arXiv subjects

Jens Malmquist

Publications and source records attributed to Jens Malmquist.

4 recordsLinked to original sources

arXiv2025

Towards a characterization of elliptic Harnack inequality for jump processes

Let $X$ be an isotropic unimodal L\'{e}vy jump process on $\mathbb{R}^d$. We develop probabilistic methods which in many cases allow us to determine whether $X$ satisfies the elliptic Harnack inequality (EHI), by looking only at the jump kernel of $X$, and its truncated second moments. Both our positive results and our negative results can be applied to subordinated Brownian motions (SBMs) in particular. We produce the first known example of an SBM that does \textit{not} satisfy EHI. We show that for many SBMs that were previously known to satisfy EHI (such as the geometric stable process, the iterated geometric stable process, and the relativistic geometric stable process), bounded perturbations of them also satisfy EHI (which was not previously clear). We show that certain SBMs with Laplace exponent $\phi(\lambda) = \tilde{\Omega}(\lambda)$ satisfy EHI, which previous methods were unable to determine.

math.PR↗

arXiv2023

Counterexamples to elliptic Harnack inequality for isotropic unimodal Lévy processes

Until now, it has been an open question whether every subordinated Brownian motion (SBM) satisfies the elliptic Harnack inequality (EHI). In this paper, we show that the answer is ``no." In our first theorem, we show that if $X=(X_t)_{t \geq 0}$ is an isotropic unimodal Lévy process, and $X$ satisfies certain criteria (involving the jump kernel of $X$ and the distribution of the location upon first exiting balls of various sizes) then $X$ does not satisfy EHI. (Note that the class of isotropic unimodal Lévy processes is larger than the class of SBMs.) We then check that many specific SBMs do indeed satisfy the criteria, and thus do not satify EHI.

math.PR↗

arXiv2021

Stability results for symmetric jump processes on metric measure spaces with atoms. Long version

Consider a symmetric Markovian jump process $\{X_t\}$ on a metric measure space $(M, d, μ)$. Chen, Kumagai, and Wang recently showed that two-sided heat kernel estimates and the parabolic Harnack inequality are both stable under bounded perturbations of the jumping measure, assuming $(M, d, μ)$ satisfies the volume-doubling and reverse-volume-doubling conditions. These results do not apply if $(M, d, μ)$ is a graph (or more generally, if $M$ contains any atoms $x$ such that $μ(x)>0$) because it is impossible for reverse-volume-doubling to hold on a space with atoms. We generalize the results of Chen, Kumagai, and Wang to a larger class of metric measure spaces, including all infinite graphs with volume-doubling. Our main tool is the construction of an "auxiliary space" that smooths out the atoms. We show that many properties transfer from $(M, d, μ)$ to the auxiliary space, and vice versa, including heat kernel estimates, the parabolic Harnack inequality, and their stable characterizations.

math.PR↗

arXiv2016

Numerical integration for fractal measures

We find estimates for the error in replacing an integral $\int f d\mu$ with respect to a fractal measure $\mu$ with a discrete sum $\sum_{x \in E} w(x) f(x)$ over a given sample set $E$ with weights $w$. Our model is the classical Koksma-Hlawka theorem for integrals over rectangles, where the error is estimated by a product of a discrepancy that depends only on the geometry of the sample set and weights, and variance that depends only on the smoothness of $f$. We deal with p.c.f self-similar fractals, on which Kigami has constructed notions of energy and Laplacian. We develop generic results where we take the variance to be either the energy of $f$ or the $L^1$ norm of $\Delta f$, and we show how to find the corresponding discrepancies for each variance. We work out the details for a number of interesting examples of sample sets for the Sierpi\'{n}ski gasket, both for the standard self-similar measure and energy measures, and for other fractals.

math.NA↗