We deal with a class of semilinear parabolic PDEs on the space of continuous functions that arise, for example, as Kolmogorov equations associated to the infinite-dimensional lifting of path-dependent SDEs. We investigate existence of smooth solutions through their representation via forward–backward stochastic systems, for which we provide the necessary regularity theory. Because of the lack of smoothing properties of the parabolic operators at hand, solutions in general will only share the same regularity as the coefficients of the equation. To conclude we exhibit an application to Hamilton–Jacobi–Bellman equations associated to suitable optimal control problems.

我们处理连续函数空间上的一类半线性抛物线型PDE，例如，与与路径依赖的SDE的无穷维提升相关的Kolmogorov方程。我们通过前向后向随机系统的表示来研究光滑解的存在性，为此我们提供了必要的规则性理论。由于手边的抛物线算子缺乏平滑特性，因此解决方案通常只会与方程的系数具有相同的规律性。总而言之，我们展示了与合适的最优控制问题相关的Hamilton–Jacobi–Bellman方程的应用。