SMALL MASS ASYMPTOTICS FOR PROBLEMS IN STOCHASTIC DIFFERENTIAL EQUATIONS

Abstract

Title of dissertation: SMALL MASS ASYMPTOTICS FOR PROBLEMS IN STOCHASTIC DIFFERENTIAL EQUATIONS Jong Jun Lee, Doctor of Philosophy, 2014 Dissertation directed by: Professor Mark Freidlin and Professor Sandra Cerrai Department of Mathematics Small mass asymptotics of the motion of a particle moving in a force field (Smoluchowski-Kramers approximation) was first studied in Smoluchowski [13] and Kramers [9]. Freidlin summarized the results and considered various asymptotic problems related to it in 2004 [4]. Recently, there have been papers from various authors on small mass asymptotics [1,5,7] after Freidlin’s work. Cerrai and Freidlin showed in 2011 [1] that a type of the Smoluchowski-Kramers approximation works in the case of the motion of a charged particle moving in a constant magnetic field and Freidlin and Hu showed in 2011 [5] that a type of the Smoluchowski-Kramers approximation works in the case of the motion of a particle moving in a space with friction coefficient dependent upon position. We summarize these results in Chapter 1. We consider generalizations of the works by Freidlin [4], Cerrai and Freidlin [1], Freidlin and Hu [5], and Gitterman [6]. We study the problem of the motion of a charged particle moving in a variable magnetic field dependent upon position [10] in Chapter 2, the Smoluchowkski-Kramers approximation in the case of linear differential operators in Chapter 3, and the small mass asymptotics in the case of random mass in Chapter 4.