טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentArcusin Nitay
SubjectOn The Asymptotic Behavior of the Principal Eigenvalue for
Diffusion Processes with Jumps
DepartmentDepartment of Mathematics
Supervisor Professor Ross Pinsky
Full Thesis textFull thesis text - English Version


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→∞.