Pub Date : 2023-11-09DOI: 10.1007/s11118-023-10114-4
Liping Li, Michael Röckner
{"title":"On the Restriction of a Right Process Outside a Negligible Set","authors":"Liping Li, Michael Röckner","doi":"10.1007/s11118-023-10114-4","DOIUrl":"https://doi.org/10.1007/s11118-023-10114-4","url":null,"abstract":"","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":" 15","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135243421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-28DOI: 10.1007/s11118-023-10110-8
Wolfram Bauer, Abdellah Laaroussi, Daisuke Tarama
Abstract We present an explicit form of the subelliptic heat kernel of the intrinsic sublaplacian $$Delta _{textrm{sub}}^5$$ Δsub5 induced by a rank 5 trivializable subriemannian structure on the Euclidean seven dimensional sphere $$mathbb {S}^7$$ S7 . This completes the heat kernel analysis of trivializable subriemannian structures on $$mathbb {S}^7$$ S7 induced by a Clifford module action on $$mathbb {R}^8$$ R8 . As an application we derive the spectrum of $$Delta _{textrm{sub}}^5$$ Δsub5 and the Green function of the conformal sublaplacian in an explicit form.
{"title":"Rank 5 Trivializable Subriemannian Structure on $$mathbb {S}^7$$ and Subelliptic Heat Kernel","authors":"Wolfram Bauer, Abdellah Laaroussi, Daisuke Tarama","doi":"10.1007/s11118-023-10110-8","DOIUrl":"https://doi.org/10.1007/s11118-023-10110-8","url":null,"abstract":"Abstract We present an explicit form of the subelliptic heat kernel of the intrinsic sublaplacian $$Delta _{textrm{sub}}^5$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mtext>sub</mml:mtext> </mml:mrow> <mml:mn>5</mml:mn> </mml:msubsup> </mml:math> induced by a rank 5 trivializable subriemannian structure on the Euclidean seven dimensional sphere $$mathbb {S}^7$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>7</mml:mn> </mml:msup> </mml:math> . This completes the heat kernel analysis of trivializable subriemannian structures on $$mathbb {S}^7$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>7</mml:mn> </mml:msup> </mml:math> induced by a Clifford module action on $$mathbb {R}^8$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>8</mml:mn> </mml:msup> </mml:math> . As an application we derive the spectrum of $$Delta _{textrm{sub}}^5$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mtext>sub</mml:mtext> </mml:mrow> <mml:mn>5</mml:mn> </mml:msubsup> </mml:math> and the Green function of the conformal sublaplacian in an explicit form.","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"1 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136233562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-19DOI: 10.1007/s11118-023-10103-7
Armen Grigoryan, Andrzej Michalski, Dariusz Partyka
Abstract Let I be a line segment in the complex plane $$mathbb C$$ C . We describe a method of constructing a bi-Lipschitz sense-preserving mapping of $$mathbb C$$ C onto itself, which is harmonic in $$mathbb Csetminus I$$ CI and coincides with a given sufficiently regular function $$f:Irightarrow mathbb C$$ f:I→C . As a result we show that a quasiconformal self-mapping of $$mathbb C$$ C which is harmonic in $$mathbb Csetminus I$$ CI does not have to be harmonic in $$mathbb C$$ C .
{"title":"Extensions of Harmonic Functions of the Complex Plane Slit Along a Line Segment","authors":"Armen Grigoryan, Andrzej Michalski, Dariusz Partyka","doi":"10.1007/s11118-023-10103-7","DOIUrl":"https://doi.org/10.1007/s11118-023-10103-7","url":null,"abstract":"Abstract Let I be a line segment in the complex plane $$mathbb C$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>C</mml:mi> </mml:math> . We describe a method of constructing a bi-Lipschitz sense-preserving mapping of $$mathbb C$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>C</mml:mi> </mml:math> onto itself, which is harmonic in $$mathbb Csetminus I$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo></mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:math> and coincides with a given sufficiently regular function $$f:Irightarrow mathbb C$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>I</mml:mi> <mml:mo>→</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> . As a result we show that a quasiconformal self-mapping of $$mathbb C$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>C</mml:mi> </mml:math> which is harmonic in $$mathbb Csetminus I$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo></mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:math> does not have to be harmonic in $$mathbb C$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>C</mml:mi> </mml:math> .","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135730415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-19DOI: 10.1007/s11118-023-10108-2
Carlos Beltrán, Víctor de la Torre, Fátima Lizarte
Abstract In this paper, we get the sharpest known to date lower bounds for the minimal Green energy of the compact harmonic manifolds of any dimension. Our proof generalizes previous ad-hoc arguments for the most basic harmonic manifold, i.e. the sphere, extending it to the general case and remarkably simplifying both the conceptual approach and the computations.
{"title":"Lower Bound for the Green Energy of Point Configurations in Harmonic Manifolds","authors":"Carlos Beltrán, Víctor de la Torre, Fátima Lizarte","doi":"10.1007/s11118-023-10108-2","DOIUrl":"https://doi.org/10.1007/s11118-023-10108-2","url":null,"abstract":"Abstract In this paper, we get the sharpest known to date lower bounds for the minimal Green energy of the compact harmonic manifolds of any dimension. Our proof generalizes previous ad-hoc arguments for the most basic harmonic manifold, i.e. the sphere, extending it to the general case and remarkably simplifying both the conceptual approach and the computations.","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"187 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135730413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-16DOI: 10.1007/s11118-023-10102-8
David Darrow
Abstract Internal DLA (IDLA) is an internal aggregation model in which particles perform random walks from the origin, in turn, and stop upon reaching an unoccupied site. Levine and Peres showed that, when particles start instead from fixed multiple-point distributions, the modified IDLA processes have deterministic scaling limits related to a certain obstacle problem. In this paper, we investigate the convergence rate of this “extended source” IDLA in the plane to its scaling limit. We show that, if $$delta $$ δ is the lattice size, fluctuations of the IDLA occupied set are at most of order $$delta ^{3/5}$$ δ3/5 from its scaling limit, with probability at least $$1-e^{-1/delta ^{2/5}}$$ 1-e-1/δ2/5 .
{"title":"A Convergence Rate for Extended-Source Internal DLA in the Plane","authors":"David Darrow","doi":"10.1007/s11118-023-10102-8","DOIUrl":"https://doi.org/10.1007/s11118-023-10102-8","url":null,"abstract":"Abstract Internal DLA (IDLA) is an internal aggregation model in which particles perform random walks from the origin, in turn, and stop upon reaching an unoccupied site. Levine and Peres showed that, when particles start instead from fixed multiple-point distributions, the modified IDLA processes have deterministic scaling limits related to a certain obstacle problem. In this paper, we investigate the convergence rate of this “extended source” IDLA in the plane to its scaling limit. We show that, if $$delta $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>δ</mml:mi> </mml:math> is the lattice size, fluctuations of the IDLA occupied set are at most of order $$delta ^{3/5}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>δ</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> </mml:math> from its scaling limit, with probability at least $$1-e^{-1/delta ^{2/5}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>δ</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> .","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136077706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1007/s11118-023-10107-3
Hausenblas, Erika, Tölle, Jonas M.
On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.
{"title":"The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem","authors":"Hausenblas, Erika, Tölle, Jonas M.","doi":"10.1007/s11118-023-10107-3","DOIUrl":"https://doi.org/10.1007/s11118-023-10107-3","url":null,"abstract":"On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"118 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135805031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1007/s11118-023-10106-4
Filippo De Mari, Matteo Monti, Maria Vallarino
Abstract In this paper we investigate some properties of the harmonic Bergman spaces $$mathcal A^p(sigma )$$ Ap(σ) on a q -homogeneous tree, where $$qge 2$$ q≥2 , $$1le p1≤p<∞ , and $$sigma $$ σ is a finite measure on the tree with radial decreasing density, hence nondoubling. These spaces were introduced by J. Cohen, F. Colonna, M. Picardello and D. Singman. When $$p=2$$ p=2 they are reproducing kernel Hilbert spaces and we compute explicitely their reproducing kernel. We then study the boundedness properties of the Bergman projector on $$L^p(sigma )$$ Lp(σ) for $$11<p<∞ and their weak type (1,1) boundedness for radially exponentially decreasing measures on the tree. The weak type (1,1) boundedness is a consequence of the fact that the Bergman kernel satisfies an appropriate integral Hörmander’s condition.
摘要本文研究了q -齐次树上的调和Bergman空间$$mathcal A^p(sigma )$$ A p (σ)的一些性质，其中$$qge 2$$ q≥2,$$1le p<infty $$ 1≤p ＆lt;∞，且$$sigma $$ σ是密度呈径向递减的树的有限测度，因此不加倍。这些空间由J. Cohen、F. Colonna、M. Picardello和D. Singman引入。当$$p=2$$ p = 2时，它们正在再现核希尔伯特空间，我们显式地计算它们的再现核。然后研究了$$1<p<infty $$ 1 ＆lt下$$L^p(sigma )$$ L p (σ)上Bergman投影的有界性;P ＆lt;∞和它们的弱型(1,1)有界性。弱型(1,1)有界性是Bergman核满足适当的积分Hörmander条件的结果。
{"title":"Harmonic Bergman Projectors on Homogeneous Trees","authors":"Filippo De Mari, Matteo Monti, Maria Vallarino","doi":"10.1007/s11118-023-10106-4","DOIUrl":"https://doi.org/10.1007/s11118-023-10106-4","url":null,"abstract":"Abstract In this paper we investigate some properties of the harmonic Bergman spaces $$mathcal A^p(sigma )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> on a q -homogeneous tree, where $$qge 2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , $$1le p<infty $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , and $$sigma $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>σ</mml:mi> </mml:math> is a finite measure on the tree with radial decreasing density, hence nondoubling. These spaces were introduced by J. Cohen, F. Colonna, M. Picardello and D. Singman. When $$p=2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> they are reproducing kernel Hilbert spaces and we compute explicitely their reproducing kernel. We then study the boundedness properties of the Bergman projector on $$L^p(sigma )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for $$1<p<infty $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>p</mml:mi> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> and their weak type (1,1) boundedness for radially exponentially decreasing measures on the tree. The weak type (1,1) boundedness is a consequence of the fact that the Bergman kernel satisfies an appropriate integral Hörmander’s condition.","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"255 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135858927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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