M.Sc Thesis
M.Sc Student Arcusin Nitay On The Asymptotic Behavior of the Principal Eigenvalue for Diffusion Processes with Jumps Department of Mathematics Professor Ross Pinsky

Abstract

Let D be a subset of R d, a smooth bounded domain, and let D0 be its closure. Let ?  be a probability measure on D, and let V(x)>0 be in C2(D0). Consider a Markov process X(t) performing Brownian motion in D, which is killed at the boundary and which while alive, jumps instantaneously at random times to a new point, according to the distribution ?. The probability that the process has not jumped by time t is given by exp{-∫0 V(X(s))ds}, γ>0. After the process jumps, it repeats the above behavior independently of the past.

The infinitesimal generator of this process is an extension of the operator -LγV,?, defined on C2(D0)∩{u:u, LγV,?u Î C0(D0)} by

LγV,?u≡-?ΔuγV(u-∫Dud?),

with the Dirichlet boundary condition. The spectrum σ(LγV,?) of LγV,? consists exclusively of eigenvalues. The operator LγV,? possesses a principal eigenvalue, λ0(γV,?); that is, λ0(γV,?) is real and simple and satisfies λ0(γV,?)=inf{Re(λ):λÎσ(LγV,?)} and λ0(γV,?)>0.

In this thesis we study the asymptotic behavior of the principal eigenvalue when γ→∞.