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logarithmic derivative of the Poisson kernel on regular domains in Riemannian mani- folds corresponding to elliptic PDOs of H ormander type. Such formulaeĀ ...
Stochastics and Stochastics Reports 61 (1997) 297{321

On the Di erentiation of Heat Semigroups and Poisson Integrals ANTON THALMAIER

Abstract:

We give a version of integration by parts on the level of local martingales; combined with the optional sampling theorem, this method allows us to obtain di erentiation formulae for Poisson integrals in the same way as for heat semigroups involving boundary conditions. In particular, our results yield Bismut type representations for the logarithmic derivative of the Poisson kernel on regular domains in Riemannian manifolds corresponding to elliptic PDOs of Hormander type. Such formulae provide a direct approach to gradient estimates for harmonic functions on Riemannian manifolds.

1. Introduction

Let M be an n-dimensional smooth manifold and, for some m 2 N , let A: M  R m ! TM ; (x; e) 7! A(x)e ; be a homomorphism of vector bundles over M . Thus, A 2 ?(R m TM ), i.e., the map A(x): R m ! Tx M is linear for x 2 M , and A( . )e 2 ?(TM ) is a smooth vector eld on M for e 2 R m . Consider the Stratonovich stochastic di erential equation dX = A(X )  dB + A0 (X ) dt (1.1) m where A0 2 ?(TM ) is an additional vector ? eld, and B an R -valued Brownian motion on a ltered probability space ; F ; P; (Ft)t2R satisfying the usual completeness conditions. There is a partial ow Xt( . );  ( . ) associated to (1.1) (see [12] for details) such that for each x 2 M the process Xt(x), 0  t <  (x), is the maximal strong solution to (1.1) with starting point X0(x) = x, de ned up to the explosion time  (x); moreover, using the notation Xt(x; !) = Xt (x)(!) and  (x; !) =  (x)(!), if Mt (!) = fx 2 M : t <  (x; !)g then there exists a set 0  of full measure such that for all ! 2 0: +

1991 Mathematics Subject Classi cation. 58G32, 60H10, 60H30. Key words and phrases. Di usion, heat semigroup, integration by parts, heat kernel, Poisson kernel.

{1{

(i) Mt (!) is open in M for each t  0, i.e.  ( . ; !) is lower semicontinuous on M. (ii) Xt ( . ; !): Mt (!) ! M is a di eomorphism onto an open? subset of M .  (iii) The map s 7! Xs( . ; !) is continuous from [0; t] into C 1 Mt(!); M with its C 1 -topology, for each t > 0. The solution processes X = X (x) to (1.1) are di usions on M with generator m P L = A0 + 21 A2i i=1

where Ai = A( . )ei 2 ?(TM ), i = 1; : : :; m. Throughout this paper we assume that the system (1.1) is non-degenerate, i.e., A(x): R m ! Tx M is surjective for each x, or equivalently that L is elliptic. This non-degeneracy provides a Riemannian metric on M such that A(x)A(x): Tx M ! Tx M is the identity on Tx M for x 2 M . In other words, A(x): Tx M ! R m de nes an isometric inclusion for each x 2 M , i.e., hu; viTxM = hA(x)u; A(x)viRm for all u; v 2 Tx M : With respect to this Riemannian metric, L = 21 M + Z where Z is of rst order, i.e. a vector eld on M . Standard examples are the gradient Brownian systems when M is immersed into some Euclidean space R m , and A(x): R m ! Tx M is the orthogonal projection; for A0 = 0 this construction gives Brownian motion on M with respect to the induced metric, see [5]. For x 2 M , let Tx Xt: Tx M ! TXt (x) M be the di erential of Xt ( . ) at x (wellde ned for all ! 2 such that x 2 Mt (!)) and Vt = Vt (v) = (Tx Xt)v the derivative process to Xt ( . ) at x in the direction v 2 Tx M . It is well-known that V on TM solves the formally di erentiated SDE (1.1), i.e., dV = (TX A) V  dB + (TX A0) V dt ; V0 = v ; (1.2) with the same lifetime as X (x), if v 6= 0. Using the metric and the corresponding Levi-Civita connection on M , equation (1.2) is most concisely written as a covariant equation along X DV = (rA) V  dB + (rA0) V dt (1.3) (see [5]); by de nition, (1.3) means dV~ = ==0?;t1 (rA) ==0;tV~  dB + ==0?;t1 (rA0 ) ==0;tV~ dt for V~t = ==0?;t1 Vt where ==0;t : TX M ! TXt M is parallel transport along the paths of X . We rst assume completeness in (1.1), i.e.  (x) = 1 a.s. for each x 2 M . Note that this does not necessarily imply the existence of a sample continuous version of the ow R+  M ! M , (t; x) 7! Xt (x). For f 2 bC 1(M ) (bounded C 1 functions with bounded rst derivative) let ?  (Pt f )(x) = E f  Xt(x) ; x 2 M; (1.4) be the semigroup associated to (1.1), and   (1.5) Pt(1)(df )x v = E (df )Xt(x) (Tx Xt ) v ; v 2 Tx M ; its formal derivative whenever the right-hand side exists. More generally, for a (bounded) di erential form 2 ?(T  M ) let Pt(1) ( ) = E [Xt ] ; (1.6) 0

{2{

provided the right-hand side of (1.6) is well-de ned; here Xt is the pullback of under the (random) map Xt: M ! M . Further, for x 2 M and I = [0; t] or I = R + let  H (I; Tx M ) = : I ! Tx M absolutely continuous, k _ k 2 L2 (I; ds) be the Cameron-Martin space and H 0 (I; TxM ) = f 2 H (I; Tx M ) : (0) = 0g. The following version of an integration by parts formula is a slight variation of a formula obtained by Elworthy-Li [6] (see also [3]); we use it to exemplify our approach to derivative formulae. Theorem 1.1 (Integration by parts formula) Assume (1.1) to be complete and non-degenerate. Let f 2 bC 1 (M ). Then Z h i h?  t

?  i E (df )Xt (x) (Tx Xt ) ht = E f  Xt (x) (Tx Xs) h_ s ; A Xs(x) dBs (1.7)

0

for each bounded adapted process h with  sample paths in H 0 ([0; t]; Tx M ) such that E sup0st d(Pt?s f )Xs (x) (Tx Xs ) hs < 1, and with the additional property that ?  R r

0 (Tx Xs ) h_ s ; A Xs (x) dBs , 0  r  t, is a martingale.

Proof Let h be an adapted bounded process with h. (!) 2 H ([0; t]; Tx M ), almost all !. It will be shown in Lemma 2.1 below that Nr = d(Pt?r f )Xr (x) (Tx Xr ) hr Z (1.8) ?  ? ?  r

(Tx Xs) h_ s; A Xs(x) dBs ? Pt?r f Xr (x) 0

provides a local martingale for 0  r  t. The additional assumptions assure that N is even a martingale; the claim follows upon taking expectation.

Remark 1.2 A canonical choice for h in equation (1.7) is hs = (s=t) v, v 2 Tx M , or more generally, hs = (s ^ "=") v with some constant 0 < "  t. Then, under the assumptions of Theorem 1.1, Z h i h? ?  i  1 "

( T X ) v; A X ( x ) dB E (df )Xt (x) (Tx Xt ) v = E f  Xt (x) x s s s : (1.9) " 0

In general, if h in Theorem 1.1 has the property that ht = v, then we get Pt(1) (df )xv for the left-hand side in (1.7) while the right-hand side represents d(Pt f )xv as will be shown in Theorem 2.4 below. Thus, in this case, d(Pt f )x = Pt(1) (df )x is already a consequence of (1.7).

Note that di erentiating (1.4) by taking derivatives under the expectation requires di erentiability of f . However, due to the smoothing property of the semigroup, Pt f is already di erentiable even if f is only measurable | a fact which is explained by formula (1.7) where the right-hand side does not involve any derivatives of f . In case system (1.1) is explosive, the minimal heat semigroup associated to (1.1) is given by  ?  (1.10) (Ptf )(x) = E f  Xt (x) 1ft 0 and any f 2 C 2 (D ) with f j@D = 0 and f > 0 in D, formula (4.1) gives a process with the required properties. ?



0

0

0

0

0

0

0

0

5. Extensions to closed di erential forms For the sake of simplicity we restrict ourselves to the case when the system (1.1) de nes Brownian motion on (M; g); generalizations to h-Brownian motion (see [6]) { 11 {

for instance are straightforward. Let (M; g) be a complete Riemannian manifold and d ?(p T  M ) ?! d ?(p+1 T  M ) ?! d ::: ::: ?! its deRham complex. Denote by  the deRham-Hodge-Laplace operator de ned as the .L2 -closureLof the operator ?(d + )2 on compactly supported elements of ?( T  M ) = p0 ?(p T  M ). Let dom   L2 -?(. T  M ) be the domain of , and . 1 = j dom  \ L2-?(T  M ) .the restriction of  to 1-forms; note . 2  2   that L -?( T M ) are the L -sections of  T M , in contrast to ?( T M ) which denotes the smooth sections. By the spectral theorem, there is a smooth semigroup Pt = e(1=2)t on L2 -?(T  M ) solving the heat equation @ P = 1 P : (5.1) @t t 2 t Note that Pt 2 ?(T  M ) for 2 RL2-?(T  M ) due to elliptic regularity. For R a di erential form 2 ?(T  M ), let X be the Stratonovich integral, and X the It^o integral ofR along XR = X (x) (see [10]). Recall Rthat?  R R 1 1 X = X + 2?R r (dX; RdX ) = X ? 2R  Xs?(x) ds :  r (dX ) = r A ( X ( x ))  dB , = In our situation, we have s s X X ( x ) 0 s 0 ? X r  ?R Rr x)) dBs . Analogously, for the ?\time-dependent" and X r = 0 Xs (x) A(Xs(?R   Rr A ( X ( x ))  dB , ( P ) P = differential forms P , we set . . s s t ? s t ? t ? X ( x ) s 0 X r  ?  Rr ?R and X Pt? . r = 0 (Pt?s )Xs (x) A(Xs(x)) dBs . The following theorem is along the lines of Elworthy-Li [6]. Theorem 5.1 Suppose that the system (1.1) de nes Brownian motion on (M; g ), possibly with nite lifetime. LetZ 2 L2 -?(T  M ) \ dom  with d = 0. Then  (I)  Z r

?  Nr = (Pt?r )Xr (x) Tx Xr hr ? Pt? . r Tx Xs h_ s ; A Xs(x) dBs ; (5.2) X 0 r 2 [0; t] \ [0;  (x)[, is a local martingale for any adapted bounded process h such that h. (!) 2 H ([0; t]; TxM ), almost all !. Proof Again the situation is reduced to Lemma 2.1. By a standard localization argument, e.g., [10], Lemma (3.5), it is enough to check the local martingale property of N on stochastic intervals contained in sets of the form fX 2 Vi g where (Vi )i2I forms an open covering of M . First, since is closed, we get dPr = Pr d = 0. Hence, for each r0 2 [0; t] and x0 2 M there is an open neighbourhood V of x0 such that Pr = dar on V for? all r in some  open interval I about t ? r0 ; moreover ar can be chosen such that @r@ + 21 M ar = 0 on I  V . We may assume that (r; x) 7! ar (x) is bounded on I  V . Now, let [;  [  I be a stochastic interval such that X j[;  [ takes values in V , then on [;  [ ?   ?R d X Pt? . = (dat?r )Xr (x)  dXr + 21 (at?r ) Xr (x) dr ?  ?  = (dat?r )Xr (x)  dXr + (@r at?r ) Xr (x) dr = d at?r  Xr (x) : Thus, N j[;  [ is a local martingale by Lemma 2.1. Corollary 5.2 For any harmonic 1-form 2 ?(T  M ) \ L2 the process Z  Z r

?  Nr = Xr (x) (Tx Xr hr ) ? r Tx Xs h_ s ; A Xs(x) dBs (5.3) X 0 de nes a local martingale, 0  r <  (x). 1

(I)

(I)

(I)

(I)

(I)

(I)

{ 12 {

R (I)

R

Proof Obviously X = X for harmonic 1-forms . R Remark 5.3 Let Qr = ? 12 0r Ps ( ) ds. Then, if  (x) = 1 a.s., we get Z

Z

   (I) Pt? . t = ? (Qt )(x) ; X X t ?  as can be seen by applying It^o's formula to (Qt?s ) Xs(x) , 0  s  t.

(5.4)

With the help of identity (5.4) it is straightforward to recover the corresponding formulae for 1-forms in [7] from Theorem 5.1.

6. Some variations of the di erentiation formulae In this section we rewrite our basic di erentiation formulae (2.6) and (3.1) in terms of the conditional derivative process, as de ned by Elworthy-Yor [9]. The resulting formulae will be intrinsic in the sense that, for xed x, the right-hand sides are given entirely in terms of the di usion X (x), starting at x; they involve no longer the derivative ow which depends on the SDE (1.1), used to obtain the di usion X (x). The idea is to lter out extraneous noise of the local martingale (2.1) by conditioning with respect to the smaller ltration generated by X (x). More precisely, for x 2 M , let Fr (x) := FrX(x)  Xs(x) : 0  s  r : (6.1) For some given v 2 Tx M consider again the derivative process Vr (v) = (Tx Xr ) v. Fix an F. (x)-stopping time  such that V (v) is integrable on [0;  ], i.e., kVr (v) 1fr gk 2 L1 (P) for each r  0, and de ne a TM -valued process W (v) along X (x) by     Wr (v) := E Fr (x) (Tx Xr ) v 1fr g  ==0;r E Fr (x) ==0?;r1 (Tx Xr ) v 1fr g (6.2) where ==0;r : Tx M ! TXr (x) M denotes parallel transport along X (x). Note that, instead of conditioning with respect to Fr (x) in (6.2), we may equivalently take expectations with respect to F (x), or F1 (x). Recall that L = 21 M + Z where Z 2 ?(TM ). Then, as in [9], it can be shown that W (v) satis es the covariant equation 8 < D W (v ) = ? 1 Ric?W (v ); . # + rZ ?W (v ) r r 2 dr r (6.3) : W0 (v) = v along X (x) for r   . (Without loss of generality we may assume that the LeviCivita on M coincides with the Le Jan-Watanabe connection associated to (1.1), D Wr (v ) = == d ==?1 Wr (v ) by de nition; moreover, if w 2 see [8]). Note that dr 0;r dr 0;r Ty M , then Ric(w; . )# 2 Ty M is determined by hRic(w; . )# ; zi = Ric(w; z) for all z 2 Ty M . Let U be a horizontal lift of X (x) to the orthonormal frame bundle : O(M ) ! M , R with respect to the Leviand Z = U0 U # the anti-development? of X (x) in Tx M   n Civita connection, see [10]; here # 2 ? T O(M ) R , #u = u?1 du, u 2 O(M ), { 13 {

is the canonical 1-form of the connection. Thus Zr = 0r ==0?;s1  dXs(x). Let Z r ?  ~ (6.4) Br = ==0?;s1 A Xs(x) dBs 0 ?  be the martingale part of Z ; then A Xs(x) dBs = ==0;s dB~s. On the other hand, it is easily seen that B~ is a Brownian motion on Tx M , stopped at the lifetime  (x) of X (x). The point is that by construction B~ is adapted to the ltration F. (x) generated by X (x). We return to the general situation of Lemma 2.1 and consider the local martingale Z ?  ?  r

?  Nr = da(r; . ) Xr (x) (Tx Xr ) hr ? a r; Xr (x) (Tx Xs ) h_ s; A Xs(x) dBs (6.5) 0 on a stochastic interval [;  [. Here a: I  M0 ! R (with I  R?+ an interval and  M0  M open) is again a transformation such? that theprocess r; Xr (x) on [;  [ takes its values almost surely in I  M0 , and a . ; X.(x) de nes a local martingale; for the required technical properties of a see the beginning of section 2. ?  Lemma 6.1 Let a r; Xr (x) ,   r <  (with  <  predictable stopping times) be a local martingale for some function a as above. Suppose that h is a bounded process with sample paths h. (!) 2 H (I; Tx M ), almost all !, which is already adapted to F. (x). Then Z ?  ?  r

~ . Nr = da(r; ) Xr (x) Wr (hr ) ? a r; Xr (x) Ws(h_ s ); ==0;s dB~s (6.6) 0 is a local martingale on [;  [ ; here W ( . ) is de ned by (6.3) and the Brownian motion B~ is given by (6.4). Proof By Lemma 2.1, the process N , as de ned in (6.5), is a local martingale. Conditioning of N with respect to F. (x) gives the claim. With the help of Lemma 6.1, i.e., by working with N~ instead of N , we can rewrite our basic formulae in an obvious way. For instance, given the assumptions of Theorem 2.4, formula (2.6) reads as Z  (x)^t h?

i  Ws (h_ s ); ==0;s dB~s : (6.7) d(Pt f )xv = ?E f  Xt(x) 1ft 0. { 15 {

Using the covariant equation (6.3) it is easy to get norm estimates for Wr = Wr (v). For instance, let c  0 be such that the following estimate holds: ? Ric (w; w) + 2 rZ (w; w)  c kwk2 ; w 2 Ty M; y 2 D ; (7.2) where rZ (w; w) = hrw Z; wi. Then d

==?1 W

2 = 2 D d ==?1 W ; ==?1 W E dr 0;r r Tx M D dr 0;r r 0;r r E  ?  ?  = 2 ==0?;r1 ? 12 Ric Wr ; . # + rZ Wr (v) ; ==0?;r1 Wr = ? Ric (Wr ; Wr ) + 2rZ (Wr ; Wr ) : In other words,

Wr 2

= kW0k2 +

r

Z

0



? Ric (Ws; Ws) + 2 rZ (Ws; Ws) ds :

Thus, if kW0 k = kvk 6= 0, we get Z

kWr k2 = kW0k2 exp

r

0



? Ric (W^ s; W^ s) + 2 rZ (W^ s; W^ s) ds



(7.3)

where W^ r = Wr =kWr k. Together with (7.2), the last equation gives kWr k2  kvk2 ecr :

(7.4)

 0, and suppose that Ric  ?Cg on D for some C  0, where g is the Riemannian metric on M . Then, for r   (x), kWr (v)k  kvk e1=2 Cr :

Example 7.2 Let Z

Given the situation of Theorem 7.1, we get a straightforward estimate for any nonnegative function u 2 C (D ) which is L-harmonic on D as follows:

(grad u)x ; v 2

h ?

 E u X (x) (x) 

 u(x) sup juj 

@D

 u(x) sup juj @D

2 i

 

 (x)

Z

E

0

 (x)

Z

E Z

E

0

2 

Ws (h_ s ); ==0;s dB~s

Ws (h_ s ) 2 ds

 (x)

0





kh_ sk2 ecs ds :

Summarizing this argument, we veri ed the following general estimate for the gradient of harmonic functions on regular domains in a Riemannian manifolds. Corollary 7.3 Let u 2 C (D ) be a nonnegative function which is L-harmonic on D. Let KZ be the smallest constant such that (7.2) holds. Then

(grad u)x ; v



 1=2  u(x)1=2 sup juj inf E h

@D

 (x)

Z

0

 1=2

kh_ sk2 eKZ s ds

(7.5)

where the in mum is taken over all bounded F. (x)-adapted processes h such that h. 2 H (R + ; Tx M ), h0 = v, and hs  0 for s   (x), a.s. We are not going to exploit formula (7.5) here further. For explicit estimates, using the described method, the reader is referred to [15]. { 16 {

8. Concluding remarks The assumptions of Theorem 2.4 can be slightly weakened when combined with the estimates for the covariant equation (6.3) as given in the previous section. More precisely, we have the following result for heat semigroups associated to (1.1). Now, we assume that M with the induced Riemannian metric is complete.

! R bounded measurable, x 2 M , and v 2 Tx M . Then, for any bounded F ( x )-adapted process h with sample paths in H (R + ; Tx M ) such .  ?R D (x)^t kh_ (s)k2 ds 1=2 2 L1 , and the property that h(0) = v, h(s) = 0 for that 0 all s  D ^ t, the following formula holds:

Theorem 8.1 Let f : M

Z D (x)^t

h

i

?  hd(Ptf )x ; vi = ?E f  Xt (x) 1ft 0 and with h replaced by h" , see part 2) in the proof of Theorem 2.3. The desired formula then follows as " ! 0. Next, by Lemma 6.1, Z ? ?  r

Nr = d(Pt?r f )Xr (x) Wr (hr ) ? Pt?r f Xr (x) Ws (h_ s); ==0;s dB~s 0

is a local martingale for 0  r < D (x) ^ t. Since kWr (v)k  kvk eKZ r=2 for r  D (x), we conclude that (Nr^D (x) ), r 2 [0; t], is already a martingale under the given assumptions; on a complete Riemannian manifold d(Psf )x is bounded for s  t, x 2 M , e.g. [2] or [15]. This implies h ?



Z D (x)^t

hd(Ptf )x ; vi = ?E f Xt (x) 1ft