I apologize in advance if this question is not suitable for MO (please let me know), but the fact is that since I am not familiar with the theory of Feynman integrals I don't know whether this is a trivial question.
In Kuo's book "White noise distribution analysis" or in Hida, et al. "White noise analysis" there are discussions regarding the connection between the Feynman integrals and the White noise theory.
In particular they discuss the informal object
$$\exp \bigg\{c\int_0^T \dot B(s)^2 ds\bigg\}.$$
Of course this object is not well defined and one must "renormalize" it.
They discuss a way of renormalizing it and define it (as a generalized random variable) by means of its S-transform:
$$S\left(\mathcal N \exp \bigg\{c\int_0^T \dot B(s)^2 ds\bigg\}\right)(\xi)=\exp\left(\frac{c}{1-2c}\int_0^T \xi(s)^2 ds\right).$$
I am curious about the fact that this renormalization does not coincide with the one obtain by considering the Wick versions, i.e. the object defined by
$$S\left(\exp^{\diamond}\bigg\{c\int_0^T \dot B(s)^{\diamond 2} ds\bigg\}\right)(\xi)=\exp\left(c\int_0^T \xi(s)^2 ds\right).$$
In general in the White noise context when we want to renormalize an object we take the Wick version, sometimes denoted using the notation $:\cdot :$, but in this case they do not, why is this?
