A note on the invariance under change of measure for stochastic test functions and distribution spaces

In this note we prove that the spaces (S), (S)∗ and G, G∗ are invariant under a certain class of translations of the underlying Brownian motion. This problem arises naturally in dealing with anticipating stochastic differential equations, in particular when the Girsanov theorem is involved. The proofs are based on a Bayes formula for second-quantization operators that was derived by Lanconelli [Lanconelli, A., 2006a. Bayes' formula for second quantization operators. Stoch. Dyn. 6 (2), 245-253] and on the properties of the translation operators.