西问题的解[公式：见正文]。假设这个方程存在一个强连续解析半群，我们使用费曼公式构造这个半群的表示（即，我们将它写成[公式：见文本]，因为积分的多重性趋于无穷大），这为我们提供了对 [公式：见文本] 上光滑圆柱函数集的一致闭包中的柯西问题的唯一解。该解决方案持续依赖于初始条件。在[公式：见正文]中一阶导数项的系数为零的情况下，我们证明强连续解析半群确实存在（这意味着存在上述类中柯西问题的唯一解)，并且柯西问题的解持续依赖于方程的系数。该解决方案持续依赖于初始条件。在[公式：见正文]中一阶导数项的系数为零的情况下，我们证明强连续解析半群确实存在（这意味着存在上述类中柯西问题的唯一解)，并且柯西问题的解持续依赖于方程的系数。该解决方案持续依赖于初始条件。在[公式：见正文]中一阶导数项的系数为零的情况下，我们证明强连续解析半群确实存在（这意味着存在上述类中柯西问题的唯一解)，并且柯西问题的解持续依赖于方程的系数。