Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states

Suppose that L is a second order elliptic differential operator on a manifold M, B is a vector field, and V is a continuous function. The paper studies by probabilistic and dynamical systems means the behavior as e{open} → 0 of the principal eigenvalue λ ^ε (V) for the operator L ^ε = e{open}L + (B, ∇) +V considered on a compact manifold or in a bounded domain with zero boundary conditions. Under certain hyperbolicity conditions on invariant sets of the dynamical system generated by the vector field B the limit as e{open} → 0 of this principal eigenvalue turns out to be the topological pressure for some function. This gives a natural transition as e{open} → 0 from Donsker-Varadhan's variational formula for principal eigenvalues to the variational principle for the topological pressure and unifies previously separate results on random perturbations of dynamical systems.