In this work, we prove a version of Hörmander’s theorem for a stochastic evolution equation driven by a trace-class fractional Brownian motion with Hurst exponent [Formula: see text] and an analytic semigroup on a given separable Hilbert space. In contrast to the classical finite-dimensional case, the Jacobian operator in typical solutions of parabolic stochastic PDEs is not invertible which causes a severe difficulty in expressing the Malliavin matrix in terms of an adapted process. Under a Hörmander’s bracket condition and some algebraic constraints on the vector fields combined with the range of the semigroup, we prove that the law of finite-dimensional projections of such solutions has a density with respect to Lebesgue measure. The argument is based on rough path techniques and a suitable analysis on the Gaussian space of the fractional Brownian motion.

中文翻译：

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分数布朗运动驱动的随机演化方程的密度存在

在这项工作中，我们证明了一个随机演化方程的 Hörmander 定理版本，该方程由具有赫斯特指数 [公式：见正文] 的迹类分数布朗运动和给定可分离希尔伯特空间上的解析半群驱动。与经典的有限维情况相比，抛物线随机偏微分方程的典型解中的雅可比算子是不可逆的，这导致很难根据适应过程来表达 Malliavin 矩阵。在霍曼括号条件和向量场上的一些代数约束下，结合半群的范围，我们证明了这种解的有限维投影定律关于勒贝格测度具有密度。