arXiv subjects
Publications and source records attributed to Paweł Sztonyk.
We derive an explicit representation for the distribution of a Lévy process with total stochastic resetting and use it to investigate the asymptotic behaviour of the corresponding transition densities. The proposed approach applies to several important classes of Lévy processes, including relativistic stable processes, and yields new asymptotic estimates for their transition densities.
We study the spatial decay behaviour of resolvent kernels for a large class of non-local Lévy operators and bound states of the corresponding Schrödinger operators. Our findings naturally lead us to proving results for Lévy measures, which have subexponential or exponential decay, respectively. This leads to sharp transitions in the the decay rates of the resolvent kernels. We obtain estimates that allow us to describe and understand the intricate decay behaviour of the resolvent kernels and the bound states in either regime, extending findings by Carmona, Masters and Simon for fractional Laplacians (the subexponential regime) and classical relativistic operators (the exponential regime). Our proofs are mainly based on methods from the theory of operator semigroups.
We construct a strong Markov process corresponding to the Dirichlet form of Servadei and Valdinoci and use the process to solve the corresponding Neumann boundary problem for the fractional Laplacian and the half-line.
We study heat kernels of Schrödinger operators whose kinetic terms are non-local operators built for sufficiently regular symmetric Lévy measures with radial decreasing profiles and potentials belong to Kato class. Our setting is fairly general and novel -- it allows us to treat both heavy- and light-tailed Lévy measures in a joint framework. We establish a certain relative-Kato bound for the corresponding semigroups and potentials. This enables us to apply a general perturbation technique to construct the heat kernels and give sharp estimates of them. Assuming that the Lévy measure and the potential satisfy a little stronger conditions, we additionally obtain the regularity of the heat kernels. Finally, we discuss the applications to the smoothing properties of the corresponding semigroups. Our results cover many important examples of non-local operators, including fractional and quasi-relativistic Schrödinger operators.
We consider the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, driven by cylindrical $α$-stable process $Z_t$ in $R^d$, where $α\in (0,1)$ and $d \ge 2$. We assume that the determinant of $A(x) = (a_{ij}(x))$ is bounded away from zero, and $a_{ij}(x)$ are bounded and Lipschitz continuous. We show that for any fixed $γ\in (0,α)$ the semigroup $P_t$ of the process $X_t$ satisfies $|P_t f(x) - P_t f(y)| \le c t^{-γ/α} |x - y|^γ ||f||_\infty$ for arbitrary bounded Borel function $f$. Our approach is based on Levi's method.
We study a spatial asymptotic behaviour at infinity of kernels $p_t(x)$ for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limits $$ \lim_{r \to \infty} \frac{p_t(rθ-y)}{t \, ν(rθ)}, \quad t \in T, \ \ θ\in E, \ \ y \in \mathbb R^d, $$ exist and can be effectively computed. Here $ν$ is the corresponding Lévy density, $T \subset (0,\infty)$ is a bounded time-set and $E$ is a subset of the unit sphere in $\mathbb R^d$, $d \geq 1$. Our results are local on the unit sphere. They apply to a wide class of convolution semigroups, including those corresponding to highly asymmetric (finite and infinite) Lévy measures. Key examples include fairly general families of stable, tempered stable, jump-diffusion and compound Poisson semigroups. A main emphasis is put on the semigroups with Lévy measures that are exponentially localized at infinity, for which our assumptions and results are strongly related to the existence of the multidimensional exponential moments. Here a key example is the evolution semigroup corresponding to the so-called quasi-relativistic Hamiltonian $\sqrt{-Δ+m^2} - m$, $m>0$. As a byproduct, we also obtain sharp two-sided estimates of the kernels $p_t$ in generalized cones, away from the origin.
We construct and estimate the fundamental solution of highly anisotropic space-inhomogeneous integro-differential operators. We use the Levi method. We give applications to the Cauchy problem for such operators.
We study small time bounds for transition densities of convolution semigroups corresponding to pure jump Lévy processes in $\mathbb{R}^{d}$, $d \geq 1$, including those with jumping kernels exponentially and subexponentially localized at infinity. For a large class of Lévy measures, non-necessarily symmetric nor absolutely continuous with respect to the Lebesgue measure, we find the optimal, both in time and space, upper bound for the corresponding transition kernels at infinity. In case of Lévy measures that are symmetric and absolutely continuous, with densities $g$ such that $g(x) \asymp f(|x|)$ for nonincreasing profile functions $f$, we also prove the full characterization of the sharp two-sided transition densities bounds of the form $$ p_t(x) \asymp h(t)^{-d} \cdot \mathbf{1}_{\left\{|x|\leq θh(t)\right\}} + t \, g(x) \cdot \mathbf{1}_{\left\{|x| \geq θh(t)\right\}}, \quad t \in (0,t_0), \ \ t_0>0, \ \ x \in \mathbb{R}^{d}. $$ 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 an interesting and surprising dichotomy of the decay properties at infinity for transition kernels of pure jump Lévy processes. All results are obtained solely by analytic methods, without use of probabilistic arguments.
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.
We derive explicitly the coupling property for the transition semigroup of a Lévy process and gradient estimates for the associated semigroup of transition operators. This is based on the asymptotic behaviour of the symbol or the characteristic exponent near zero and infinity, respectively. Our results can be applied to a large class of Lévy processes, including stable Lévy processes, layered stable processes, tempered stable processes and relativistic stable processes.
We derive upper estimates of transition densities for Feller semigroups with jump intensities lighter than that of the rotation invariant stable Levy process
We consider Feller semigroups of operators determinated by systems of jumps dominated by the rotation invariant stable Lévy measure. Using an approximation schema we prove the existence and obtain estimates of corresponding heat kernels.
Estimates of densities of convolution semigroups of probability measures are given under specific assumptions on the corresponding Lévy measure and the Lévy--Khinchin exponent. The assumptions are satisfied, e.g., by tempered stable semigroups of J. Rosi{ń}ski.
We prove that bounded harmonic functions of anisotropic fractional Laplacians are Hölder continuous under mild regularity assumptions on the corresponding Lévy measure. Under some stronger assumptions the Green function, Poisson kernel and the harmonic functions are even differentiable of order up to three.
We characterize those homogeneous translation invariant symmetric non-local operators with positive maximum principle whose harmonic functions satisfy Harnack's inequality. We also estimate the corresponding semigroup and the potential kernel.