Kamil Kaleta

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

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

{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}.

Fractional P(ϕ)1-processes and Gibbs measures

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.

Integrated density of states for Poisson–Schrödinger perturbations of subordinate Brownian motions on the Sierpiński gasket

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

Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval ☆

We study the spatial decay of eigenfunctions of non-local Schrödinger operators whose kinetic terms are generators of symmetric jump-paring Lévy processes with Kato-class potentials decaying at infinity. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps. We find that the decay rates of eigenfunctions depend on the process via specific preference rates in particular jump scenarios, and depend on the potential through the distance of the corresponding eigenvalue from the edge of the continuous spectrum. We prove that the conditions of the jump-paring class imply that for all eigenvalues the corresponding positive eigenfunctions decay at most as rapidly as the Lévy intensity. This condition is sharp in the sense that if the jump-paring property fails to hold, then eigenfunction decay becomes slower than the decay of the Lévy intensity. We furthermore prove that under reasonable conditions the Lévy intensity also governs the upper bounds of eigenfunctions, and ground states are comparable with it, i.e., two-sided bounds hold. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the Lévy intensity is varied from sub-exponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition Lévy intensity-driven decay becomes slower than the rate of decay of the Lévy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.

Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman–Kac semigroups for a class of Lévy processes

The time evolution of random variables with Lévy statistics has the ability to develop jumps, displaying very different behaviors from continuously fluctuating cases. Such patterns appear in an ever broadening range of examples including random lasers, non-Gaussian kinetics, or foraging strategies. The penalizing or reinforcing effect of the environment, however, has been little explored so far. We report a new phenomenon which manifests as a qualitative transition in the spatial decay behavior of the stationary measure of a jump process under an external potential, occurring on a combined change in the characteristics of the process and the lowest eigenvalue resulting from the effect of the potential. This also provides insight into the fundamental question of what is the mechanism of the spatial decay of a ground state.

Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians

We define and prove existence of fractional P(ϕ)1-processes as random processes generated by fractional Schrodinger semigroups with Kato-decomposable potentials. Also, we show that the measure of such a process is a Gibbs measure with respect to the same potential. We give conditions of its uniqueness and characterize its support relating this with intrinsic ultracontractivity properties of the semigroup and the fall-off of the ground state. To achieve that we establish and analyse these properties first.