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

Suppose thatL is a second order elliptic differential operator on a manifoldM, B is a vector field, andV is a continuous function. The paper studies by probabilistic and dynamical systems means the behavior asɛ → 0 of the principal eigenvalueλ ^ε (V) for the operatorL ^ε = ɛL + (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 fieldB the limit asɛ → 0 of this principal eigenvalue turns out to be the topological pressure for some function. This gives a natural transition asɛ → 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. This work was supported by US-Israel Binational Science Foundation.