A Note on the Smoluchowski-Kramers Approximation for the Langevin Equation with Reflection

Details A Note on the Smoluchowski-Kramers Approximation for the Langevin Equation with Reflection A Note on the Smoluchowski-Kramers Approximation for the Langevin Equation with Reflection

2010-04-18

arxiv.org/abs/1004.3043v1

Stochastics and Dynamics, Vol. 7, No. 2, June 2007 , 141-153

Mathematics

Probability

14 pages, 2 figures

Scientific paper

10.1142/S0219493707002001

According to the Smoluchowski-Kramers approximation, the solution of the equation ${\mu}\ddot{q}^{\mu}_t=b(q^{\mu}_t)-\dot{q}^{\mu}_t+{\Sigma}(q^{\mu}_t)\dot{W}_t, q^{\mu}_0=q, \dot{q}^{\mu}_0=p$ converges to the solution of the equation $\dot{q}_t=b(q_t)+{\Sigma}(q_t)\dot{W}_t, q_0=q$ as {\mu}->0. We consider here a similar result for the Langevin process with elastic reflection on the boundary.

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