Potential theory on Hilbert space

Backward Ornstein-Uhlenbeck Transition Operators and Mild Solutions of Non-Autonomous Hamilton-Jacobi Equations in Banach Spaces

“…In section 2 we recall some basic facts on Gaussian measures in Banach spaces, reproducing kernel Hilbert spaces and the Cameron-Martin formula. We present an alternative proof of a well-known result on regularizing property of Gaussian convolutions which first appeared in the seminal paper Gross (1967). In section 3 we review some results from van Neerven and Weis (2005a) on stochastic integration of deterministic operator valued functions with respect to a cylindrical Wiener process.…”

“…6, No. 4;2014 The following regularizing property is a classical result proved by L. Gross in his seminal paper (Gross 1967, Proposition 9) using directly the notion of Fréchet derivative. Here we present an alternative proof based on Gâteaux differentiability.…”

Application of the functional calculus to solving of infinite dimensional heat equation

“…In 1967 L. Gross [4] introduced Laplacian ∆ G on an abstract Wiener space as a natural infinite dimensional analogue of the finite dimensional Laplacian and studied potential theory associated with ∆ G . Within the white noise framework, the Gross Laplacian has been formulated by Kuo in [8] as a continuous linear operator acting on test white noise functions.…”

Quasi-differentiable functions of Banach spaces

“…Let B be a real separable Banach space with norm || • ||. Gross [3] has shown that there exists a continuous linear embedding e of a real separable Hilbert space into B such that the range of e is dense and such that the function ra(x) = ||ex|| is a measurable norm on the Hilbert space. The concept of a measurable norm is defined in [3, Definition 4, p. 34] and it is shown there that the space B carries a family of Borel probability measures {pt}, t>0, which are characterized by the following property: Any element X of B* is a Gaussian random variable with mean zero relative to the measure pt.…”