Abstract
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.
Original language | English |
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Pages (from-to) | 1-47 |
Number of pages | 47 |
Journal | Israel Journal of Mathematics |
Volume | 70 |
Issue number | 1 |
DOIs | |
State | Published - Feb 1990 |