Estimates of densities for Lévy processes with lower intensity of large jumps

Abstract

We obtain general lower estimates of transition densities of jump Lévy processes. We use them for processes with Lévy measures having bounded support, processes with exponentially decaying Lévy measures for large times and for processes with high intensity of small jumps for small times.

Construction and heat kernel estimates of generalstable-like Markov processes

“…This estimate is the generalization of the results for the transition probability density of a Lévy process, obtained in a series of papers by Sztonyk [74,75,76,77], and Kaleta & Sztonyk [38,39,40]; similar results for certain not necessarily stable Lévy processes and related Lévy-type processes are discussed in [23], [24]. In general, even in the Lévy case it is impossible to get a lower bound with the same rate as the upper bound, without additional assumptions on the spectral measure σ(d ) (see [77]). In the recent works of Kulczycki, Ryznar & Sztonyk [55,56,57] systems of SDEs of the type dX t = A(X t− ) dZ t , driven by cylindrical α-stable processes are studied.…”

Strong Feller Property for SDEs Driven by Multiplicative Cylindrical Stable Noise

“…The main tools to prove Proposition 3.9 are Lemma 3.6 and the estimate (18). This key estimate (18) is proven using the techniques and results from [23], [22] and [35]. After constructing the transition density u(t, x, y) we use the technique developed by Knopova and Kulik [19] to show that U t f (x) := R d u(t, x, y)f (y) dy satisfies the appropriate heat equation in the so-called approximate setting.…”

Strong Feller property for SDEs driven by multiplicative cylindrical stable noise

“…The main tools to prove Proposition 3.9 are Lemma 3.6 and the estimates (13). These key estimates (13) are proven using the techniques and results from [23], [22] and [33]. After constructing the transition density u(t, x, y) we use the technique developed by Knopova and Kulik [19] to show that u(t, x, y) satisfies the appropriate heat equation in the so-called approximate setting.…”