Estimates of perturbation series for kernels
For integral kernels on space-time we indicate a class of nonnegative Schrödinger perturbations which produce comparable integral kernels.
math.FA↗
arXiv subjects
Tomasz Jakubowski
Publications and source records attributed to Tomasz Jakubowski.
Estimates of gradient perturbation series
We give upper and lower bounds of perturbation series for transition densities, corresponding to additive gradient perturbations satisfying certain space-time integrability conditions.
math.PR↗
Fractional Laplacian with singular drift
For $α\in (1,2)$ we consider the equation $\partial_t u = Δ^{α/2} u - r b \cdot \nabla u$, where $b$ is a divergence free singular vector field not necessarily belonging to the Kato class. We show that for sufficiently small $r>0$ the fundamental solution is globally in time comparable with the density of the isotropic stable process
math.PR↗
Estimates of the Green function for the fractional Laplacian perturbed by gradient
The Green function of the fractional Laplacian of the differential order bigger than one and the Green function of its gradient perturbations are comparable for bounded smooth multidimensional open sets if the drift function is in an appropriate Kato class.
math.AP↗
On exit time of stable processes
We study the exit time $τ=τ_{(0,\infty)}$ for 1-dimensional strictly stable processes and express its Laplace transform at $t^α$ as the Laplace transform of a positive random variable with explicit density. Consequently, $τ$ satisfies some multiplicative convolution relations. For some stable processes, e.g. for the symmetric $\frac23$-stable process, explicit formulas for the Laplace transform and the density of $τ$ are obtained as an application.
math.PR↗
On Wiener-Hopf factors for stable processes
We give a series representation of the logarithm of the bivariate Laplace exponent $κ$ of $α$-stable processes for almost all $α\in (0,2]$.
math.PR↗
Time-dependent Schrödinger perturbations of transition densities
We construct the fundamental solution of $\partial_t-Δ_y- q(t,y)$, for functions $q$ with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure spaces. We also discuss specific applications to Schrödinger perturbations of the fractional Laplacian in view of the fact that the 3P Theorem holds for the fundamental solution corresponding to the operator.
math.FA↗