Paweł Sztonyk

Estimates of transition densities and their derivatives for jump Lévy processes

Abstract We give upper and lower estimates of densities of convolution semigroups of probability measures under explicit assumptions on the corresponding Levy measure and the Levy–Khinchin exponent. We obtain also estimates of derivatives of densities.

Small-time sharp bounds for kernels of convolution semigroups

We study small-time bounds for transition densities of convolution semigroups corresponding to pure jump Lévy processes in Rd, d ≥ 1, including the processes with jump measures which are exponentially and subexponentially localized at ∞. For a large class of Lévy measures, not necessarily symmetric or absolutely continuous with respect to Lebesgue measure, we find the optimal upper bound in both time and space for the corresponding heat kernels at ∞. In case of Lévy measures that are symmetric and absolutely continuous with densities g such that g(x) ≍ f(|x|) for non-increasing profile functions f, we also prove the full characterization of the sharp two-sided transition densities bounds of the form

Harnack inequality for symmetric stable processes on fractals

{p_t}\left( x \right) \asymp h{\left( t \right)^{ - d}} \cdot {1_{\left\{ {\left| x \right| \leqslant \theta h\left( t \right)} \right\}}} + tg\left( x \right) \cdot {1_{\left\{ {\left| x \right| \geqslant \theta h\left( t \right)} \right\}}},t \in \left( {0,{t_0}} \right),{t_0} > 0,x \in {\mathbb{R}^d}.

Estimates of the potential kernel and Harnack's inequality for the anisotropic fractional Laplacian

pt(x)≍h(t)−d⋅1{|x|≤θh(t)}+tg(x)⋅1{|x|≥θh(t)},t∈(0,t0),t0>0,x∈ℝd. This is done for small and large x separately. Mainly, our argument is based on new precise upper bounds for convolutions of Lévy measures. Our investigations lead to a surprising dichotomy correspondence of the decay properties at ∞ for transition densities of pure jump Lévy processes. All results are obtained solely by analytic methods, without use of probabilistic arguments.

Potential theory for L?evy stable processes

We derive upper estimates of transition densities for Feller semigroups with jump intensities lighter than that of the rotation invariant stable Lévy process.

Boundary potential theory for stable Lévy processes

Abstract We study nonnegative harmonic functions of symmetric α-stable processes on d-sets F. We prove the Harnack inequality for such functions when α∈(0,2/dw)∪(ds,2). Furthermore, we investigate the decay rate of harmonic functions and the Carleson estimate near the boundary of a region in F. In the particular case of natural cells in the Sierpinski gasket we also prove the boundary Harnack principle. To cite this article: K. Bogdan et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 59–63.