Variational Formula for Dirichlet Forms and Estimates of Principal Eigenvalues for Symmetric α-stable Processes

In this paper, we prove a variational formula for Dirichlet forms generated by general symmetric Markov processes. As its applications, we obtain lower bound estimates of the bottom of spectrum for symmetric Markov processes. Moreover, for a positive measure μ charging no set of zero capacity, we give a new proof of an L2(μ)-estimate of functions in Dirichlet spaces. Finally, we calculate the principal eigenvalues for absorbing and time changed α-stable processes and obtain conditions for some measures being gaugeable.