We develop cohomological interpretations for several types of automorphic forms for Hecke triangle groups of infinite covolume. We then use these interpretations to establish explicit isomorphisms between spaces of automorphic forms, cohomology spaces and spaces of eigenfunctions of transfer operators. These results show a deep relation between spectral entities of Hecke surfaces of infinite volume and the dynamics of their geodesic flows.
{"title":"Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume","authors":"R. Bruggeman, A. Pohl","doi":"10.1090/memo/1423","DOIUrl":"https://doi.org/10.1090/memo/1423","url":null,"abstract":"We develop cohomological interpretations for several types of automorphic forms for Hecke triangle groups of infinite covolume. We then use these interpretations to establish explicit isomorphisms between spaces of automorphic forms, cohomology spaces and spaces of eigenfunctions of transfer operators. These results show a deep relation between spectral entities of Hecke surfaces of infinite volume and the dynamics of their geodesic flows.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42665184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the self-dual Chern-Simons-Schrödinger equation (CSS), also known as a gauged nonlinear Schrödinger equation (NLS). CSS is L2L^{2}-critical, admits solitons, and has the pseudoconformal symmetry. These features are similar to the L2L^{2}-critical NLS. In this work, we consider pseudoconformal blow-up solutions under mm-equivariance, m≥1mgeq 1. Our result is threefold. Firstly, we construct a pseudoconformal blow-up solution uu with given asymptotic profile z∗z^{ast }: [
我们考虑自对偶Chern-Simons-Schrödinger方程（CSS），也称为规范非线性薛定谔方程（NLS）。CSS是L2L^｛2｝-临界的，包含孤立子，并且具有伪共形对称性。这些特征类似于L2L^｛2｝-临界NLS。在这项工作中，我们考虑m-等变，m≥1mgeq1下的伪共形爆破解。我们的结果有三个方面。首先，我们构造了一个具有给定渐近轮廓z＊z^｛ast｝的伪共形爆破解u：[[u（t，r）−1|t|Q（r|t|）e−i r 2 4|t|]e i mθ→ H1Big[u（t，r）-frac｛1｝｛|t|｝QBig→ 0−t到0^｛-｝，其中Q（r）e i mθQ（r。其次，我们证明了这种爆破解决方案在合适的类别中是独特的。最后，但最重要的是，我们展示了u u的不稳定性机制。我们构造了一个连续的解族u（η）u^{（eta）}，0≤η≪1 0leqetalll 1，使得u（0）=
{"title":"On Pseudoconformal Blow-Up Solutions to the Self-Dual Chern-Simons-Schrödinger Equation: Existence, Uniqueness, and Instability","authors":"Kihyun Kim, Soonsik Kwon","doi":"10.1090/memo/1409","DOIUrl":"https://doi.org/10.1090/memo/1409","url":null,"abstract":"<p>We consider the self-dual Chern-Simons-Schrödinger equation (CSS), also known as a gauged nonlinear Schrödinger equation (NLS). CSS is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L squared\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">L^{2}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-critical, admits solitons, and has the pseudoconformal symmetry. These features are similar to the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L squared\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">L^{2}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-critical NLS. In this work, we consider pseudoconformal blow-up solutions under <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\">\u0000 <mml:semantics>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">m</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-equivariance, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m greater-than-or-equal-to 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mgeq 1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Our result is threefold. Firstly, we construct a pseudoconformal blow-up solution <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u\">\u0000 <mml:semantics>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">u</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with given asymptotic profile <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"z Superscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>z</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">z^{ast }</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>: <disp-formula content-type=\"math/mathml\">\u0000[\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket u left-parenthesis t comma r right-parenthesis minus StartFraction 1 Over StartAbsoluteValue t EndAbsoluteValue EndFraction upper Q left-parenthesis StartFraction r Over StartAbsoluteValue t EndAbsoluteValue EndFraction right-parenthesis e Supe","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47513575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work is devoted to the study of rates of convergence of the empirical measures μn = 1 n ∑n k=1 δXk , n ≥ 1, over a sample (Xk)k≥1 of independent identically distributed real-valued random variables towards the common distribution μ in Kantorovich transport distances Wp. The focus is on finite range bounds on the expected Kantorovich distances E(Wp(μn, μ)) or [ E(W p p (μn, μ)) ]1/p in terms of moments and analytic conditions on the measure μ and its distribution function. The study describes a variety of rates, from the standard one 1 √ n to slower rates, and both lower and upperbounds on E(Wp(μn, μ)) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, logconcavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.
{"title":"One-dimensional empirical measures, order\u0000 statistics, and Kantorovich transport\u0000 distances","authors":"S. Bobkov, M. Ledoux","doi":"10.1090/memo/1259","DOIUrl":"https://doi.org/10.1090/memo/1259","url":null,"abstract":"This work is devoted to the study of rates of convergence of the empirical measures μn = 1 n ∑n k=1 δXk , n ≥ 1, over a sample (Xk)k≥1 of independent identically distributed real-valued random variables towards the common distribution μ in Kantorovich transport distances Wp. The focus is on finite range bounds on the expected Kantorovich distances E(Wp(μn, μ)) or [ E(W p p (μn, μ)) ]1/p in terms of moments and analytic conditions on the measure μ and its distribution function. The study describes a variety of rates, from the standard one 1 √ n to slower rates, and both lower and upperbounds on E(Wp(μn, μ)) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, logconcavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73427125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quiver Grassmannians of Extended Dynkin type\u0000 𝐷 Part 1: Schubert Systems and Decompositions Into\u0000 Affine Spaces","authors":"Oliver Lorscheid, Thorsten Weist","doi":"10.1090/memo/1258","DOIUrl":"https://doi.org/10.1090/memo/1258","url":null,"abstract":"","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72877915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Degrijse, M. Hausmann, W. Luck, Irakli Patchkoria, S. Schwede
This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective ‘proper’ alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective ‘genuine’ indicates that the theory comes with appropriate transfers and Wirthmüller isomorphisms, and the resulting equivariant cohomology theories support the analog of an R O ( G ) R O(G) -grading. Our model for genuine proper G G -equivariant stable homotopy theory is the category of orthogonal G G -spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of G G . This class of π ∗ pi _* -isomorphisms is part of a symmetric monoidal stable model structure, and the associated tensor triangulated homotopy category is compactly generated. Consequently, every orthogonal G G -spectrum represents an equivariant cohomology theory on the category of G G -spaces. These represented cohomology theories are designed to only depend on the ‘proper G G -homotopy type’, tested by fixed points under all compact subgroups. An important special case of our theory are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties in the sense of geometric group theory; for example, the G G -sphere spectrum is a compact object in our triangulated equivariant homotopy category if the universal space for proper G G -actions has a finite G G -CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper G G -CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by G G -vector bundles. Via this description, we can identify the previously defined G G -cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal G G -spectra.
{"title":"Proper Equivariant Stable Homotopy Theory","authors":"D. Degrijse, M. Hausmann, W. Luck, Irakli Patchkoria, S. Schwede","doi":"10.1090/memo/1432","DOIUrl":"https://doi.org/10.1090/memo/1432","url":null,"abstract":"This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective ‘proper’ alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective ‘genuine’ indicates that the theory comes with appropriate transfers and Wirthmüller isomorphisms, and the resulting equivariant cohomology theories support the analog of an \u0000\u0000 \u0000 \u0000 R\u0000 O\u0000 (\u0000 G\u0000 )\u0000 \u0000 R O(G)\u0000 \u0000\u0000-grading.\u0000\u0000Our model for genuine proper \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000-equivariant stable homotopy theory is the category of orthogonal \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000-spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000. This class of \u0000\u0000 \u0000 \u0000 π\u0000 ∗\u0000 \u0000 pi _*\u0000 \u0000\u0000-isomorphisms is part of a symmetric monoidal stable model structure, and the associated tensor triangulated homotopy category is compactly generated. Consequently, every orthogonal \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000-spectrum represents an equivariant cohomology theory on the category of \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000-spaces. These represented cohomology theories are designed to only depend on the ‘proper \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000-homotopy type’, tested by fixed points under all compact subgroups.\u0000\u0000An important special case of our theory are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties in the sense of geometric group theory; for example, the \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000-sphere spectrum is a compact object in our triangulated equivariant homotopy category if the universal space for proper \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000-actions has a finite \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000-CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000-CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000-vector bundles. Via this description, we can identify the previously defined \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000-cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000-spectra.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47408720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1925 Elie Cartan introduced the principal of triality specifically for the Lie groups of type D4, and in 1935 Ruth Moufang initiated the study of Moufang loops. The observation of the title was made by Stephen Doro in 1978 who was in turn motivated by work of George Glauberman from 1968. Here we make the statement precise in a categorical context. In fact the most obvious categories of Moufang loops and groups with triality are not equivalent, hence the need for the word “essentially.” Received by the editor 20 June 2016. 2010 Mathematics Subject Classification. Primary 20.
{"title":"Moufang Loops and Groups with Triality are\u0000 Essentially the Same Thing","authors":"J. Hall","doi":"10.1090/MEMO/1252","DOIUrl":"https://doi.org/10.1090/MEMO/1252","url":null,"abstract":"In 1925 Elie Cartan introduced the principal of triality specifically for the Lie groups of type D4, and in 1935 Ruth Moufang initiated the study of Moufang loops. The observation of the title was made by Stephen Doro in 1978 who was in turn motivated by work of George Glauberman from 1968. Here we make the statement precise in a categorical context. In fact the most obvious categories of Moufang loops and groups with triality are not equivalent, hence the need for the word “essentially.” Received by the editor 20 June 2016. 2010 Mathematics Subject Classification. Primary 20.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76861766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Algebraic Geometry over 𝐶^{∞}-rings","authors":"D. joyce","doi":"10.1090/MEMO/1256","DOIUrl":"https://doi.org/10.1090/MEMO/1256","url":null,"abstract":"","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83640972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the wave maps problem with domain R2+1mathbb {R}^{2+1} and target S2mathbb {S}^{2} in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from R2mathbb {R}^{2} to S2mathbb {S}^{2}, with polar angle equal to Q1(r)=2arctan
We study the moduli space of trace-free irreducible rank 2 connections over a curve of genus 2 and the forgetful map towards the moduli space of under- lying vector bundles (including unstable bundles), for which we compute a natural Lagrangian rational section. As a particularity of the genus 2 case, connections as above are invariant under the hyperelliptic involution : they descend as rank 2 logarithmic connections over the Riemann sphere. We establish explicit links between the well-known moduli space of the underlying parabolic bundles with the classical approaches by Narasimhan-Ramanan, Tyurin and Bertram. This allow us to explain a certain number of geometric phenomena in the considered moduli spaces such as the classical (16, 6)-configuration of the Kummer surface. We also recover a Poincare family due to Bolognesi on a degree 2 cover of the Narasimhan-Ramanan moduli space. We explicitly compute the Hitchin integrable system on the moduli space of Higgs bundles and compare the Hitchin Hamiltonians with those found by vanGeemen-Previato. We explicitly describe the isomonodromic foliation in the moduli space of vector bundles with sl(2,C)-connection over curves of genus 2 and prove the transversality of the induced flow with the locus of unstable bundles.
{"title":"Flat Rank Two Vector Bundles on Genus Two\u0000 Curves","authors":"Viktoria Heu, F. Loray","doi":"10.1090/MEMO/1247","DOIUrl":"https://doi.org/10.1090/MEMO/1247","url":null,"abstract":"We study the moduli space of trace-free irreducible rank 2 connections over a curve of genus 2 and the forgetful map towards the moduli space of under- lying vector bundles (including unstable bundles), for which we compute a natural Lagrangian rational section. As a particularity of the genus 2 case, connections as above are invariant under the hyperelliptic involution : they descend as rank 2 logarithmic connections over the Riemann sphere. We establish explicit links between the well-known moduli space of the underlying parabolic bundles with the classical approaches by Narasimhan-Ramanan, Tyurin and Bertram. This allow us to explain a certain number of geometric phenomena in the considered moduli spaces such as the classical (16, 6)-configuration of the Kummer surface. We also recover a Poincare family due to Bolognesi on a degree 2 cover of the Narasimhan-Ramanan moduli space. We explicitly compute the Hitchin integrable system on the moduli space of Higgs bundles and compare the Hitchin Hamiltonians with those found by vanGeemen-Previato. We explicitly describe the isomonodromic foliation in the moduli space of vector bundles with sl(2,C)-connection over curves of genus 2 and prove the transversality of the induced flow with the locus of unstable bundles.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90068584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including n-dimensional cube [0, 1] are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on [0, 1], density of the medium is homogeneous and represented by the Lebesgue measure. Our study includes densities which are singular to the homogeneous one. We establish a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on [0, 1] and self-similar measures. We are going to show the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, we obtain diagonal lower and upper estimates of the heat kernel as time tends to 0. In particular, to express the principal part of the lower diagonal heat kernel estimate, we introduce “protodistance” associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal subGaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown. 2010 Mathematics Subject Classification. Primary , 31E05, 60J35, 60J60; Secondary 28A80, 30L10, 43A99, 60J65, 80A20.
{"title":"Time Changes of the Brownian Motion: Poincaré\u0000 Inequality, Heat Kernel Estimate and\u0000 Protodistance","authors":"Jun Kigami","doi":"10.1090/MEMO/1250","DOIUrl":"https://doi.org/10.1090/MEMO/1250","url":null,"abstract":"In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including n-dimensional cube [0, 1] are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on [0, 1], density of the medium is homogeneous and represented by the Lebesgue measure. Our study includes densities which are singular to the homogeneous one. We establish a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on [0, 1] and self-similar measures. We are going to show the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, we obtain diagonal lower and upper estimates of the heat kernel as time tends to 0. In particular, to express the principal part of the lower diagonal heat kernel estimate, we introduce “protodistance” associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal subGaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown. 2010 Mathematics Subject Classification. Primary , 31E05, 60J35, 60J60; Secondary 28A80, 30L10, 43A99, 60J65, 80A20.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75760331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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