arXiv subjects

Radosław K. Wojciechowski

Publications and source records attributed to Radosław K. Wojciechowski.

14 recordsLinked to original sources

arXiv2026

On the accretivity and m-accretivity of Laplacians and porous medium-type operators on graphs

We study the accretivity and m-accretivity of Laplacian and porous medium-type operators on weighted graphs. In particular, we give several conditions that imply these properties for maximal operators and investigate when these operators agree with various restrictions. For porous medium-type operators on $\ell^1$ and for Laplacians on $\ell^p$ for $p \in [1,\infty)$, we show that there always exists a dense subset of the domain on which the maximal operator is m-accretive. As a consequence, we establish that accretivity, m-accretivity and injectivity of the shifted operator are all equivalent for these maximal operators. Under additional conditions on the graph, we then prove that the maximal operators are m-accretive on the entire domain, not just a dense subset. We also investigate minimal operators and show that they are m-accretive if and only if the minimal and maximal operators agree and the maximal operator is accretive. We then give some conditions that imply this agreement. Furthermore, for the minimal Laplacian on $\ell^p$, we show that accretivity and m-accretivity are not equivalent. For the $\ell^2$ case, we give connections to Markov uniqueness and essential self-adjointness. For the $\ell^\infty$ case, we establish the equivalence of stochastic completeness at infinity, m-accretivity for the maximal Laplacian on $\ell^\infty$, and m-accretivity of the minimal Laplacian on $\ell^1$.

math.FA↗

arXiv2026

Nonlinear parabolic characterizations of stochastic completeness at infinity on weighted graphs

We prove a nonlinear parabolic characterization of stochastic completeness at infinity for weighted graphs. For the filtration equation \[ (\partial_t + ΔΦ)u =0 \] where $Δ$ is the non-negative formal graph Laplacian and $Φu =ϕ\circ u$ with $ϕ\colon \R\to\R$ nonconstant, continuous and increasing, stochastic completeness at infinity is equivalent to uniqueness of bounded pointwise solutions for every bounded initial datum. For $Φ=\id$, this recovers the classical heat equation characterization of stochastic completeness at infinity, and of stochastic completeness when the killing term is trivial, i.e., when $κ=0$. If stochastic completeness at infinity fails, then every bounded initial datum admits infinitely many bounded pointwise solutions of the filtration equation. Admissible nonlinearities include the signed porous medium and fast diffusion powers $ϕ(s)=s|s|^{m-1}$ for all $m>0$, as well as many others. Stochastic completeness at infinity is further characterized by a generalized mass balance: the total mass of a positive pointwise solution at time $t$, augmented by the mass $\int_0^t\sum_{x}κ(x)ϕ(u(s,x)) \dd s$ dissipated by the killing term $κ$, equals the initial mass. This balance holds for every bounded positive solution on graphs of finite measure and for bounded finite-mass data on graphs of arbitrary measure under the sharp condition $\limsup_{r\to0^+}ϕ(r)/r<\infty$. It also extends to positive pointwise solutions in $\ell^1$ that are bounded on every positive time interval. When the killing term is trivial, stochastic completeness at infinity reduces to stochastic completeness and generalized balance to conservation of mass.

math.PR↗

arXiv2026

The generalized porous medium equation on graphs: well-posedness, extinction, and mass conservation

We study the Cauchy problem for the generalized porous medium equation on infinite weighted graphs. For a general nonlinearity, we establish Dirichlet comparison and weak maximum principles on finite subgraphs and, through an exhaustion argument, construct minimal and maximal pointwise solutions for arbitrary $\ell^\infty$ initial data, controlled by explicit, possibly time-dependent, barriers. For the porous nonlinearity $ϕ(s)=s|s|^{m-1}$, assuming a $ν$-Sobolev inequality with $ν>2$, we derive quantitative energy estimates for $\ell^1$-mild solutions. These yield finite-time extinction in the fast diffusion range $0 2/ν$. Interestingly, we recover the Euclidean critical exponent for several model graphs. Finally, we prove an exact generalized mass balance for nonnegative $\ell^1$-mild solutions on graphs that are stochastically complete at infinity, allowing for an arbitrary killing term. The same balance is established for suitable classical and bounded pointwise solutions. In the absence of killing, these reduce to conservation of mass.

math.AP↗

arXiv2025

Form uniqueness for graphs with weakly spherically symmetric ends

We give characterizations for the failure of form uniqueness on weakly spherically symmetric graphs. The first characterization is in terms of the graph structure, the second involves the capacity of a Cauchy boundary. We also discuss the stability of form uniqueness and give characterizations for graphs which, following the removal of a set, consist of a disjoint union of weakly spherically symmetric graphs.

math-ph↗

arXiv2024

Essential self-adjointness of the Laplacian on weighted graphs: harmonic functions, stability, characterizations and capacity

We give two characterizations for the essential self-adjointness of the weighted Laplacian on birth-death chains. The first involves the edge weights and vertex measure and is classically known; however, we give another proof using stability results, limit point-limit circle theory and the connection between essential self-adjointness and harmonic functions. The second characterization involves a new notion of capacity. Furthermore, we also analyze the essential self-adjointness of Schrödinger operators, use the characterizations for birth-death chains and stability results to characterize essential self-adjointness for star-like graphs, and give some connections to the $\ell^2$-Liouville property.

math.FA↗

arXiv2023

Neumann semigroup, subgraph convergence, form uniqueness, stochastic completeness and the Feller property

We study heat kernel convergence of induced subgraphs with Neumann boundary conditions. We first establish convergence of the resulting semigroups to the Neumann semigroup in $\ell^2$. While convergence to the Neumann semigroup always holds, convergence to the Dirichlet semigroup in $\ell^2$ turns out to be equivalent to the coincidence of the Dirichlet and Neumann semigroups while convergence in $\ell^1$ is equivalent to stochastic completeness. We then investigate the Feller property for the Neumann semigroup via generalized solutions and give applications to graphs satisfying a condition on the edges as well as birth-death chains.

math.FA↗

arXiv2021

Essential self-adjointness and the $L^2$-Liouville property

We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Green's function and when it gives a non-constant harmonic function which is square integrable.

math.SP↗

arXiv2020

Stochastic completeness of graphs: bounded Laplacians, intrinsic metrics, volume growth and curvature

The goal of this article is to survey various results concerning stochastic completeness of graphs. In particular, we present a variety of formulations of stochastic completeness and discuss how a discrepancy between uniqueness class and volume growth criteria in the continuous and discrete settings was ultimately resolved via the use of intrinsic metrics. Along the way, we discuss some equivalent notions of boundedness in the sense of geometry and of analysis. We also discuss various curvature criteria for stochastic completeness and discuss how weakly spherically symmetric graphs establish the sharpness of results.

math.MG↗

arXiv2018

Coverings and the heat equation on graphs: stochastic incompleteness, the Feller property and uniform transience

We study regular coverings of graphs and manifolds with a focus on properties of the heat equation. In particular, we look at stochastic incompleteness, the Feller property and uniform transience; and investigate the connection between the validity of these properties on the base space and its covering. For both graphs and manifolds, we prove the equivalence of stochastic incompleteness of the base and that of its cover. Along the way we also give some new conditions for the Feller property to hold on graphs.

math.FA↗

arXiv2014

Note on uniformly transient graphs

We study a special class of graphs with a strong transience feature called uniform transience. We characterize uniform transience via a Feller-type property and via validity of an isoperimetric inequality. We then give a further characterization via equality of the Royden boundary and the harmonic boundary and show that the Dirichlet problem has a unique solution for such graphs. The Markov semigroups and resolvents (with Dirichlet boundary conditions) on these graphs are shown to be ultracontractive. Moreover, if the underlying measure is finite, the semigroups and resolvents are trace class and their generators have $\ell^p$ independent pure point spectra (for $1 \leq p \leq \infty$). Examples of uniformly transient graphs include Cayley graphs of hyperbolic groups as well as trees and Euclidean lattices of dimension at least three. As a surprising consequence, the Royden compactification of such lattices turns out to be the one-point compacitifcation and the Laplacians of such lattices have pure point spectrum if the underlying measure is chosen to be finite.

math.FA↗

arXiv2013

Graphs of finite measure

We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of `relative compactness' for such graphs and study sufficient and necessary conditions for this property in terms of various metrics. We then equip graphs satisfying this property with a finite measure and investigate the associated Laplacian and its semigroup. In this context, our results include the trace class property for the semigroup, uniqueness and existence of solutions to the Dirichlet problem with boundary arising from the natural compactification, an explicit description of the domain of the Dirichlet Laplacian, convergence of the heat semigroup for large times as well as stochastic incompleteness and transience of the corresponding random walk in continuous time.

math.MG↗

arXiv2013

A note on self-adjoint extensions of the Laplacian on weighted graphs

We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.

math.FA↗

arXiv2012

Cheeger inequalities for unbounded graph Laplacians

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

math.SP↗

arXiv2012

Volume growth and bounds for the essential spectrum for Dirichlet forms

We consider operators arising from regular Dirichlet forms with vanishing killing term. We give bounds for the bottom of the (essential) spectrum in terms of exponential volume growth with respect to an intrinsic metric. As special cases we discuss operators on graphs. When the volume growth is measured in the natural graph distance (which is not an intrinsic metric) we discuss the threshold for positivity of the bottom of the spectrum and finiteness of the bottom of the essential spectrum of the (unbounded) graph Laplacian. This threshold is shown to lie at cubic polynomial growth.

math.FA↗