arXiv subjects

Jonathan Mui

Publications and source records attributed to Jonathan Mui.

6 recordsLinked to original sources

arXiv2026

Localization of generators of Markov processes via the Dynkin operator

In this article, we consider a Feller process whose resolvent has the strong Feller property. Using the Dynkin operator of this process, we define local realizations of the generator on an open subset $U$ of its state space. In particular, this allows us to describe the generator of the process killed upon leaving a regular, open subset $U$ as a realization of a maximal operator subject to Dirichlet boundary conditions. Besides the definition via the Dynkin operator, we also consider other possible definitions of such a local realization: a definition in terms of the unkilled semigroup, a definition based on approximation with elements of the generator and a probabilistic definition using martingales. Under mild additional assumptions, we prove that all these definitions are equivalent.

math.FA↗

arXiv2026

Smoothing of operator semigroups under relatively bounded perturbations

We investigate a smoothing property for strongly-continuous operator semigroups, akin to ultracontractivity in parabolic evolution equations. Specifically, we establish the stability of this property under certain relatively bounded perturbations of the semigroup generator. This result yields a spectral perturbation theorem, which has implications for the long-term dynamics of evolution equations driven by elliptic operators of second and higher orders. In particular, a new perturbation theorem for so-called eventually positive semigroups is derived as a consequence of the general results.

math.AP↗

arXiv2026

Non-Positivity of the heat equation with non-local Robin boundary conditions

We study heat equations $\partial_t u - \operatorname{div}(A\nabla u) = 0$ on bounded Lipschitz domains $Ω$, where $-\operatorname{div}(A\nabla\,\cdot\,)$ is a second-order uniformly elliptic operator with generalised Robin boundary conditions. These boundary conditions are formally given by $ν\cdot A\nabla u + Bu=0$, where $B\in\mathcal{L}(L^2(\partialΩ))$ is a general operator. In contrast to large parts of the literature on non-local Robin boundary conditions, we also allow for operators $B$ that destroy the positivity preserving property of the solution semigroup. Nevertheless, we obtain ultracontractivity of the semigroup under quite mild assumptions on $B$. For a certain class of operators $B$ we demonstrate that the semigroup is in fact eventually positive rather than positivity preserving.

math.AP↗

arXiv2025

Elliptic operators with non-local Wentzell-Robin boundary conditions

In this article, we study strictly elliptic, second-order differential operators on a bounded Lipschitz domain in $\mathbb{R}^d$, subject to certain non-local Wentzell-Robin boundary conditions. We prove that such operators generate strongly continuous semigroups on $L^2$-spaces and on spaces of continuous functions. We also provide a characterisation of positivity and (sub-)Markovianity of these semigroups. Moreover, based on spectral analysis of these operators, we discuss further properties of the semigroup such as asymptotic behaviour and, in the case of a non-positive semigroup, the weaker notion of eventual positivity of the semigroup.

math.AP↗

arXiv2022

Spectral properties of locally eventually positive operator semigroups

This paper considers strongly continuous semigroups of operators on Banach lattices which are locally eventually positive, a property that was first investigated in the context of concrete fourth-order evolution equations. We construct a simple example to show that the typical assumptions on the spectrum of the semigroup generator considered currently in the literature are far from necessary in the more general setting of local eventual positivity. Under minimal additional assumptions, we obtain results on the asymptotic behaviour of orbits, as well as necessary conditions on the peripheral point spectrum of locally eventually positive semigroups.

math.FA↗

arXiv2021

Local uniform convergence and eventual positivity of solutions to biharmonic heat equations

We study the evolution equation associated with the biharmonic operator on infinite cylinders with bounded smooth cross-section subject to Dirichlet boundary conditions. The focus is on the asymptotic behaviour and positivity properties of the solutions for large times. In particular, we derive the local eventual positivity of solutions. We furthermore prove the local eventual positivity of solutions to the biharmonic heat equation and its generalisations on Euclidean space. The main tools in our analysis are the Fourier transform and spectral methods.

math.AP↗