  We prove $L^{p}$-uniqueness of Dirichlet operators for Gibbs measures on the
path space $C(\mathbb R, \mathbb R^{d})$ associated with exponential type
interactions in infinite volume by extending an SPDE approach presented in
previous work by the last two named authors. We also give an SPDE
characterization of the corresponding dynamics. In particular, we prove
existence and uniqueness of a strong solution for the SPDE, though the
self-interaction potential is not assumed to be differentiable, hence the drift
is possibly discontinuous. As examples, to which our results apply, we mention
the stochastic quantization of $P(\phi)_{1}$-, ${\rm exp}(\phi)_{1}$-, and
trigonometric quantum fields in infinite volume. In particular, our results
imply essential self-adjointness of the generator of the stochastic dynamics
for these models. Finally, as an application of the strong uniqueness result
for the SPDE, we prove some functional inequalities for diffusion semigroups
generated by the above Dirichlet operators. These inequalities are improvements
of previous work by the second named author.
