[{"project":[{"_id":"357","name":"TRR 358: Ganzzahlige Strukturen in Geometrie und Darstellungstheorie"}],"_id":"65073","external_id":{"arxiv":["2603.11232"]},"user_id":"100325","language":[{"iso":"eng"}],"type":"preprint","publication":"2603.11232","abstract":[{"lang":"eng","text":"We study the large-time behavior of the continuous-time heat kernel and of solutions to the heat equation on homogeneous trees. First, we derive sharp asymptotic formulas for the heat kernel as $t\\to\\infty$. Second, using them, we show that solutions with initial data in weighted $\\ell^1$ classes, asymptotically factorize in $\\ell^p$ norms, $p\\in[1,\\infty]$, as the product of the heat kernel, times a $p$-mass function, dependent on the initial condition and $p$. The  $p$-mass function is described in terms of boundary averages associated with Busemann functions for $p<2$, while for $p\\ge 2$, it is expressed through convolution with the ground spherical function. For comparison, the case of the integers shows that a single constant mass determines the asymptotics of solutions to the heat equation for all $p$, emphasizing the influence of the graph geometry on heat diffusion."}],"status":"public","date_updated":"2026-03-20T17:55:30Z","date_created":"2026-03-20T17:55:24Z","author":[{"first_name":"Efthymia","last_name":"Papageorgiou","full_name":"Papageorgiou, Efthymia","id":"100325"}],"title":"Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees","year":"2026","citation":{"short":"E. Papageorgiou, 2603.11232 (2026).","bibtex":"@article{Papageorgiou_2026, title={Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees}, journal={2603.11232}, author={Papageorgiou, Efthymia}, year={2026} }","mla":"Papageorgiou, Efthymia. “Long-Time Asymptotics for the Heat Kernel and for Heat Equation Solutions on Homogeneous Trees.” <i>2603.11232</i>, 2026.","apa":"Papageorgiou, E. (2026). Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees. In <i>2603.11232</i>.","ieee":"E. Papageorgiou, “Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees,” <i>2603.11232</i>. 2026.","chicago":"Papageorgiou, Efthymia. “Long-Time Asymptotics for the Heat Kernel and for Heat Equation Solutions on Homogeneous Trees.” <i>2603.11232</i>, 2026.","ama":"Papageorgiou E. Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees. <i>260311232</i>. Published online 2026."}},{"status":"public","abstract":[{"lang":"eng","text":"Let $\\mathbb{H}^n$ be the $n$-dimensional real hyperbolic space, $Δ$ its nonnegative Laplace--Beltrami operator whose bottom of the spectrum we denote by $λ_{0}$, and $σ\\in (0,1)$.\r\n  The aim of this paper is twofold. On the one hand, we determine the Fujita exponent for the fractional heat equation \\[\\partial_{t} u + Δ^σu = e^{βt}|u|^{γ-1}u,\\] by proving that nontrivial positive global solutions exist if and only if $γ\\geq 1 + β/ λ_{0}^σ$. On the other hand, we prove the existence of non-negative, bounded and finite energy solutions of the semilinear fractional elliptic equation \\[\r\n  Δ^σ v - λ^σ v - v^γ=0 \\] for $0\\leq λ\\leq λ_{0}$ and $1<γ< \\frac{n+2σ}{n-2σ}$. The two problems are known to be connected and the latter, aside from its independent interest, is actually instrumental to the former.\r\n  \\smallskip\r\n  At the core of our results stands a novel fractional Poincaré-type inequality expressed in terms of a new scale of $L^{2}$ fractional Sobolev spaces, which sharpens those known so far, and which holds more generally on Riemannian symmetric spaces of non-compact type. We also establish an associated Rellich--Kondrachov-like compact embedding theorem for radial functions, along with other related properties."}],"type":"preprint","publication":"arXiv:2509.12349","language":[{"iso":"eng"}],"user_id":"100325","project":[{"name":"TRR 358: Ganzzahlige Strukturen in Geometrie und Darstellungstheorie","_id":"357"}],"external_id":{"arxiv":["2509.12349"]},"_id":"64267","citation":{"ieee":"T. Bruno and E. Papageorgiou, “Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces,” <i>arXiv:2509.12349</i>. 2025.","chicago":"Bruno, Tommaso, and Effie Papageorgiou. “Blow-up Exponents and a Semilinear Elliptic Equation for the Fractional Laplacian on Hyperbolic Spaces.” <i>ArXiv:2509.12349</i>, 2025.","ama":"Bruno T, Papageorgiou E. Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces. <i>arXiv:250912349</i>. Published online 2025.","apa":"Bruno, T., &#38; Papageorgiou, E. (2025). Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces. In <i>arXiv:2509.12349</i>.","short":"T. Bruno, E. Papageorgiou, ArXiv:2509.12349 (2025).","bibtex":"@article{Bruno_Papageorgiou_2025, title={Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces}, journal={arXiv:2509.12349}, author={Bruno, Tommaso and Papageorgiou, Effie}, year={2025} }","mla":"Bruno, Tommaso, and Effie Papageorgiou. “Blow-up Exponents and a Semilinear Elliptic Equation for the Fractional Laplacian on Hyperbolic Spaces.” <i>ArXiv:2509.12349</i>, 2025."},"year":"2025","title":"Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces","author":[{"first_name":"Tommaso","last_name":"Bruno","full_name":"Bruno, Tommaso"},{"last_name":"Papageorgiou","full_name":"Papageorgiou, Effie","first_name":"Effie"}],"date_created":"2026-02-19T11:42:22Z","date_updated":"2026-02-19T11:43:16Z"},{"date_updated":"2026-02-19T11:43:53Z","author":[{"first_name":"Effie","last_name":"Papageorgiou","full_name":"Papageorgiou, Effie"},{"full_name":"Trojan, Bartosz","last_name":"Trojan","first_name":"Bartosz"}],"date_created":"2026-02-19T11:41:25Z","title":"Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings","year":"2025","citation":{"chicago":"Papageorgiou, Effie, and Bartosz Trojan. “Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings.” <i>ArXiv:2506.17042</i>, 2025.","ieee":"E. Papageorgiou and B. Trojan, “Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings,” <i>arXiv:2506.17042</i>. 2025.","ama":"Papageorgiou E, Trojan B. Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings. <i>arXiv:250617042</i>. Published online 2025.","bibtex":"@article{Papageorgiou_Trojan_2025, title={Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings}, journal={arXiv:2506.17042}, author={Papageorgiou, Effie and Trojan, Bartosz}, year={2025} }","mla":"Papageorgiou, Effie, and Bartosz Trojan. “Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings.” <i>ArXiv:2506.17042</i>, 2025.","short":"E. Papageorgiou, B. Trojan, ArXiv:2506.17042 (2025).","apa":"Papageorgiou, E., &#38; Trojan, B. (2025). Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings. In <i>arXiv:2506.17042</i>."},"project":[{"_id":"357","name":"TRR 358: Ganzzahlige Strukturen in Geometrie und Darstellungstheorie"}],"external_id":{"arxiv":["2506.17042"]},"_id":"64266","user_id":"100325","language":[{"iso":"eng"}],"type":"preprint","publication":"arXiv:2506.17042","abstract":[{"lang":"eng","text":"We study the large-time asymptotic behavior of solutions to the discrete-time heat equation, i.e., caloric functions, on affine buildings, including those without transitive group actions. For each $p \\in [1, \\infty]$, we introduce a notion of a $p$-mass function and prove that caloric functions with initial data belonging to certain weighted-$\\ell^1$ spaces or to the radial $\\ell^1$ class, asymptotically decouple as the product of this mass function and the heat kernel. These results extend classical analogues from Euclidean spaces and symmetric spaces of non-compact type to the non-Archimedean setting, and remain valid even for exotic buildings beyond the Bruhat--Tits framework. We characterize the spatial concentration of heat kernels in $p$-norms and describe the geometry of associated critical regions. Our results highlight substantial differences in the asymptotic regimes depending on the value of $p$, and clarify the interplay between volume growth and heat diffusion."}],"status":"public"},{"type":"journal_article","status":"public","_id":"56717","project":[{"name":"TRR 358 - Ganzzahlige Strukturen in Geometrie und Darstellungstheorie","_id":"357"}],"department":[{"_id":"555"}],"user_id":"73664","article_type":"original","file_date_updated":"2026-02-19T14:14:39Z","has_accepted_license":"1","publication_status":"published","related_material":{"link":[{"relation":"research_paper","url":"https://arxiv.org/abs/2410.10364"}]},"page":"1671-1694","intvolume":"        36","citation":{"chicago":"Langen, Lukas, and Margit Rösler. “Multiresolution Analysis on Spectra of Hermitian Matrices.” <i>Indagationes Mathematicae</i> 36, no. 6 (2025): 1671–94.","ieee":"L. Langen and M. Rösler, “Multiresolution analysis on spectra of hermitian matrices,” <i>Indagationes Mathematicae</i>, vol. 36, no. 6, pp. 1671–1694, 2025.","ama":"Langen L, Rösler M. Multiresolution analysis on spectra of hermitian matrices. <i>Indagationes Mathematicae</i>. 2025;36(6):1671-1694.","mla":"Langen, Lukas, and Margit Rösler. “Multiresolution Analysis on Spectra of Hermitian Matrices.” <i>Indagationes Mathematicae</i>, vol. 36, no. 6, Elsevier, 2025, pp. 1671–94.","bibtex":"@article{Langen_Rösler_2025, title={Multiresolution analysis on spectra of hermitian matrices}, volume={36}, number={6}, journal={Indagationes Mathematicae}, publisher={Elsevier}, author={Langen, Lukas and Rösler, Margit}, year={2025}, pages={1671–1694} }","short":"L. Langen, M. Rösler, Indagationes Mathematicae 36 (2025) 1671–1694.","apa":"Langen, L., &#38; Rösler, M. (2025). Multiresolution analysis on spectra of hermitian matrices. <i>Indagationes Mathematicae</i>, <i>36</i>(6), 1671–1694."},"date_updated":"2026-02-19T14:16:43Z","volume":36,"author":[{"first_name":"Lukas","full_name":"Langen, Lukas","id":"73664","last_name":"Langen"},{"last_name":"Rösler","full_name":"Rösler, Margit","id":"37390","first_name":"Margit"}],"main_file_link":[{"url":"https://doi.org/10.1016/j.indag.2025.03.009"}],"publication":"Indagationes Mathematicae","abstract":[{"lang":"eng","text":"We establish a multiresolution analysis on the space $\\text{Herm}(n)$ of\r\n$n\\times n$ complex Hermitian matrices which is adapted to invariance under\r\nconjugation by the unitary group $U(n).$ The orbits under this action are\r\nparametrized by the possible ordered spectra of Hermitian matrices, which\r\nconstitute a closed Weyl chamber of type $A_{n-1}$ in $\\mathbb R^n.$ The space\r\n$L^2(\\text{Herm}(n))^{U(n)}$ of radial, i.e. $U(n)$-invariant $L^2$-functions\r\non $\\text{Herm}(n)$ is naturally identified with a certain weighted $L^2$-space\r\non this chamber.\r\n  The scale spaces of our multiresolution analysis are obtained by usual dyadic\r\ndilations as well as generalized translations of a scaling function, where the\r\ngeneralized translation is a hypergroup translation which respects the radial\r\ngeometry. We provide a concise criterion to characterize orthonormal wavelet\r\nbases and show that such bases always exist. They provide natural orthonormal\r\nbases of the space $L^2(\\text{Herm}(n))^{U(n)}.$\r\n  Furthermore, we show how to obtain radial scaling functions from classical\r\nscaling functions on $\\mathbb R^{n}$. Finally, generalizations related to the\r\nCartan decompositions for general compact Lie groups are indicated."}],"file":[{"content_type":"application/pdf","success":1,"relation":"main_file","date_updated":"2026-02-19T14:14:39Z","date_created":"2026-02-19T14:14:39Z","creator":"llangen","file_size":443262,"file_id":"64288","file_name":"MSA_hermitsch_published.pdf","access_level":"closed"}],"external_id":{"arxiv":["2410.10364"]},"ddc":["510"],"language":[{"iso":"eng"}],"issue":"6","year":"2025","publisher":"Elsevier","date_created":"2024-10-22T09:31:19Z","title":"Multiresolution analysis on spectra of hermitian matrices"}]