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We study measures in Banach space which arise as the skew convolution productof two other measures where the convolution is deformed by a skew map. This isthe structure that underlies both the theory of Mehler semigroups and operatorself-decomposable measures. We show how that given such a set-up the skew mapcan be lifted to an operator that acts at the level of function spaces anddemonstrate that this is an example of the well known functorial procedure ofsecond quantisation. We give particular emphasis to the case where the productmeasure is infinitely divisible and study the second quantisation process insome detail using chaos expansions when this is either Gaussian or is generatedby a Poisson random measure.

Second quantisation for skew convolution products of measures in Banach spaces