A one-dimensional C^3 example shows that vanishing-viscosity limits for mechanical Hamiltonian ergodic problems may not exist.
Anantharaman,On the zero-temperature or vanishing viscosity limit for Markov processes arising from Lagrangian dynamics, J
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Nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic problems
A one-dimensional C^3 example shows that vanishing-viscosity limits for mechanical Hamiltonian ergodic problems may not exist.