Abstract
Let d ≥ 1 and Z be a subordinate Brownian motion on Rd with infinitesimal generator Δ + ψ(Δ), where ψ is the Laplace exponent of a one-dimensional non-decreasing Lévy process (called subordinator). We establish the existence and uniqueness of fundamental solution (also called heat kernel) pb(t, x, y) for non-local operator ℒb = Δ + ψ(Δ) + b · ∇, where b is an Rd-valued function in Kato class Kd,1. We show that pb(t, x, y) is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for (Lb,C∞c (Rd)) and X is a weak solution of (Formula presented. ). Moreover, we prove that the above stochastic differential equation has a unique weak solution.
Original language | English |
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Pages (from-to) | 239-260 |
Number of pages | 22 |
Journal | Science China Mathematics |
Volume | 59 |
Issue number | 2 |
DOIs | |
Publication status | Published - 1 Feb 2016 |
Externally published | Yes |
Keywords
- Feller process
- Kato class
- Lévy system
- gradient perturbation
- heat kernel
- martingale problem
- stochastic differential equation
- subordinate Brownian motion