Abstract
Suppose that α ∈ (0, 2) and that X is an α-stable-like process on Rd. Let μ be a signed measure on Rd belonging to the class Kd,α and Aμt be the continuous additive functional of X associated with μ. In this paper we show that the Feynman-Kac semigroup (Tμt : t ≥ 0) defined by Tμtf(x) = Ex(e−Aμt f(Xt)) has a density qμ and that there exist positive constants c1, c2, c3, c4 such that c1e−c2tt−d/α (1 ⋀ t1/α/|x − y|)d+α ≤ qμ(t, x, y) ≤ c3ec4tt−d/α (1 ⋀ t1/α/|x − y|)d+α for all (t, x, y) ∈ (0, ∞)×Rd×Rd. We also provide similar estimates for the densities of two other kinds of Feynman-Kac semigroups of X.
Original language | English (US) |
---|---|
Pages (from-to) | 146-161 |
Number of pages | 16 |
Journal | Electronic Journal of Probability |
Volume | 11 |
DOIs | |
State | Published - Jan 1 2006 |
Keywords
- Continuous additive functionals
- Continuous additive functionals of zero energy
- Feynman-Kac semigroups
- Kato class
- Purely discontinuous additive functionals
- Stable processes
- Stable-like processes
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty