Abstract
We study measures in Banach space which arise as the skew convolution product of two other measures where the convolution is deformed by a skew map. This is the structure that underlies both the theory of Mehler semigroups and operator self-decomposable measures. We show how that given such a set-up the skew map can be lifted to an operator that acts at the level of function spaces and demonstrate that this is an example of the well known functorial procedure of second quantisation. We give particular emphasis to the case where the product measure is infinitely divisible and study the second quantisation process in some detail using chaos expansions when this is either Gaussian or is generated by a Poisson random measure.
Highlights
In recent years there has been considerable interest in skew-convolution semigroups of probability measures in Banach spaces and the so-called Mehler semigroups that they induce on function spaces
In this paper we focus on the representation of Mehler semigroups as second quantised operators
When the semigroups act on Hilbert spaces, the desired second quantisation representation was recently obtained in [28] in the pure jump case using chaotic decomposition techniques from [20], under the assumption that the Ornstein-Uhlenbeck process has an invariant measure
Summary
In recent years there has been considerable interest in skew-convolution semigroups of probability measures in Banach spaces and the so-called Mehler semigroups that they induce on function spaces These objects arise naturally in the study of infinite dimensional Ornstein-Uhlenbeck processes driven by Banach-space valued Lévy processes. In this paper we focus on the representation of Mehler semigroups as second quantised operators Such a result has been known for a long time in the Gaussian case. When the semigroups act on Hilbert spaces, the desired second quantisation representation was recently obtained in [28] in the pure jump case using chaotic decomposition techniques from [20], under the assumption that the Ornstein-Uhlenbeck process has an invariant measure. If both choices are permitted we write Bb(E)
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