We consider the small mass asymptotic (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The friction coefficient is assumed to be vanishing within certain region. We introduce a regularization for this problem and study the limiting motion for the 1-dimensional case and a multidimensional model problem. The limiting motion is a Markov process on a projected space. We specify the generator and the boundary condition of this limiting Markov process and prove the convergence.
Bibliographical noteFunding Information:
This work is supported in part by NSF Grants DMS-0803287 and DMS-0854982 .
- Boundary theory of Markov processes
- Diffusion processes
- Smoluchowski-Kramers approximation
- Weak convergence