Semigroup properties of solutions of SDEs driven by Levy processes with independent coordinates

We study the stochastic differential equation dX(t) = A(Xt-)dZ(t), X-0 = x, where Z(t) = (Z(t)((1)), ..., Z(t)((d)))(T) and Z(t)((1)), ..., Z(t)((d)) are independent one-dimensional Levy processes with characteristic exponents psi(1), ..., psi(d). We assume that each psi(i) satisfies a weak lower scaling condition WLSC(alpha, 0, (C) under bar), a weak upper scaling condition WUSC(beta, 1, (C) over bar) (where 0 < alpha <= beta < 2) and some additional regularity properties. We consider two mutually exclusive assumptions: either (i) all psi(1), ..., psi(d) are the same and alpha, beta are arbitrary, or (ii) not all psi(1), ..., psi(d) are the same and alpha > (2/3)beta. We also assume that the determinant of A(x) = (a(ij)(x)) is bounded away from zero, and a(ij)(x) are bounded and Lipschitz continuous. In both cases (i) and (ii) we prove that for any fixed gamma is an element of (0, alpha) boolean AND (0,1] the semigroup P-t of the process X satisfies vertical bar P(t)f(x)-P(t)f(y)vertical bar <= c(t)(-gamma/alpha )vertical bar x-y vertical bar(gamma) parallel to f parallel to(infinity) for arbitrary bounded Borel function f. We also show the existence of a transition density of the process X. (C) 2020 Elsevier B.V. All rights reserved.