## Abstract

Suppose that α ∈ (0, 2) and that X is an α-stable-like process on R^{d}. Let μ be a signed measure on Rd belonging to the class K_{d,α} and A^{μ}_{t} be the continuous additive functional of X associated with μ. In this paper we show that the Feynman-Kac semigroup (T^{μ}_{t} : t ≥ 0) defined by T^{μ}_{t}f(x) = E_{x}(e^{−Aμt} f(X_{t})) has a density q^{μ} and that there exist positive constants c_{1}, c_{2}, c_{3}, c_{4} such that c_{1}e^{−c2t}t^{−d/α} (1 ⋀ t^{1/α}/|x − y|)^{d+α} ≤ q^{μ}(t, x, y) ≤ c_{3}e^{c4t}t^{−d/α} (1 ⋀ t^{1/α}/|x − y|)^{d+α} for all (t, x, y) ∈ (0, ∞)×R^{d}×R^{d}. We also provide similar estimates for the densities of two other kinds of Feynman-Kac semigroups of X.

Original language | English (US) |
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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