Some Topics in Probability and Analysis (Cbms Regional Conference Series in Mathematics) 0821807218, 9780821807217

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SOME TOPICS IN PROBABILITY AND ANALYSIS

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http://dx.doi.org/10.1090/cbms/070

Conference Board of the Mathematical Sciences REGIONAL CONFERENCE SERIES IN MA THEM A TICS supported by the National Science Foundation

Number 70

SOME TOPICS IN PROBABILITY AND ANALYSIS Richard F. Gundy

Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island

Lectures given a t DePau l University , Chicago, Illinois , July 1 4-1 8 , 1 98 6 Supported b y the Conferenc e Boar d of th e Mathematica l Science s Research supporte d b y National Scienc e Foundation Gran t DMS-8602950 . 1980 Mathematics Subject Classifications (1 98 5 Revision). 60G44 , 26D1 5 , 26B15, 44A15, 47D05, 60J05, 60J25.

Library of Congress Cataloging-in-Publication Dat a Some topics i n probability an d analysi s / Richar d F . Gundy , p. cm . —(Conference Boar d o f th e Mathematica l Science s Regional conferenc e serie s in mathematics; no. 70.) Bibliography: p . ISBN 0-821 8-0721 - 8 (alk . paper ) 1. Inequalities (Mathematics ) 2 . Harmonic functions . I . Series: Regiona l conference serie s in mathematics; no . 70. QA1.R33 no . 70 [Q A 295]15 0 s—dcl91 [5 2.9'7 ] 89-30 3 CI P

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Contents

Introduction The Barlow-Yor Inequalitie s 2 The Density o f th e Area Integra l 6 Norm Inequalitie s fo r D 9 Local Estimates

1

Terminal Expectation 1 s an d Singula r Integral s

8

The Ornstein-Uhlenbeck Semigrou p 3 1 P. A. Meyer's Riesz Transform Inequalitie s 3

7

The Ornstein-Uhlenbeck Proces s for th e Rademacher Function s 4

4

References 4

7

v

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Introduction In thes e lecture s I concentrate d o n thre e topics : (1 ) Loca l tim e theor y for Brownia n motio n an d som e geometrical inequalitie s fo r harmoni c func tions i n th e uppe r half-plan e R+ +1 . (2 ) A probabilisti c treatmen t o f Ries z transforms i n R+ +1 an d semimartingal e inequalities . (3 ) A discussion o f th e Ornstein-Uhlenbeck semigrou p an d P . A. Meyer's extension o f the Riesz inequalities fo r th e infinite-dimensiona l versio n o f thi s semigroup, introduce d by Malliavin (se e [26, 27]). Regarding topic (1), we sketch a proof of the inequalities obtained by Barlow and Yor in [1 ] for th e maximal loca l time functional. Thes e inequalitie s led th e autho r t o som e ne w inequalitie s fo r a geometri c functional , define d on harmonic function s i n R+ +1 , called the density of the area integral Topic (2) is a probabilistic approach to the Riesz transform inequalitie s in R". Thi s metho d o f proo f wa s first introduced b y the autho r an d Varopou los [24] . Th e metho d wa s elaborate d furthe r b y th e autho r wit h Silverstei n [23]. W e sho w tha t th e sam e idea s ar e effectiv e i n provin g semimartingal e inequalities o f the type usually obtained fo r martingales . The final topic in the series is a discussion of the Ornstein-Uhlenbeck semigroup. W e give a proof o f Nelson' s hypercontractivit y inequality , followin g the ideas of Neveu [29] . Then, we present a proof o f P . A. Meyer's inequal ities, the analogues of the classical Riesz transform inequalitie s i n which th e role o f th e Laplacia n A i s replace d b y th e Ornstein-Uhlenbec k generator , A-X-V. Thi s approach, foun d i n [21 ] , is a direct extension o f the method s discussed i n the previous sectio n (topi c (2)) . The autho r woul d lik e t o expres s hi s warmest gratitud e t o th e organizer s of th e Conferenc e a t DePau l University , th e participants , an d especially , t o Roger Jones for hi s enthusiastic hospitality .

I

The Barlow-Yor Inequalitie s For //^-theor y o f harmonic function s i n R+ +1 , two functionals hav e been studied i n detail : th e nontangentia l maxima l functio n an d th e Lusi n are a function. Thes e two geometric objects ar e analogs of two probabilistic func tionals associate d wit h continuou s martingales . I f X = {X tJ > 0} is a continuous martingale , the n X* - su p \Xt\ an d S(X) = (X)

in

t

(where (X) is the quadratic variation of X), Th e basic relation between S(X) and X* i s well known now: \\S{X)\\P*\\X*\\P

for al l 0 < p < oo. Unti l th e appearance o f Barlow and Yor's pape r [1 ] , no other functiona l "o f significance" ha d been found . The y discovere d anothe r functional tha t is , in som e sense , i n between th e maximal functio n an d the quadratic variation : th e maxima l loca l time . Suppos e fo r simplicit y tha t X i s Brownia n motio n ru n u p to a stoppin g tim e r . Th e local tim e o f the Brownian motio n ma y be defined a s follows. Fo r each trajectory co conside r the mappin g X(co) : [0,T ]— • R1, an d th e "push-forward " ma p X*. Tha t is, X*(dt) i s the image measur e o f Lebesgue measur e o n [0 , T] on R 1 , unde r the mappin g X. Th e measur e X*(dt) o n R 1 i s absolutel y continuous , b y a result du e t o P . Levy , an d it s densit y L(r) = L{r,o),x) i s calle d th e local time at r (u p to time t fo r the trajectory co). Conside r L(r) a s a process in the rea l parameter r . O f course, the filtration, based o n the space paramete r r, i s completely differen t fro m th e tim e filtration. Now , take th e maxima l function L * = sup r L(r). We seek to prove a good-A inequality o f the for m (1) P{L*

> 0X,S(X) < SX) < s{pfS)P{L* > X)

with fi > 1 , S < 1 , valid for all k > 0. Fro m this we can obtain a n inequality for norm s of the for m

W\\p 1 , by now standard argument s [7] . The converse inequalit y i s obtained by another good- A inequality wit h the roles of L* and S(X) interchanged . 2

3

THE BARLOW-YO R INEQUALITIE S

Because L* is a maximal functio n acros s a different filtration than the one giving time evolution , Barlo w an d Yo r appealed t o th e Ray-Knigh t theore m which specifie s th e structur e o f L(r) a s a process . However , durin g thes e lectures, Burges s Davi s observe d tha t thi s wa s no t necessary . Th e scalin g properties o f thes e functiona l alread y foretel l th e good- A inequalitie s an d their consequences . A s i t turn s out , Richar d Bas s ha d observe d th e sam e thing some months before. Thei r articles appear i n [3 , 11]. (Tha t the scaling and Marko v propert y implie s th e good- A inequalit y betwee n S(X) an d X* already was observed b y Burkholder [8]. ) Suppose w e wis h t o prov e (1 ) . Le t X 1 b e th e proces s X stoppe d a t a stopping time t. Th e key observation is that, for constant times t, the random variable L*(X l) scale s like Brownian motion :

L^X^-kL*^2) and, lik e (X 1 )* an d S(X f), i t i s a n increasing , subadditiv e functiona l i n t (since L(X l,r) i s additiv e fo r eac h fixed r) . No w th e proo f o f th e good- A inequality i s standard: Le t li = in f {t:L*(X l) = X] , u

= mf{t:L*(Xl) =

pk) .

If X^y X v an d 6^ ar e th e stoppe d Brownia n motions , an d shif t operators , respectively, the n L*((X o 6^'^) = {p - \)k o n th e se t {pt < oo}. No w suppose i n addition, tha t r 1 /2 < Sk. The n the lifetime o f (X -Op) i s bounded by (Sk) 2, s o that P(L* > 0k, t 1 / 2 < Sk) = P ( I * ( ( I o ^ ) ^) = (fi - \)k,x {l2 {P-l)k,ix
(P-l)X,/i(P- l)k,

fi I^\\x^p{fi C||X^ ) i s uniformly smal l i n X^ a s C tend s t o infinity. Thus , P(L* > $k,T 1 '2 < Sk) < o(-jp-j\p(L* >

k)

as desired. Th e convers e inequalit y i s obtained i n muc h th e sam e way. Fo r the details , see [11].

4

THE BARLOW-YO R INEQUALITIE S

REMARKS. Wha t i s L* goo d for ? Th e answer lies in the theorems on e can prove with the new device. (1) Factorizations, and ratios. The most straightforwar d applicatio n i s the following observation . Suppos e X i s a continuou s martingale . Followin g what w e did fo r Brownia n motion , w e have a measure d(X) o n [0 , (X)) an d a push-forwar d X*(d(X)) = L(r)dr, wher e -X* < r < X*. By definition , fx *

(X)= / L{r)dr 0. Th e apertur e i s sometimes indicated wit h a subscript a > 0. Thus ,

Ta(x0) = {(x, y): \x - x 0\ < ay} ; if a = 1 , we omit th e subscript . If (X) , th e quadrati c variatio n o f a martingale , ha s a n associated lo cal time , on e i s le d t o inquir e whethe r A 2(u) ca n b e treate d i n th e sam e manner. Loca l tim e fo r X i s th e "push-forward " X*(d(X)). I n th e sam e spirit, replac e X b y the harmonic functio n u: consider u(x, y) a s a mapping u:T(xo) — > R 1 . The n u+(y l~n\X7u\2(x, y)dxdy) i s the 'push-forward " o f th e indicated measur e to anothe r measur e o n R 1 . Th e specification o f this mea sure may be accomplished i n two ways: (1) The coarea formula (elementar y version). Le t u(x, y) b e a C°° -function on R+ +1 to R 1 an d y/ } f b e a pair of functions wit h compact suppor t i n R+ +1 and R , respectively . Suppos e tha t o n th e suppor t o f y/, |Vw | ^ 0 . The n by th e standar d chang e o f variable s theore m fro m advance d calculus , w e can find function s v if i = 1 ,2,... , n, define d o n R" +1 suc h tha t th e n + 1 vector (v(x, y), u(x, y)) form s a coordinate system with the property that the Jacobian \det(d(v,u)/d(x,y))\ = \Vu\. This translates integral s i n the followin g way : / /

f(u{x, y))yt(x t y)\Vu(x, y)\ 2 dx dy = / / f(u)(y/ • \Vu\)(x(v, u),y(v,u))dvdu.

The loss of a power on | Vu\2 goin g from lef t to right comes from th e interpretation of | Vw| as the Jacobian o f a transformation. I n this context, we do not 6

THE DENSITY OF THE AREA INTEGRAL

7

have to worry about the complications of the general coarea formul a o f Fed erer [1 2 ] since the presence of the "extra" |Vw | on the right-hand sid e (|Vw| 2 as opposed t o |Vw| ) means that w e can avoi d discussio n o f critica l values of the transformation (v(x, y), u(x, y)). Finally , notice that for y/ with compact support i n R+ +1 , f(y/ • \Vu\)(x(v,r),y(v,r))dv i s a continuous functio n o f the parameter r , indexin g the level surfaces {(x, y): u(x, y) = r}. Now we apply these considerations t o th e area integra l /

y\Vu\2y/y(xo - x)dxdy =

A 2(U)(XQ)

where y/ y i s the dilated "bump " function give n above. W e have rN(x J\{X00))r

/ y\Vu\(x,y)if/

y(xo-x)ar(dxdy)dr

/ -N(xo) J fN(x0)

= / D(u;r)(x

0)dr

-N(xo) J-N(Xn)

where a r(dxdy) i s the n-dimensional Hausdorf f surfac e measur e of the level set u = r. Notic e tha t D(u\r) i s lower semicontinuou s a s a function o f r , s o that the supremum an d the essential supremum (ove r r) coincide. W e define D(U)(XQ)

= supD(u;r)(xo). r

(2) A second change of variables formula. I n the previous change of variables formula, w e have assumed tha t u was smooth, but no t necessaril y har monic. No w we suppose u to be harmonic. W e wish to prov e D(u; r)(x0) = / / yy/ y(x0 - x)A(u - r) +(dx, dy). Write

oo

/

(s-r)+f(r)dr

-OO

so that F"(s) = f(s), an d i f u(x, y) i s harmonic, AF[u] = f(u)\Vu\ 2. Agai n for an y y/(x,y) wit h compact support strictly in the interior of R" +1 , w e have / / y/(x,y)AF[u](x,y)dxdy = = JJJ(Ay/)f(r)(u -

Ay/F[u]dxdy

r) + dr dx dy

= ff v(x, y)f(r)A(u - r) +(dx dy) dr. Now form a n integral using the dilated function y/ y(x) a s before, an d F(r) = r2. We have / yVy(xo - x)A(u - r) +(dxdy) =

D(u;r)(xo) a.e .

8

THE DENSITY OF THE AREA INTEGRAL

as define d abov e sinc e th e tw o chang e o f variable s formula s giv e th e sam e result whe n integrate d i n r. W e would lik e t o establis h tha t th e tw o proce dures giv e th e sam e resul t fo r all r. W e conjecture d tha t A( w - r) + neve r charges the set | Vw| = 0 , but we could not prove this. Nevertheless , by Sard's theorem, th e se t o f r suc h tha t u~ l{r) contain s point s {(JC , y): Vu(x, y) - 0 } has Lebesgue measure zero . Furthermore , th e integra l / /

yVy(*o ~ x)A(u - r)

+

(dxdy)

is lowe r semicontinuou s a s a functio n o f r , s o tha t again , th e supremu m over r i s equal t o th e essentia l supremu m ove r r . Therefore , th e supremu m functional D(u)(xo) ma y be defined b y procedure (1 ) or procedure (2) . The question tha t w e could not answer , that is , whether A(w - r) + charge s the se t wher e |Vw | = 0 has recently bee n answere d b y Jean Brossar d [5] . I n fact, Brossar d shows, using the Weierstrass preparation theorem , that locally, {|Vw| = 0 } is a surface o f dimension a t most n — 1 i n R w+1 . Thi s means that this surface is of capacity zero (see Carleson [10, p. 28] ) and therefore, canno t support an y measur e wit h finit e potential , suc h a s A(u - r) + . Conclusion: A(u-r)+ doe s not charge {|Vw|.= 0}. Here is an alternative proof of the fac t that, at most, the set {Vu = 0} is of dimension n-1 i n R++1. Firs t of all, since u is assumed to be nonconstant on R++1, there is a partial derivative v of some finite orde r suc h tha t th e se t {Vu = 0} is containe d i n {Vv = 0;D2v ^ 0} . Observe that the latter surface ha s dimension a t most n — 1 : The matrix D 2v is symmetric and of trace zero. Certainl y {Vv = 0 } has dimension at most n, the exact dimension give n by the number o f independent row s in the matri x D2v. However , sinc e D 2v i s not trivial , an d o f trac e zero , there mus t b e a t least tw o independent rows . (T o see this, simply diagonaliz e th e matrix. )

Norm Inequalities fo r D With the maximal density D defined in the most natural way, it is tempting to ask whether the Barlow-Yor inequalities hold for D. Tha t is what we wish to show . THEOREM. \\D\\

P

^ \\A\\ P for 0 < p < oo .

There ar e tw o proof s know n fo r thi s se t o f inequalities . Th e first , give n in [1 8] , relie d o n th e Barlow-Yo r inequalitie s togethe r wit h a probabilisti c argument relatin g the local time of the martingale u(B t) t o the area integral . The secon d proo f [23 ] relies on a technique o f Barlo w and Yo r and th e ma chinery o f th e Calderon-Zygmun d theory . I t i s this metho d tha t w e presen t here. Th e proof i s not quit e perfected. Fo r example , the good-A inequalitie s proved b y Barlow and Yo r for L* hav e escaped us , so that th e most genera l integral inequalitie s o f the for m / ®(D)(x 0) dxo * f 9(A)(xo) dx

0

are still open . The original selling point for the D functio n (se e [18]) was a ratio theorem of the for m \\A2(u)/N(u)\\p*\\A\\p, 0 jW(fy whe n |x | = | an d tha t y/(x) ha s integra l one . Fo r y > 0, let y/ y{x) = y~ ny/{x/y). Ou r version of the Lusin area integral is then A2(U)(XQ)

= / I y/y(xo-x)y\Vu\

2

(x,y)dxdy

with associate d densitie s D(u\r)(xo)= /

/ y/

y(xo-x)yA(u-r)

+

(dxdy).

Here we have an abuse of notatio n wit h D define d wit h respect to cones, and with respec t to the smooth approximatio n o f the identity y/. As will become clear i n th e proof , th e precis e shap e o r sprea d o f y/ is no t important , i n s o far a s th e suppor t o f y/ is compact . Thu s D (conical) ca n b e majorize d b y D(smooth) an d conversely. Therefore , th e boundedness of D i s independent of the exact shap e o r sprea d o f the approximate identit y tha t define s it .

Local Estimate s The expositio n i s base d o n [23] ; however , sinc e tha t pape r wa s written , Murai an d Uchiyam a [28 ] have improve d th e techniqu e fo r doin g suc h in equalities. We introduce W 0, a subset of R", and associated with it the union of cones W = \J x€Wo ^(JC) anc * a * so ^ e u n * o n ° f large r cones W a = \J xeW ^ M w ^ a > 1 fixed once and for all . I n the rest of this section we work with the "cut down" functions N a(Wa)(xo) = sup ra(jCo) \u\Iw a,

D(W,t)(xo)= /

/ y/

y(xo-x)Iw(x,y)yA(u-t)

+

(dx,dy)

with I Wn, Iw denotin g th e indicator s (characteristi c function s t o nonproba bilists) o f W, W a. LEMMA 1

. For each t eR and for 1 < p < oo

f(D(W,t)(x))pdx
0 depending only on p and the dimension n. PROOF. W e ca n obtai n th e inequalit y a s a consequenc e o f th e metho d of Mura i an d Uchiyam a [28] . The y introduce d a new technique fo r obtain ing the good- A inequalit y tha t significantl y improve d th e estimate s give n i n [13]. (Th e method allowed Uchiyama to complete McConnelPs subharmoni c function inequality . Se e [34].) W e must establis h a good-A inequality o f th e form

(1) m(D(W

tt)

> /tt,Na(Wa)
0 an d P > 1 sufficiently large . Fo r a fixed p > 1 , thi s inequalit y implies

(2) J\D{W,t)\e = fii>J\2ypl\ < P pp I A Jo

+ fi"p I X Jo

P

p l

- rn(D(W,t) > p l

- m{Na(Wa) >

< Cex p (- (cj\) P

r X"-

pk,Na(Wa)
X)dX

+ (§)' j\Na{Wa)\? = c e x p ( - ( c | ) ) / F j\D{W,t)\" +

c(^y J\N

a(Wa)\P

If p i s large, c exp(-c/3/d)(lp < 1/2, s o that, i f the integrals are all finite, we may subtrac t th e right-han d sid e from th e left t o finish the proof. I n general we can truncat e W an d replac e u to be a harmonic functio n wit h boundar y function i n C c^m(R"), s o tha t th e righ t sid e o f (2 ) i s finite, argu e a s above, and the n pas s to th e limi t t o establis h (5 ) fo r th e give n u an d W. Thu s th e proposition wil l follow i f we prove (1). Let Q be an arbitrar y cub e in R" an d V0 = {x: D(W,t)> pA,N a(Wa)dx^ = sup / | rfx#i P/2' -1/P

X

/

dx

°\l I

Iwl//

+

2

y(xo - x)y\V((u - r) - (u - s)+)| 1

The firs t facto r i s bounde d b y C||p|| g fro m th e vector-value d result s i n [4, las t paragraph]. Thus, (7) | | J y*A x{(u - r) + - (u - s) +} dx rfy|< C (j(H(x0)p/2 dx

0)

l/p

^

LOCAL ESTIMATES

15

where H{x0)= I

+

IwyVy(xo-x)\V{(u-r)

(u - s) +}\2 dx dy.

-

Here th e V is the ful l gradient . Not e tha t i f r < s, V{(u - r) + - (u - s) +} = Vu • I{r-componen t o f th e Laplacian , integratin g by part s with respec t to the variable y: I f f

d2

I

/ / 1 , a s w e shal l see . Th e secon d i s precisel y th e ter m w e ha d before i n the examination o f I. Therefore, give n the boundedness o f the firs t factor, w e ma y appl y Holder' s inequalit y t o finis h th e proof . Wha t abou t that firs t factor ? I t i s an archtypica l Littlewood-Pale y functional : I f y/j(x) i s

1 LOCAL ESTIMATE S

7

compactly supporte d an d ha s mea n zero , b y takin g Fourie r transforms , w e see that

/M-(/l(^*/l2W^)

1/2

is a bounded operato r in L 2 , and by the Calderon-Zygmund theory , bounde d in H p, p > 1 . So , by Jensen's inequality , we know that th e norm o f the first factor

]{ffv(xo-x)\®j\2ydxy dx

0 A, N a(Wa) < A) + m(N a(Wa) > A) Let W 0 = {D> A , Na(Wa) > A}, and

w a = ( J r Q.(x0), Then o n W 0, w e hav e N a{Wa) = N a(u). Wit h Lemm a 2 and th e GR R in equality, w e obtain fD

p

(u)dx < CX pl2 [ N

p 2

/ (Wa)dx.

Therefore,

m(W0) A).

J{Nn{u) A) < CX-pl2 [ N

p 2

l {Wa) + Cm(N a(u) > A).

If / ? > 4, w e ca n multipl y bot h side s b y th e appropriat e powe r o f A , then integrate wit h respec t t o A to obtain th e inequality state d i n the theorem .

Terminal Expectations and Singular Integrals In thi s sectio n w e show how th e Ries z transfor m inequalitie s ma y be obtained fro m Burkholder' s martingal e inequalitie s via a projection argument . The projection argument itself is based upon Bochner's idea of subordinatio n of semigroups . Le t us review this idea first. Bochner's subordination. I f P t i s a semigrou p o f Marko v transitio n op erators suc h tha t eac h P t i s a symmetric operato r o n L 2{dm), Her e dm i s some referenc e measure . Fro m th e fac t tha t P t - P t/2 • Pt/i w e s e e tha t P t is nonnegativ e definit e o n L 2(dm), an d tha t th e generato r L i s nonpositiv e definite. Therefore , th e semigroup P t has a spectral squar e roo t Qy = exp(->>(-L) 1 /2 ) = / cxp(-yX Jo

l 2

^ )dF(X)

where dF(X) i s the projection value d spectra l measur e fo r th e semigrou p P t. Bochner's observation is that this square root may be obtained in another way. Let y(t), 0 < t < oo, be a one-dimensional Brownia n motion, independent o f X. T o simplif y th e constant s i n subsequen t computations , w e shall assum e that X t an d y t ar e ru n a t spee d 2 ; that i s E\X t\2 = Ey} = 2t. I n thi s way , the generator o f X t i s A, that o f y t i s d 2/dy2. Conside r the first passage time process for th e y-Brownia n motio n T ( 0 = inf{s : y(s) = t;t>0}. This process has independent increments , s o the corresponding measure s

f(y,dt) = P(x(y)edt) form a convolution semigroup . B y Brownian scaling , A1/2 max y (s) = max y(As) s 0. Usin g the stron g Markov property , w e see tha t th e famil y o f measure s P y generate s a finitel y additiv e measur e P on path s whos e y-componen t tend s t o +oo . Tha t th e measur e P is , in fact , (T-additive was first proved by M. Weil [35] who treated the class of processes introduced b y Hunt, [25] , called "approximate Marko v processes" . Notice that ther e is no natural tim e scal e for th e process corresponding t o P: a t each level y > 0, the time i s reset to zero. Sinc e P i s the simultaneou s extension o f eac h measur e P y9 w e se e that P i s a measure o n "roa d maps " in Feller' s language, o r o n equivalenc e classe s of to , where co = co' if co(t) = co'{c - f /) fo r som e constan t c = c(a> f). Nevertheless, w e shall conside r P a s

20 TERMINA

L EXPECTATIONS AND SINGULAR INTEGRALS

a measur e o n path s wit h th e tim e t < 0 when th e pat h i s i n R" +1 an d suc h that th e y-componen t o f co equal s zer o at tim e 0 . Thi s amounts t o selectin g a representativ e co fro m eac h equivalence class . The basic formula. Le t Q(f)(y, X) b e the Poisson integral of/, a suitably nice functio n define d o n R" . Tha t is , Q(f)(y,X) i s th e harmoni c exten sion o f / t o the uppe r half-plan e R+ +1 . Th e Poisson integral s Q(y, • ) form a semigroup, th e squar e roo t semigrou p correspondin g t o th e hea t semigrou p generated b y th e Laplacian . Th e basi c formul a tha t w e propos e t o prov e is th e following . Le t {y s,Xs) b e th e backgroun d radiation . Th e functio n Q(f)(y, X) , composed wit h (y s, X s) i s then a local martingale fro m (y®, Xo), a fixed startin g poin t i n R+ +1 . Tha t is , w e hav e a harmoni c functio n com posed with an (n + 1 )-dimensional Brownia n motion , ru n up to the stoppin g time T = inf{t: y t = 0} . I f X s wer e anothe r Marko v proces s wit h a sym metric generato r L , Bochner' s representatio n o f th e squar e roo t semigrou p would yiel d a function Q(f)(y, X) i n th e sam e way. Th e background radia tion woul d the n b e (y s,Xs)9 an d Q(f)(y s,Xs) agai n a local martingale. Thi s martingale ha s an It o integra l representatio n Q(f)(yt,Xt) -

Q(f)(y 0, X 0) = f VQ(f)(y s, X Jo

s)

• d(y s, X s).

Now assume that Q(f)(yo, Xo) tends to zero as yo tends to infinity. The n we have the formal representatio n

f(XQ) =

/

VG(/)(K J — oo

, X s) •

d(ys, X

s),

where th e backgroun d radiatio n pat h i s chose n s o tha t t = 0 whe n th e y~ process makes its first passage through zero. I n the obvious sense, the stochastic differential for m tha t appear s unde r the integra l is exact. If , however , we take any component o f this form (fo r example , consider ^Q(f)(y s,Xs)dys) we no longer hav e a n exac t form . Tha t is , the rando m variabl e

f^Q{f){ys,x5)dys is no longer measurable with respect to the variable Xo. However , the Bochner subordination principl e an d martingal e dualit y allo w u s t o prov e ou r basi c formula:

E (f^ ^Q(f)(y

s,Xs)dys\\x0)

=

l

-f(X0).

TERMINAL EXPECTATIONS AND SINGULAR INTEGRALS

21

To see this, take the expectation, relative to the background radiation process, of the product E (f^ ^Q{f)(y

s,Xs)dysj°^

= E(J ^Q(f)(y

^Q(g)(y

s,Xs)dysj

s,Xs)-^Q(g)(ys,Xs)ds}

Q{f)

)(l-yQ{8))ydydx-

= H{§-y

These computations are justified b y taking the usual expectation of a Brownian process starting at (yo, Xo), yo > 0 in R"+1, integratin g XQ, then passing to the limit as yo tends to infinity. Th e occupation density lim f°° f P t(y0, y\ X0 - X) dX0 dt = y

yo^oo j0

j

as one can easily verify. Fro m here we use the fact that Q = exp (—y(—A)1 /2) to obtain / /

^Q(f)^Q(g)ydydX = - jj[^ 2 cxp(-yX

1 2

/ )]2 d{F{X)f F{X)g)y dy

= \ j f{X)g{X)dX Since this holds for every suitable g, the sample path interpretation is that E ( £° ^Q(f)(ys,X s)dys\\x0^ =

i/(x„);

that is, we may take g — dXo i n the above computation. The Riesz transforms. Fro m here we can obtain a formula fo r the Riesz transforms. Th e /t h Ries z transform i s given by the operator Rj = d/dXi o (-A)1/2. Notic e that d/dXj commutes with A and therefore, with (-A),1/2 and the corresponding semigroup Q. So , by the basic formula,

\R'^'E{l~hQ{U-Ar"lf)dyAX°)

-*tt

°° d Q{f)dy s\\Xo dXt

since f yQ = (-A)1/2 • Q.

THEOREM. The vector of Riesz transforms (R\(f),..., R n(f)) is bounded in Lp, p > 1 , with a constant that is independent of the dimension n.

22

TERMINAL EXPECTATIONS AND SINGULAR INTEGRALS

PROOF. Fo r each

/= 1 ,2,...,« , we have from th e above formula

\\\wm = kS\E{S^ k

Q{f)dysUo

) dx

0

AfE(\Fw,QWy.\'M)* 0, we use a time-reversa l argumen t give n i n [22] . W e first conside r th e cas e wher e tp{x) = (p{\x\), that is , where (p i s radial.

24

TERMINAL EXPECTATION S AN D SINGULAR INTEGRAL S

In orde r t o comput e th e termina l conditiona l expectation s o f martingale like objects , ther e ar e tw o method s available . Th e firs t i s duality, whic h we used above . Th e secon d i s time reversal, whic h w e now explain . An arbitrar y martingale , subjecte d t o a time-reversa l transformatio n t— • (-*), doe s no t retai n it s martingal e characte r i n general ; indeed , th e reversed process is not even adapted since the "past" becomes "future". How ever, the "little martingale" in question here is not arbitrary; it has additional structure o f a Marko v character . Th e integran d o f / ^ ^{Z s)dys wher e Zs = (y StXs) depend s o n th e pas t Z s*9 s f < s, onl y throug h th e presen t Z s, so that f£(Z 5 ) ma y as well be considered t o be a function o f the future Z s*9 s' >s. To see the effect o f time reversal on the martingales in question, let f(x, y) be a smoot h functio n define d o n R+ +1 an d conside r a "segment" o f th e for ward proces s Z T+5, 0 < s < t, wher e r i s the tim e o f a level crossing for th e process Z s whos e time parameter begins at — oo. Le t us suppress the variable r fo r th e momen t an d writ e

Mt= f f(Z s)dys; Jo

recall that M t i s the limit o f sum s

M? = J2f{Z Sl_^ySi i=\

with Ay Si = y Si - y Si.r Reversin g time ha s the effect o f reversin g the succession 0 = x o < s\ < • • • < s n = t. Th e reverse d proces s y* i s parametrized b y the interva l -t < s' < 0 with - / = s' 0 < s[ < • • • < s' n = 0 and

W = yt - y;,_ , = y*-^., - vU)*- M = A^-/+iIf w e set n- i — j , the n

Mt" = '£nzSl_1)AySi = 22AZ Sl)AySM i=l

1=0

= iZAZs^Ay^ =-J2f{Z;

l)Ay*s,.

j=n 7= 1

As it stands , M tn doe s not approximat e a semimartingale wit h respec t to the process Z * sinc e /(Z*.) , j = 1 ,2 , , n, doe s not hav e the correc t measura bility. However ,

/(z;,) = r{zs,_s) + vf{zsl][) •

AZ;,

+ O(AZ;;)

so that

M? = -JZf^JAy;, -J2vf{Z;,JAy* j=\ '

'

j=\

'

Sj+Y,o{AZ*Si)Ay*s, 7= 1

25

TERMINAL EXPECTATIONS AND SINGULAR INTEGRALS

Now le t n ten d t o infinit y an d tak e accoun t o f th e stochasti c differentia l equation definin g th e Besse l (3 ) proces s y*: dy* = dBt + —dt. We obtain

Hm MP = - £ f( Zp)dyp - £ |(Z;, ) ds' = - f° f{z*,)dB s, - f* f(z;,)( y;,rl ds' - J* ^(z^ds 1 . Now let u s write everythin g i n th e origina l tim e scale . Th e forwar d proces s begins at time T < 0 , the time of a first crossing of some level y > 0; and continues unti l th e y-componen t Brownia n motio n crosse s zero . B y definition , this final crossin g takes place at time zero. Wit h this change of notation, th e above computation become s j*+'f(Zs)dys =

- j * t f{Z*)dBs- j^f{Z*

s){y*sT'ds-

£

^{Z*)ds.

The sam e procedur e ma y b e applie d t o eac h componen t o f th e X s proces s with th e result :

j™ f{z s)dx\ = -f f(z*s)dx\

- £ J£(z;) ds.

The effec t o f thes e transformation s i s t o facilitat e th e computatio n o f th e conditional expectatio n given the terminal position XQ. The reversed proces s on th e righ t i s a semimartingal e whos e initial positio n i s XQ. Therefore , th e conditional expectatio n give n xo ma y b e treate d a s unconditiona l relativ e to a proces s startin g a t Xo. Wit h sufficien t regularity , w e ma y le t r ten d t o "infinity" (i n this case, - t ) an d comput e th e expectatio n explicitly : th e firs t term, a stopped, centered martingale, has expectation zero, and the second is a "potential " term whic h w e can evaluat e i n the cas e at hand . I n fact , i f we carry out this computation w e obtain

E U° f{Z s)dys\\xA = -EXo Qf (f{Z*){y;)-' + ^(Z,*)) y)Ky(xo-x )dx'dy

EU°^(Zs)dys\\x0) =-JJ?l(

d2y)yKy{xo - x f)dxf dy. 2 - / / : dy Take th e Fourier transfor m wit h respec t t o the Xo variable . Th e above expression become s poo

- / (Kfl(P')O'lf l + y\Z\2tf)(y\e\))e-yw dy. /«?) Jo

where (#')> (#") a re derivative s relative to the radial variable. Now , a change of variable s y' = |£| y allow s u s to sho w tha t th e integral i s independent o f the value of |{| . I n other words, /•OO

- / (TO)(y|£l ) + y\Z\\r){y\$\)e-m)dy =

c,

Jo and i t remains for us to verify tha t C ^ 0 . The above integral is y - / m'{y) + y(y} +

y^(

x

>ynKy(x)dxdy'

As before, we take the Fourier transform relativ e to the variable xo and obtain

(

2

W)

B 7(0

K£S *H- f£('£ )'-' '' m

f°° d® C°°

ft

= -Jo Mye)fyy\Z\e-

m

)dyf{Z).

The multiplie r o f /(£ ) ma y be compute d a s i n th e proo f o f Theore m 1 : change variables y — • y\ 0. Fo r eac h poin t x eR n, le t X t b e a stationary, Gaussian Markov process from x wit h variance parameter one . Le t Xo als o be distribute d a s a standar d Gaussia n variable . With thes e requirements , th e covarianc e structur e o f X t i s determine d an d the process is known a s the Ornstein-Uhlenbeck (velocity ) process . Contras t this wit h th e previou s section . There , w e considere d (R n,dx)\ th e proces s Xt, t > 0 , wa s Brownia n motion , wit h th e stationar y initia l measur e take n as Lebesgue measur e dx. I n th e present situation , dx i s replaced b y dg, th e product o f standar d Gaussia n measure s o n eac h o f th e coordinates , an d th e Ornstein-Uhlenbeck proces s plays the sam e rol e as the Brownia n motio n o f the previou s section . Th e chang e o f invarian t measur e fro m Lebesgu e t o Gaussian produces a crucial advantage: ther e exists an extension of the finite product measure s dg t o the infinite-dimensional setting . Tha t is, we can now consider (R°°,dg), an d it s associated scal e of Z/-spaces . The charm o f (R°° , dg) an d it s L 2-space i s more eviden t whe n we realize the connectio n wit h Brownia n motion . Wher e doe s Brownian motio n com e in? I t is instructive to look at the dyadic, and /?-adic martingale spaces. I n the dyadic space , 2° ° th e measur e is , of course , infinit e produc t measur e (\)°°, with a simila r statemen t fo r p°°, (~)°°- Thes e examples o f infinit e produc t probability space s have tw o easil y recognizabl e feature s tha t shoul d aler t u s when looking at R°° , (dg): (a) Eac h spac e (p°°, (^) ) i s measure-theoretically identica l t o the uni t in terval ([0,1 ] , dx) vi a th e /?-adi c expansion o f element s JC : 0 < x < 1 . Thu s L2[p°°, (^) ] = £ 2 ([0,1 ], dx), th e point bein g that th e latter i s a richer spac e than th e sequenc e space . Th e analogous isomorphis m fo r R°°(dg) i s due t o Wiener: L 2 (R°°, (dg)) = L2(C[0, oc) , dW). Her e (C[0, oo), dW) i s the richer space. Th e proof o f Wiener's theorem i s far fro m evident . (b) Th e spac e L 2 (2°°, (j)) ma y b e decompose d int o a direc t su m o f or thogonal subspace s in two interesting ways. (i) Chaos decomposition. Wherei n th e spac e L 2(2°° ,(\)) i s decompose d into the direct sum of infinite-dimensional orthogona l subspaces CQ©C I 0 • •. 31

32

THE ORNSTEIN-UHLENBEC K SEMIGROUP

Here C o is the spac e of constant s (th e exceptional finite-dimensional space) , C\ i s the spac e generate d b y th e Rademache r functions , C2 by al l product s of two Rademacher functions , C 3 by all products o f three Rademacher func tions, etc. B y "Rademacher functions " I mean an y sequence of independen t ± 1 variables . Ther e is nothing canonical about the ordering or representation. (ii) The martingale decomposition. Her e the products of Rademacher func tions are given their Pale y orderin g (PN = r nxrni • • • rnK wit h N = 2 n" + • • • + 2 n*. The decompositio n L 2 (2 o o ,(i) o o ) = p 0 e P i e - - where Po are constants; P k ar e all products (p^ with 2 k < N< 2 k+l. In this decomposition w e have the interesting fac t tha t th e projections P/v onto PO(B--®PN for m a martingale. Sinc e each block P k consists of products with th e commo n facto r r k an d al l othe r factor s r, , j < k, w e se e tha t an y f e P k ma y b e writte n a s v krk wit h v k = v k(r\,..., r k_{). Thi s importan t feature here is the "unrolling" of the product a-field generated by Po©- • -®PN> N > 0. Th e passag e fro m P 0 © • • • © PN t o P 0 © • • • © PN+I i s via th e singl e new variable r^+ i enterin g a s the stochastic differential . These two decompositions persis t i n the case of (R°°,dg). Polynomial chaos decomposition L 2 (R 1 , dg) ha s an orthonormal basis consisting o f Hermit e polynomials , an d L 2(R°° ,dg) ha s a n orthonorma l basi s consisting of finite products o f Hermite functions . Th e polynomial chao s of order N consist s of L 2-closure of all finite products of Hermite polynomials , &>N = {P

nx{Xl)---Pnk{Xk)}

such tha t JC I , . . ., x k ar e al l distinc t (no t necessaril y th e first k coordinates) , and th e tota l degre e i s N. I t i s a purel y forma l matte r t o verif y tha t L2(R°°, dg) = ^b © &\ © • *• • Here, however , th e passag e fro m E L o ^ t 0 Ylk=Q ^k i s n ot vi a multiplication b y a single function a s in the dyadic case, nor even a finite number of functions a s in the p-adic case. Th e new space is obtained b y multiplications involvin g the infinite sequenc e of Hermit e poly nomials. Tha t is , the product structur e (R°°,dg) doe s not "unroll " as slowly as i n th e dyadi c o r p-adi c case . T o obtai n a martingale representation of / G L2(Rn,dg) on e mus t find a ne w filtration. Thi s ne w filtration shoul d allow the informatio n i n th e sequenc e {x\ ,X2,...} = (R°°,dg) t o enter , no t as i t di d i n th e coordinat e filtration, bu t gradually , i n independen t Gaus sian increment s smal l enoug h t o obtai n a martingal e representatio n fo r an y L2 functional . Th e solution : conside r no t just {x\, X2, ...} bu t all Gaussian variables define d o n (R°° , dg), i.e . th e subspac e o f al l variables o f th e for m J2T=rakxk' Cal l this J ?2 . Tak e any unitary operator U from L 2[0,00)— • Sf 1 . Now "index" 2?1 b y the family of characteristic functions o f intervals J^[0, t)

THE ORNSTEIN-UHLENBECK SEMIGROU P

33

in [0, oo) and define a filtration by the process £/[^fo.o l = B t. Th e martingal e representation fo r / i n L 2(R°°, dg) i s then given as follows: On e proves that Bt, considere d a s a n elemen t i n 3 ?1 c L 2(R°°,dg) i s continuou s i n t wit h probability 1 , and is , in fact , Brownia n motion . Therefore , w e can associat e (map) almos t ever y element i n R°° to a continuous function . Th e adjoint o f this map carries dg t o a measure on C[0 , oo), which we call standard Wiene r measure. Wit h respec t t o thi s filtration o n (R°° ,dg) a theore m o f It o give s the martingal e representatio n / = / v(s)dB Jo

s

for an y / e L 2(R°°,dg) = L 2(C[0,oc),dW). Thus , w e hav e a continuous time analogu e o f th e Pale y martingal e PN referred t o above. To do analysi s o n Wiener spac e [C[0 , oo), dW\ w e must hav e a substitut e for th e Laplacia n o n [R n,dx]. Fo r (R n, dg], th e versio n o f A is the su m o f "uncoupled" harmoni c oscillato r operator s

The operator L n i s symmetric on (R n}dg) sinc e each component is symmetric on (R 1 , dg). Formally , we may let this sum go to infinity to obtain the infinitedimensional operato r actin g o n Wiener functional s vi a th e isomorphis m re ferred t o above. A n important observatio n her e is that the eigenspaces for L are the polynomial chao s spaces ^. Thi s fact i s an immediate consequenc e of th e one-dimensiona l result : th e Hermit e polynomial s ar e eigenfunction s of th e harmonic oscillato r operator . If w e see k a coordinate-fre e approac h i t i s mor e convenien t t o construc t directly th e semigrou p generate d b y L. Thi s ha s bee n don e i n William s [36]. Tak e th e Brownia n shee t X st suc h tha t X 0t = X s0 = 0 , an d le t X.(0) be anothe r Brownia n motion , independen t o f {X st;s, t > 0} . Le t Z.(t) = e~'(X.(0) + X te2t_\). The n fo r eac h fixed s, Z s(t) i s a one-dimensiona l Ornstein-Uhlenbeck proces s i n t, an d wit h s free , Z.(i) i s a C[0 , oo)-valued process wit h continuou s paths . Fro m thi s proces s w e obtain th e infinit e di mensional semigrou p correspondin g t o L . Let u s specializ e again , an d regar d th e one-dimensiona l cas e b y settin g 5 = 1 . The n

Zl(t) = e-tXl(0) + Xi(e

2t

-l)

where X\ s = X\(s). I n distribution , Z{(t) = e-'X(O) + (e 2t - \)

X2

' X{\)

where X(0) an d X(l) ar e independent Gaussia n variables .

34

THE ORNSTEIN-UHLENBECK SEMIGROU P

The poin t i s that fo r eac h fixed t, Z\(t) = Z(t) i s distribute d a s a linea r combination o f independen t norma l variables. Thi s allows us to comput e Mt(f)(x) = E(f(Z(t))\\Z(0) = x) =

j ~ f{e~ lx + ( 1 - e~ 2ty))cxp (-2^) (2*r

l2

' dy.

This transform , calle d th e Mehle r transform , shoul d b e compare d wit h th e Gaussian transfor m Gt{f){x) = E(f(X(t)\\X(0) =

x) = J~ f{x

+ t l'2y)cxp (^j

(2n)-V

2

dy.

The latter is a symmetric average, while the former i s asymmetric. I t is easily checked tha t Gt:Lp(Rl,dx)-+L°°(Rl) and i n several variable s Gt:Lp(Rnfdx)-+L°°(Rl) with a bound

\\Gt\\p.oo = 0(r«2). That is , G t is instantly "improvin g of infinite order" , but the bound fo r t < 1 tends to infinity a s the dimension increases . The semigroup M t exhibit s entirely different behavior : M t i s not a bounded map from L p(Rnfdg) - • L«(R ntdg) (tha t is, it is not "instantly improving" ) except for a n interva l o f values o f q dependin g o n both p an d t. Certainly , Jensen's inequalit y show s that wit h respec t t o dg, \\W)\\P < II/IIP for al l 1 < p < oo; the surprise i s that

(*)

W t(f)\u < 11/11,

for a n interva l o f q > p give n by the inequalit y (p-l)>e-2t(q~l). Note that the operator nor m of M t i s one within this interval and, as it turn s out, i s infinite outsid e thi s interval ! The proo f o f Nelson' s inequalit y shoul d be straightforward; everythin g i s explicit, an d a s w e sho w below , i t suffice s t o conside r th e one-dimensiona l case only . However , th e know n proofs , no w manifold , ar e subtle . A spectacular proof ha s been obtained by Neveu [29] , whose idea is to interpolate a supermartingale betwee n the right and left side s of (*) . Thi s is natural a posteriori: Inequality (* ) is between two integrals, and supermartingales are expectation-decreasing. Thus , th e interpolate d supermartingal e S x start s a t time t = 0 with a constant So = ||/|| p an d finishes at time r = 1 as a random variable with expectatio n ||Af/||^ .

35

THE ORNSTEIN-UHLENBECK SEMIGROUP

Proof of Nelson's inequality, Neveu [27] . Th e first step is to write (* ) i n a symmetric form . Le t p = e~ l an d Z = pX(0) + ( 1 - p 2)l'2X{\) wher e X{0) and X(l) ar e independent , standar d Gaussia n variables . I n fact , thin k o f a pair o f independen t Brownia n motion s (Xs(0),X$(l)) an d a new Brownia n motions Z ps = pX s(0) + ( 1 - /? 2 ) 1 / 2 X s (l) wit h / ? = e~ l. No w conside r th e symmetric for m o f (*) : E(f(Z) • *(*) ) < ||/(Z)|| P • \\g(X)\\q* wher e 4 ' = (1 - l/q) depend s o n /> , and Z = Zf , X = JTi(O) . (I f / > = ±1 , |Z | = |X| , $' = P f, P - Q\ i f / ? = 0 , then E(f(Z) •

*(*)) = E(f(Z))E(g(X))