There is work on infinite-dimensional exponential families of measures which might be what you are looking for.
There are these possible references:
- Scricciolo (2006). Convergence rates for Bayesian density estimation of infinite-dimensional exponential families
- Rivoirard and Rousseau (2012). Posterior concentration rates for infinite dimensional exponential families
- Deuschel (1987). Infinite-dimensional diffusion processes as Gibbs measures on $C \left[ 0, 1 \right]^{Z^d}$
The first paper provides such distributions on Sobolev spaces. Also, the references in that paper are helpful about using infinite-dimensional entropy optimizing-based measures in order to construct priors for use in Bayesian non-parametric statistics.
Edit: I added an additional reference that is also relevant to the topic.
