科研之友
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摘要
Let D subset of R(d) be a bounded domain and let P(D) denote the space of probability measures on D. Consider a Brownian motion in D which is killed at the boundary and which, while alive, jumps instantaneously according to a spatially dependent exponential clock with intensity gamma V to a new point, according to a distribution mu is an element of P(D). From its new position after the jump, the process repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator -L(gamma,mu), defined by
L(gamma,mu)u -1/2 Delta u + gamma V C(mu)(u),
with the Dirichlet boundary condition, where C(mu) is the "mu-centering" operator defined by
C(mu)(u) = u - integral(D) u d mu.
The principal eigenvalue, lambda(0)(gamma, mu), of L(gamma,mu) governs the exponential rate of decay of the probability of not exiting D for large time. We study the asymptotic behavior of lambda(0)(gamma, mu) as gamma -> infinity. In particular, if mu possesses a density in a neighborhood of the boundary, which we call mu, then
lim(gamma ->infinity) gamma(-1/2) lambda(0)(gamma, mu) = integral(partial derivative D) mu/root Vd sigma/root 2 integral(D) 1/V d mu.
If mu and all its derivatives up to order k-1 vanish on the boundary, but the kth derivative does not vanish identically on the boundary, then lambda(0)(gamma, mu) behaves asymptotically like c(k) gamma(1-k/2), for an explicit constant c(k).
- 出版日期2011-12
