Large Deviations for $\nu_\epsilon = Z_\epsilon\exp\left(-\frac{1}{\epsilon}\Phi(x)\right)d\mu$

Question

Given a probability measure of the form $$\nu_\epsilon=Z_\epsilon\exp\left(-\frac{1}{\epsilon}\Phi(x)\right)d\mu$$ with $Z$ being the normalizing constant.

Under which conditions on $\mu$ and $\Phi$ does $\nu_\epsilon$ satisfies a large deviation principle or concentrate around the minimas of $\Phi$.

Thank you

Answer 1

Look up Varadhan's lemma in any large deviations text. Specifically, Deuschel-Stroock, Exercise 2.1.24