H−C1 Maps and elliptic SPDEs with polynomial and exponential perturbations of Nelsonʹs Euclidean free field

Elliptic stochastic partial differential equations (SPDE) with polynomial and exponential perturbation terms defined in terms of Nelsonʹs Euclidean free field on Rd are studied using results by S. Kusuoka and A.S. ـstünel and M. Zakai concerning transformation of measures on abstract Wiener space. SPDEs of this type arise, in particular, in (Euclidean) quantum field theory with interactions of the polynomial or exponential type. The probability laws of the solutions of such SPDEs are given by Girsanov probability measures, that are non-linearly transformed measures of the probability law of Nelsonʹs free field defined on subspaces of Schwartz space of tempered distributions.