List of Footnotes

1 We will often use the shortened term stochastic gravity as there is no confusion as to the nature and source of stochasticity in gravity being induced from the quantum fields and not a priori from the classical spacetime.
2 It is well-known that several regularization methods can work equally well for the removal of ultraviolet divergences in the stress-energy tensor of quantum fields in curved spacetime. Their mutual relations are known, and discrepancies explained. This formal structure of regularization schemes for quantum fields in curved spacetime should remain intact when applied to the regularization of the noise kernel in general curved spacetimes; it is the meaning and relevance of regularization of the noise kernel which is more of a concern (see comments below). Specific considerations will of course enter for each method. But for the methods employed so far, such as zeta-function, point separation, dimensional, smeared-field, applied to simple cases (Casimir, Einstein, thermal fields) there is no new inconsistency or discrepancy.
3 By “Lorentz invariant” we mean invariant under the transformations of the orthochronous Lorentz subgroup; see [137] for more details on the definition and properties of these tensor distributions.