Dirichlet Forms: Lectures given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Varenna, Italy, June 8–19, 1992 PDF
Preview Dirichlet Forms: Lectures given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Varenna, Italy, June 8–19, 1992
Lecture Notes in Mathematics 3651 Editors: A. Dold, Heidelberg B. Eckmann, Zarich E Takens, Groningen Subseries: Fondazione C.I.M.E., Firenze Adviser: Roberto Conti E. Fabes M. Fukushima L. Gross C. Kenig M. R6ckner D.W. Stroock Dirichlet Forms Lectures given at the 1st Session of the Centro Intemazionale Matematico Estivo (C.I.M.E.) held in Varenna, Italy, June 8-19, 1992 Editors: G. Dell'Antonio, U. Mosco galreV-regnirpS Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona tsepaduB Authors Eugene Fabes Department of Mathematics, University of Minnesota Minneapolis, MN 55455, USA Masatoshi Fukushima Department of Mathematical Science, Faculty of Engineering Science Osaka University, Toyonaka, Osaka, Japan Leonard Gross Department of Mathematics, Cornell University Ithaca, NY 14853, USA Carlos Kenig Department of Mathematics, University of Chicago Chicago, IL 60637, USA Michael R6ckner Institut ftir Angewandte Mathematik, Universitat Bonn Wegelerstrasse 6, D-53115 Bonn, Germany Daniel .W Stroock M.I.T., Rm 2-272 Cambridge, MA 02139, USA Editors Gianfausto Dell'Antonio Umberto Mosco Dipartimento di Matematica, Universit~t "La Sapienza" Piazzale Aldo Moro, 5, 1-00185 Roma, Italy Mathematics Subject Classification (1991): Primary: 46-xx Secondary: 31-xx, 35-xx, 60-xx ISBN 3-540-57421-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57421-2 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. 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(cid:14)9 Springer-Verlag Berlin Heidelberg 1993 Printed in Germany Typesetting: Camera-ready by author/editor 46/3140-543210 - Printed on acid-free paper Preface In recent years the theory of Dirichlet forms has witnessed some very important developments both in its theoretical foundations and in its applica- tions, which have ranged from the theory of Stochastic PrQcesses to Quantum Field Theory to the theory of highly inhomogeneous materials(cid:12)9 It was therefore felt that it was due time to have on this subject a CIME school, in which leading experts in the field would present both the basic theoretical aspects and some of the recent applications, pointing to areas of active research(cid:12)9 The school was organized in six courses, covering - Foundations of the theory and connection with Potential Theory and Stochastic Processes (M. Fukushima, M. Roeckner) - Regularity results and a-priori estimates for solutions of elliptic equations in general domains (E. Fabes, C. Kenig) - Hypercontractivity of semigroups and relation with spectral properties (L. Gross) Logarithmic Sobolev inequalities and connections with Statistical Mechanics (D. - Stroock) In the afternoons of the last three days a workshop was held, with financial support from the CNR project "Irregular Variational Problems", on further applica- tions of the theory of Dirichlet Forms, also in the noncommutative setting. The seminars were given by (cid:12)9 L. Accardi (Roma II) Noncommutative Stochastic Processes and applications (cid:12)9 S. Albeverio (Bochum) Construction of infinite-dimensional processes (cid:12)9 M(cid:12)9 Biroli (Pol. Milano) Asymptotic Dirichlet Forms (cid:12)9 F. Guerra (Roma I) Annealing and Cavitation in Spin Glass Models (cid:12)9 M. Lindsay (Nottingham) Noncommutative Semigroups (cid:12)9 S.R.S. Varadhan (Courant Institute) Hydrodynamic Limit and Entropy Estimates I, II (cid:12)9 Ms Zhiming (Academia Sinioa) Nonsymmetric Diriehlet Forms The lectures of the School were followed with constant and active par- ticipation by many young researchers, both from Italy and from abroad(cid:12)9 We believe that the School was successful in reaching its aims, and we wish to express here our appreciation to the speakers for the high quality of their lectures and for their availability for discussions during the School(cid:12)9 We would also like to thank prof. R. Conti and the CIME Scientific Committee for the invitation to organize the School, and prof. P. Zecca and the staff of the Centro Volta in Como for their very effective help. Gianfausto Dell'Antonio Umberto Mosco TABLE OF CONTENTS E.B. FABES, Gaussian Upper Bounds on Fundamental Solutions of Parabolic Equations: the Method of Nash .......................... 1 M. FUKUSHIMA, Two Topics Related to Dirichlet Forms: Quasi Everywhere Convergences and Additive Functionals ................................ 21 L. GROSS, Logarithmic Sobolev Inequalities and Contractive Properties of Semigroups ....... 54 C.E. KENIG, Potential Theory of Non-Divergence Form Elliptic Equations ......................... 83 M. R0CKNER, General Theory of Dirichlet Forms and Applications .......................... 129 D.W. STROOCK, Logarithmic Sobolev Inequalities for Gibbs States .......................... 194 Gaussian Upper Bounds on Fundamental Solutions of Parabolic Equations; the Method of Nash by E.B. Fabes Table of Contents Introduction Part I. The Nondegenerate Case. Section .1 The First Moment Bound Section 2. Iteration of Moment Bounds Section 3. The Gaussian Upper Bound Part II. The Nash Method Applied to Heat Kernels on Riemannian Manifolds. Section 4. Moment Bounds Section 5. The Gaussian Upper Bound Introduction In the past decade tremendous progress has been made in understanding the pointwise behavior of fundamental solutions of parabolic operators in divergence form. One can see this progress in reading the monograph of E.B. Davies 3. The interest and progress continue to this day. (See, for example, the related recent work of Saloff-Coste 9,10.) Historically the first paper establishing Gaussian upper and lower bounds for the fundamental solution F(x, t; y, s) (x and y e R n, t > s > 0) was D.G. Aronson's paper 1. Here, in the nondegenerate case, Aronson proved there exists a constant C > 0 such that C-a(t - s)-'*/2 e -~ _ F(x, t; y, .s) < C(t - s)-'*/2 e -~ I~-ul2/(t-s) for all x, y, t, s with t > s. Ideas in this paper continue to influence recent work on estimating F in the case of degenerate parabolic operators with nonsmooth coeffi- cients. (See 5). Almost 10 years before Aronson's work the celebrated paper of John Nash appeared (8). In this work Nash established the HSlder continuity of weak solutions of (nondegenerate) parabolic equations in divergence form. Nash's method in fact concentrated on the fundamental solution of the parabolic opera- tor, on establishing estimates for the fundamental solution that did not depend on the smoothness of the coefficients. From these apriori estimates came the H61der continuity of solutions. Later in 1964 and 1967 Moser proved a Harnack inequality for nonnegative solutions and from the Harnack inequality obtained Nash's result on the H61der continuity of weak solutions (6,7). It was with the help of the Harnack inequality of Moser that Aronson established the Gaussian lower bound for the fundamental solution (1). In 1985 D.W. Stroock and I returned to Nash's paper and found that his original ideas could be used to obtain a Gaussima lower bound on the fundamental solution (4). Our proof relied on some apriori spatial decay at infinity of the fundamental solution. A Gaussian upper bound certainly sufficed and such decay had been established by D.G. Aronson in 1 and by E.B. Davies in 3, p. 89 without the need of Harnack's inequality. Hence a proof of upper and lower bounds for the fundamental solution in terms of Gaussians could be achieved without Harnack's inequality and, indeed, such estimates on the fundamental solution could be used to prove Moser's Harnack inequality. (See 4.) The methods to obtain the upper bound estimates for the fundamental solution developed by Aronson and Davies are powerful and can be used to obtain upper bounds of fundamental solutions associated with certain degenerate parabolic op- erators. (See 3,5.) The direct relationship of these methods with those of Nash is not clear. Certainly the initial estimate of Nash on the fundamental solution F, namely, r(x, t; ~, s) _< c(t - s) -"/2 plays an important role in both methods; but the direct connection stops here and the methods diverge into interesting new directions. Of particular note is that neither method pays attention to what Nash calls "The moment bound" and refers to it as "essential to all subsequent parts of this paper." In Part I, will review Nash's proof of the moment bound for the fundamental solution and then show that his ideas can be extended to estimate all moments of the fundamental solution. Moreover the estimates for the moments will be such that they will imply the Gaussian decay. In Part II, the ideas of Nash will be applied to obtain a Ganssian upper bound on the fundamental solution of the heat equation for a class of complete Riemannian manifolds which includes those with a lower bound hypothesis on the Ricci curvature. In this regard the reader should also see the recent work of L. Saloff-Coste (9,101). Before beginning the main body of this paper, I would like to express my thanks to Noah Herschel Moss Fabes for his encouragement and active help in preparing this manuscript. Part I. The Nondegenerate Case The notation we will use for Sections 1,2, and 3 of Part I is rather standard. The letter z, y,z, and 4 will denote points in the Euclidean space R n, n > 3. The letters r,s, and t will denote real numbers. We consider parabolic operators of the form Lu(x,t) = ~ D~,(a~j(x,t)Dxju(x,t)) - Dtu(x,t) i,j-~l where the matrix a(x, t) - (aii(x, t)) is symmetric, measurable, and uniformly non- degenerate, i.e. There exists A > 0 such that for all (x,t) E R"+I,~ E R", 1~1~ ~ < ~ J~,~)~,x(J,~ < 1~I-~ .~ i,j=l We let F = F(x, t; y, s) denote the fundamental solution corresponding to the operator L. At times it will be necessary to show the correspondence of the funda- mental solution with the operator L through the matrix a(z, t). At such times we will use the notation Fa(x, t; y, s). The letter C will stand for a constant which depends only on the e11ipticity parameter A and the spatial dimension F. The constant need not be the same at each occurrence. 4 We will often write ~-~ D,, (aij (x, t)D,j u(x, t)) = div a(Vu) i,j=l where we understand that div and 7~ denote only the spatial divergence and spatial gradient. Section 1. The Fist Moment Bound. In this section we will review Nash's proof of the following moment bound: Theorem 1. There exists a constant C depending only on A and n, such that for s < t and all x, "R xI -- yF(z,t;y,s)dy < C(t - s) 1/2. We begin the proof with the basic pointwise bound which serves as a good estimate along the diagonal. Lemma 1.1. F(x, t;y,s) < C(t - s) -"/~, s < t. Proof For t > 0 set p(z,t) = r(z,t;0,0) and u(t) = fp2(x,t)dz. Then u'(t) = 2 fpp'dx = 2 fp div (aXYp)dx and so u'(t) < --2A /I(cid:127)pI2dx. Applying the interpolation inequality ( / p2)l+2/n < Cn( / Vp2)( / p)4/il and remembering that f pdx = 1, we have u'(t) _< - ~(t) ~+21n, i.~. (~-2.) _> -c . We integrate this last inequality from 7 t to t and obtain (1.2) u(t) - i r(x, t; 0, O)2dx __< Ct--~/2. Now for s < t r~(x, t; y,s) = ro(x + w,t - u;x + z,t - v)l ...... 0 and F~(z + w, t - u; x + z, t - v) = Fa(z+.,t-.)(z, v; w, u). As a consequence of (1.2) we have (1.3) f ra(x, t; y, s)2dy _< C(t- s) -"/2. J Similarly ra(z,t;y,s) ra(z,t - v)l %_7=o = u;y,t - = F~(.,t-.)(y, t - s; x, 0) and from (1.3) it also follows .2/n--)q~ (1.4) / Fa(x,t;y,s)2dx < C(t - Finally the reproducing property of F. implies (for s < t) t+s t+s ro(x, t; y, s) = fro(x,t;z, T)Fo(z,-~;y,s)dt. Schwartz's inequality together with (1.3) and (1.4) give the conclusion of Lemma (1.1). Lemma (1.5): (The first moment bound) There exists a constant C depending only on t and n such that /I x ylr(x, t; ,Y ~)dx _< C(t s),/2. - - Proof. As in the proof of Lemma (1.1) we may take y = 0 and s = 0. Now set g. Ml(t) = Ixb(x,t)dx where again p(x, t) = F(x, t; 0, 0). Mi(t)-- Ixldiv a(Vp)dz =- ~ < C(/IVP(x' t)12 dx) .e/~ v(x, t) - To estimate the interesting quantity on the right-hand side of the above inequality Nash introduces the function f ,x(p logp(x, t)dx. Q(t) : - t) This function is introduced since Q'(t) = / a(Vp)" VP dx, i.e. P Q'(t) is equivalent to f I vpl~- p ax. Hence 1 M~(t) < C(Q'(t)) 2/1 _< CtUZ(Q'(t) + 7).
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