Drift perturbation of subordinate Brownian motions with Gaussian component

Abstract

Let d ≥ 1 and Z be a subordinate Brownian motion on R^d 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) p ^b(t, x, y) for non-local operator ℒ^b = Δ + ψ(Δ) + b · ∇, where b is an R^d-valued function in Kato class K _d,1. We show that p ^b(t, x, y) is jointly continuous and derive its sharp two-sided estimates. The kernel p ^b(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 (L^b,C ^∞ _c (R^d)) and X is a weak solution of $X_t = X_0 + Z_t + \int_0^t {b(X_s )ds,} t \geqslant 0$. Moreover, we prove that the above stochastic differential equation has a unique weak solution.