Let μ be a probability measure on a locally convex Hausdorff space E and A(μ) be the quasi-invariant set of μ. If μ^*(A(μ))>0, then there exist a finite-dimensional subspace L, a thick subgroup G of L and a countable subgroup {x_i} such that A(μ)=∪\limits_i=1^∞(G+x_i). If E is Souslin, then A(μ) is a Borel subset. If E is Souslin and if μ (A(μ))>0, then A(μ)=∪\limits_i=1^∞(L+x_i).

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We develop a stochastic integration theory with respect to creation, annihilation and gauge operators on the full Fock space. This is done by using a kernel representation for a large class of bounded operators on the full Fock space. It is shown that the kernels form a Banach algebra. Having established the definition of processes and stochastic integrals we go on to prove an Ito formula and use this for examining stochastic evolutions and constructing dilations of special completely positive semigroups. Explicit solutions of the corresponding stochastic differential equations are given.

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