Table Of ContentLecture 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
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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
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(cid:14)9 Springer-Verlag Berlin Heidelberg 1993
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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).
