Abstract
Heat kernel of supercritical nonlocal operators with unbounded drifts
Highlights
Throughout this paper we fix α ∈ (0, 2)
Κ−1 1I(t, x) κ1I, |a(t, x) − a(t, y)| κ1|x − y|γ, where a∗ stands for the transpose of a and I is the identity matrix
We can first mention the seminal work of Kolokoltsov [22] from which one can derive that for an stochastic differential equation (SDE) driven by a symmetric stable process with smooth nondegenerate spectral measure, Lipschitz non-degenerate diffusion coefficient and non trivial Lipschitz bounded drifts when α > 1, two sided estimates for the density of the type: (1.6)
Summary
Throughout this paper we fix α ∈ (0, 2). Let L(α) be a d-dimensional rotationally invariant α-stable process. The operator Ls is called supercritical for α ∈ (0, 1) since in this case, the drift term plays a dominant role. This precisely means that the fluctuations induced by the noise are smaller than the typical order of the drift term in (1.1). For the remaining case α = 1, the noise and drift both have the same typical order and the operator Ls is called critical. Let us indicate that there is a quite large literature concerning stable driven SDEs. We can first mention the seminal work of Kolokoltsov [22] from which one can derive that for an SDE driven by a symmetric stable process with smooth nondegenerate spectral measure, Lipschitz non-degenerate diffusion coefficient and non trivial Lipschitz bounded drifts when α > 1, two sided estimates for the density of the type:.
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