Approximation of plurisubharmonic functions by logarithms of Gaussian analytic functions

[2402.11833] Approximation of plurisubharmonic functions by logarithms of Gaussian analytic functions Skip to main content Search Submit Donate Log in Mathematics > Complex Variables arXiv:2402.11833 (math) [Submitted on 19 Feb 2024 ( v1 ), last revised 25 Aug 2026 (this version, v5)] Title: Approximation of plurisubharmonic functions by logarithms of Gaussian analytic functions Authors: Kiyoon Eum View PDF HTML (experimental) Abstract: Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^N$, and let $u$ be a continuous plurisubharmonic function on $\Omega$. We construct a sequence of Gaussian analytic functions $f_n$ on $\Omega$, associated with $u$, such that $\frac{1}{n}\log|f_n|$ converges to $u$ in $L^1_{loc}(\Omega)$ almost surely, as $n\rightarrow\infty$. Consequently, the normalized zero currents of $f_n$ converge weakly to $dd^c u$. More generally, for every $1\leq k\leq N$, we prove that the normalized currents of simultaneous zeros of $k$ independent copies of $f_n$ converge almost surely to the Bedford--Taylor product $(dd^c u)^k$. We also give a probabilistic proof of the well-known fact that normalized logarithms of the moduli of holomorphic functions are $L^1_{loc}$-dense in the space of plurisubharmonic functions on a pseudoconvex domain. Comments: 15 pages. An error in Section 5 of the previous version has been corrected; added new result on convergence of zeros Subjects: Complex Variables (math.CV) ; Probability (math.PR) MSC classes: 32U05, 60G15 Cite as: arXiv:2402.11833 [math.CV] (or arXiv:2402.11833v5 [math.CV] for this version) https://doi.org/10.48550/arXiv.2402.11833 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Kiyoon Eum [ view email ] [v1] Mon, 19 Feb 2024 04:55:51 UTC (12 KB) [v2] Thu, 9 May 2024 04:40:19 UTC (12 KB) [v3] Sun, 2 Jun 2024 05:25:44 UTC (12 KB) [v4] Thu, 20 Mar 2025 07:40:17 UTC (13 KB) [v5] Tue, 25 Aug 2026 03:44:08 UTC (15 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.CV < prev | next > new | recent | 2024-02 Change to browse by: math math.PR References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

Related documents

• Approximation of plurisubharmonic functions by logarithms of Gaussian analytic functions#670272
• A counterexample to Kenig's conjecture for the Laplace double-layer operator#985351
• Pogorelov interior estimates and a Liouville theorem for the complex Monge--Ampère equation#1012461
• Discrete Hyperbolic Secant Distributions#998831
• Differential Harnack Estimates and Frequency Monotonicity for Filtration Equations on Riemannian Manifolds#1036762