arXiv subjects

Niels Jacob

Publications and source records attributed to Niels Jacob.

4 recordsLinked to original sources

arXiv2019

On Adjoint Additive Processes

Starting with an additive process $(Y_t)_{t\geq0}$, it is in certain cases possible to construct an adjoint process $(X_t)_{t\geq0}$ which is itself additive. Moreover, assuming that the transition densities of $(Y_t)_{t\geq0}$ are controlled by a natural pair of metrics $\mathrm{d}_{ψ,t}$ and $δ_{ψ,t}$, we can prove that the transition densities of $(X_t)_{t\geq0}$ are controlled by the metrics $δ_{ψ,1/t}$ replacing $\mathrm{d}_{ψ,t}$ and $\mathrm{d}_{ψ,1/t}$ replacing $δ_{ψ,t}$.

math.FA↗

arXiv2017

Higher Order Eigenvalues for Non-Local Schrödinger Operators

Two-sided estimates for higher order eigenvalues are presented for a class of non-local Schrödinger operators by using the jump rate and the growth of the potential. For instance, let $L$ be the generator of a Lévy process with Lévy measure $ν(d z):= ρ(z) d z$ such that $ρ(z)=ρ(-z)$ and $$c_1 |z|^{-(d+α_1)}\le ρ(z)\le c_2|z|^{-(d+α_2)},\ \ |z|\le κ$$ for some constants $κ, c_1,c_2>0$ and $α_1,α_2\in (0,2),$ and let $c_3|x|^{θ_1} \le V(x)\le c_4|x|^{θ_2}$ for some constants $θ_1,θ_2, c_3,c_4>0$ and large $|x|$. Then the eigenvalues $λ_1\le λ_2\le\cdots λ_n\le \cdots $ of $-L+V$ satisfies the following two-side estimate: for any $p>1$, there exists a constant $C>1$ such that $$C n^{\frac{θ_2α_2}{d(θ_2+α_2)}}\ge λ_n \ge C^{-1} n^{\frac{θ_1α_1}{d(θ_1+α_1)}},\ \ n\ge 1.$$ When $α_1$ is variable, a better lower bound estimate is derived.

math-ph↗

arXiv2006

Feller Semigroups Obtained by Variable Order Subordination

For certain classes of negative definite symbols $q(x,ξ)$ and state space dependent Bernstein function $f(x,s)$ we prove that $-p(x,D)$, the pseudo-differential operator with symbol $-p(x,ξ)=-f(x,q(x,ξ))$, extends to the generator of a Feller semigroup. Our result extends previously known results related to operators of variable (fractional) order of differentiation, or variable order fractional powers. New concrete examples are given.

math.FA↗

arXiv2006

Some thoughts on multiparameter stochastic processes

We suggest to investigate certain non-standard (pseudo-)differential operators in order to construct and to study multi-parameter processes. Our approach will include "classical" multi-parameter Markov processes but will go eventually far beyond.

math.PR↗