On the differentiation of heat semigroups and Poisson integrals

On the Dierentiation 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 dieren- tiation 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 mani- folds 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^2N, let A: M ^{Rm !}TM ; (x;e)^7!A(x)e;

be a homomorphism of vector bundles over M. Thus, A²?(^Rm TM), i.e., the map A(x): ^{Rm !} TxM is linear for x ² M, and A(

.

_)e ² _?(TM) is a smooth vector eld on M for e ^{2 Rm}. Consider the Stratonovich stochastic dierential equation

dX =A(X)^dB+A⁰(X)dt (1.1) where A^{0 2} ?(TM) is an additional vector eld, and B an ^Rm-valued Brownian motion on a ltered probability space ^?;^F;^P;(^Ft)t^2R+
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 notationXt(x;!) =Xt(x)(!) and

(x;!) =(x)(!), if

Mt(!) =^fx²M :t < (x;!)^g

then there exists a set ^{0 } of full measure such that for all ! ²⁰:

1991 Mathematics Subject Classication. 58G32, 60H10, 60H30.

Key words and phrases. Diusion, heat semigroup, integration by parts, heat kernel, Poisson kernel.

{1{

(i) M_t(!) is open in M for each t ^ 0, i.e. (

.

_;!) is lower semicontinuous on M.

(ii) Xt(

.

_;!):Mt(!)^!_M is a dieomorphism onto an open subset of M. (iii) The map s^7!Xs(

.

_;!) is continuous from [0;t] intoC^1?Mt(!);M
with its

C¹-topology, for each t >0.

The solution processes X =X(x) to (1.1) are diusions on M with generator L =A⁰+ ^{12 Pm}

i⁼¹A²_i

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): ^{Rm !} TxM is surjective for each x, or equivalently thatL is elliptic. This non-degeneracy provides a Rieman- nian metric on M such that A(x)A(x)^: TxM ^!TxM is the identity on TxM for x²M. In other words, A(x)^: TxM ^!Rm denes an isometric inclusion for each x²M, i.e.,

hu;vⁱ_TxM =^hA(x)^u;A(x)^v^iRm for all u;v ²TxM :

With respect to this Riemannian metric, L= ¹²
_M +Z where Z is of rst order, i.e. a vector eld on M. Standard examples are the gradient Brownian systems whenM is immersed into some Euclidean space ^Rm, andA(x): ^{Rm !}TxM 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 TxXt: TxM ^! TXt⁽x⁾M be the dierential of Xt(

.

) at x (well- dened for all ! ² such that x ² Mt(!)) and Vt = Vt(v) = (TxXt)v the derivative process to Xt(

.

) at x in the direction v ² TxM. It is well-known that V on TM solves the formally dierentiated SDE (1.1), i.e.,

dV = (T_XA)V ^dB+ (T_XA⁰)V dt; V⁰ =v ; (1.2) with the same lifetime as X(x), if v ⁶= 0. Using the metric and the correspond- ing 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?1_;t(^rA)==⁰_;tV^{~ }dB+==^0?1_;t(^rA⁰)==⁰_;tV dt^~

for ~Vt ===^0?1_;tVt where ==⁰_;t: TX⁰M ^!TXtM 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). Forf ²bC¹(M) (boundedC¹ functions with bounded rst derivative) let

(Ptf)(x) =^E?f ^Xt(x)
^; x²M; (1.4) be the semigroup associated to (1.1), and

P_t⁽¹⁾(df)_xv =^E(df)_Xt⁽_x⁾(T_xX_t)v^; v ²T_xM ; (1.5) its formal derivative whenever the right-hand side exists. More generally, for a (bounded) dierential form ²?(T^M) let

P_t⁽¹⁾() =^E[X_t^]; (1.6) {2{

provided the right-hand side of (1.6) is well-dened; here X_t^ is the pullback of under the (random) map Xt: M ^!M.

Further, for x²M and I = [0;t] orI =^R+ let

H(I;T_xM) =^ : I ^!T_xM absolutely continuous, ^k _^k2L²(I;ds) be the Cameron-Martin space and ^H0(I;TxM) =^{f 2H}(I;TxM) : (0) = 0^g. 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 ²bC¹(M). Then

E

h(df)_Xt⁽_x⁾(TxXt)ht

i=^Eh?f^Xt(x)

Z t

0

(TxXs) _hs;A^?Xs(x)
dBs

i (1.7) for each bounded adapted processhwith sample paths in^H0([0;t];TxM) such that

E

sup^0_s^_td(P_t^?_sf)_Xs⁽_x⁾(T_xX_s)h_s^<¹, and with the additional property that

Rr

0

(TxXs) _hs;A^?Xs(x)
dBs, 0^r^t, is a martingale.

Proof Let h be an adapted bounded process with h

.

^{(!) 2 H([0;t];TxM), almost}

all !. It will be shown in Lemma 2.1 below that Nr =d(Pt^?rf)_Xr⁽_x⁾(TxXr)hr

?

?Pt^?rf
^?Xr(x)
^{Z r}

0

(TxXs) _hs;A^?Xs(x)
dBs (1.8) 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 TxM, or more generally,h_s = (s^{^}"=")vwith some constant 0 < "^t. Then, under the assumptions of Theorem 1.1,

E

h(df)Xt⁽x⁾(TxXt)vⁱ=^Eh?f ^Xt(x)
1

"

Z "

0

(TxXs)v;A^?Xs(x)
dBs

i: (1.9) In general, ifhin Theorem 1.1 has the property thatht =v, then we getP_t⁽¹⁾(df)xv for the left-hand side in (1.7) while the right-hand side represents d(P_tf)_xv as will be shown in Theorem 2.4 below. Thus, in this case,d(P_tf)_x =P_t⁽¹⁾(df)_x is already a consequence of (1.7).

Note that dierentiating (1.4) by taking derivatives under the expectation requires dierentiability of f. However, due to the smoothing property of the semigroup, Ptf is already dierentiable even iff 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

(Ptf)(x) =^E?f ^Xt(x)
1^f_t<⁽_x^)g (1.10) where dierentiation under the integral is no longer possible even for smooth f. An appropriate generalization of (1.5) is

P_t⁽¹⁾(df)xv=^E(df)_Xt⁽_x⁾(TxXt)v1^f_t<⁽_x^)g; v²TxM : (1.11) {3{

From a stochastic point of view, there seems to be no obvious reason why (1.11) should be the derivative of (1.10) in the direction v, i.e.,

dPtf =P_t⁽¹⁾(df): (1.12) Of course, formula (1.12) cannot hold forf ^1 unless the system (1.1) is complete (non-explosive).

Even more fundamental problems occur when dealing with boundaries where the process needs to be stopped when exiting a given domain. The situation is best illustrated in the case of the Dirichlet problem. Suppose that D is an open (rela- tively compact) domain in M with D⁶=M. Let

u(x) =^E[f ^X⁽_x⁾(x)] (1.13) where(x) denotes the rst exit time ofX(x) fromD. Thenuis dierentiable (and L-harmonic) on D whereas x ^7!X⁽_x⁾(x) is not even continuous with probability one. The non-continuity follows from purely topological reasons, since there is no continuous retraction of D to the boundary @D. Thus, there is denitely no way of dierentiating (1.13) by taking derivatives under the integral.

In this paper we shall extend integration by parts and derivative formulae in var- ious directions to cover situations where nite lifetime or stopping times resulting from boundary conditions are involved. Specically, we develop formulae for the dierentiation of (1.10) and (1.13) not involving any derivatives of f. Analogously to Bismut type formulae for the logarithmic derivative of the heat kernel, we get similar formulae for the Poisson kernel.

Our methods are inspired by the notion of quasiderivatives in the sense of Krylov [11]. The following fact is elementary but crucial for our approach: If a local mar- tingale depends on a parameter and is dierentiable with respect to this parameter in probability uniformly on compact time intervals, then its derivative is also a local martingale.

2. A basic formula for the derivative of a heat semigroup

We start by explaining our basic strategy for proving integration by parts and derivative formulae; see also [7]. Let X be again the partial ow associated to the non-degenerate system (1.1). Suppose that, for somex ²M, a process of the form Y(x):Y_r(x) = a^?r;X_r(x)
provides a local martingale. We assume that Y(x) is dened on a stochastic interval [;[ such that

?r;Xr(x;!)
:(!)^r < (!) ^I^M⁰; almost all !,

wherea: I ^M^{0 !R} such thata(t;

.

₎²_C¹_(M⁰_{) for t}²_I with jointly continuous derivative (t;x)^7!da(t;

.

_{)x; here I} ^R+ is an interval and M^{0 }M open.

Such a situation is typically given whenais someC² time-space harmonic function so that @ra+La= 0. In this paper we have mainly two cases in mind, namely

(i)  ^t, and a(r;y) = (Pt^?rf)(y), for some bounded measurable f on M, (ii)  ^(x), where(x) is the rst exit time ofX(x) from a bounded domainD,

a(r;y) =u(y) withu ²C²(D) and Lu= 0.

{4{

Lemma 2.1 (Integration by parts on the local martingale level) Let a^?r;X_r(x)
,

 ^r <  (with <  predictable stopping times) be a local martingale for some function a having the above properties. Then

Nr =^?da(r;

.

_)
Xr⁽_x⁾_(TxXr)hr^?_a^?_r;Xr(x)
^{Z r}

0

(TxXs) _hs;A^?Xs(x)
dBs (2.1) ( ^ r < ) is also a local martingale for any bounded adapted process h with sample paths h

.

^(!)2H(I;TxM) for almost all !.

Proof For r ^0, letH_"r:M ^!M be the pathwise dened solution to

( @@"H_"r(x) =A^?H_"r(x)
A(x)^hr

H_r⁰(x) =x: ^(2.2)

Set X_"r(x) = Xr^?H_"r(x)
; then in particular X_r⁰(x) = Xr(x). The perturbed process X^" satises

dX^"=A(X^")^dB+A⁰(X^")dr+ (TXr)dH_"r with dH_"r=^? _@@rH_"r
dr = _H_"rdr, see [12]. Hence

dX^"(x) =A^?X^"(x)
^dB+A^?X^"(x)
^(T_H"r⁽_x⁾Xr)dH_"r(x)^+A^0?X^"(x)
dr : We observe that this is an SDE of the same type as (1.1) but with the perturbed driving process dB^"(x) = dB +A^?X^"(x)
^(TH_"r⁽x⁾Xr)dH_"r(x). Roughly speak- ing, the next step is to compensate this perturbation by changing the measure according to Girsanov-Maruyama. More precisely, set

M_"r=^{?Z r}

0

A(X_"s)^(T_H"sX_s) _H_"s;dB_s (2.3) andG_"r = exp^?M_"r^?21[M^"]r
. Then, for any stopping time <  with the property that the exponential^?G_"r^{^}_(x)
_r^0 is a martingale,B^"(x)^j[0;] is a Brownian mo- tion on [0;] with respect to the measureG_"(x)
^P. Hence, by pathwise uniqueness of solutions to (1.1), if Yr(x) = a^?r;Xr(x)
is a (local) martingale on [;[ then also Y_"r(x) :=a(r;X_"r(x))G_"r(x) is a local martingale on [;[, both with respect to the measure ^P. Consequently, also

@"@





"⁼⁰Y_"r(x) =^?da(r;

.

_)
Xr⁽_x⁾_(TxXr)hr+a^?_r;Xr(x)
@"^@ _"

=0

G_"r(x); for  ^r < , is a local martingale. Using _H_s⁰ = 0 and @@s @@""⁼⁰H_"s = _hs, we get

@"@





"⁼⁰G_"r(x) =^? @

@"





"⁼⁰

Z r

0

(T_H"s⁽_x⁾Xs) _H_"s(x);A^?X_"s(x)
dBs

=^?

Z r

0

(TxXs) @

@"





"⁼⁰H__"s(x);A^?Xs(x)
dBs

=^?

Z r

0

(TxXs) _hs;A^?Xs(x)
dBs:

(2.4)

Thus Nr = _@@""⁼⁰Y_"r(x) where Nr is dened by (2.1). This shows that N is a local martingale on [;[.

We shall exploit Lemma 2.1 for various choices of transformations a and proces- sesh. An essential observation is that (before taking expectations in (2.1) with an appropriateh) there is still the possibility of applying the optional sampling theo- rem to the local martingale (2.1). This fact allows one to deal with stopping times in the derivative formulae which take into account given boundary conditions.

{5{

Remark 2.2 In the notation of Krylov [11] the local martingale property of (2.1) means that r := (TxXr)hr is a quasiderivative of Xr(x) in the direction h⁰ =v, and ⁰_r :=^?R0r(TxXs) _hs;A^?Xs(x)
dBs its adjoint process.

Now, let f ²B(M) (bounded measurable functions on M), and suppose that the local martingaleNr, 0^r^t, as given by (1.8), is already a martingale; moreover suppose that h⁰ =v ²TxM andht = 0. Then ^EN⁰ =^ENt, in other words,

d(Ptf)xv=^?Eh?f^Xt(x)

Z t

0

(TxXs) _hs;A^?Xs(x)
dBs

i: For instance, choosing h_r =^?1^?r^{^}"="
v (where 0^" ^t), we get

d(Ptf)xv=^Eh?f^Xt(x)
1

"

Z "

0

(TxXs)v;A^?Xs(x)
dBs

i

provided (1.8) is actually a martingale for this choice ofh. The latter question can be reduced to integrability conditions on the derivative processTxXr (see [6]). Ob- viously also an appropriate choice forh may be helpful to make (1.8) a martingale.

We follow this idea in the next theorem.

Theorem 2.3 Assume that (1.1) is complete and non-degenerate. Letf ²B(M), t >0. Then

d(P_tf)_xv=^?Eh?f^X_t(x)

Z t

0

(T_xX_s) _h_s;A^?X_s(x)
dB_sⁱ (2.5) holds for any bounded adapted process h with sample paths in ^H(^R+;TxM) such that ^{?R0(x)^tk}h_s^k2ds
^{1=2 2} L^1+" for some " > 0, and the property that h⁰ = v, h_s = 0 for all s ^ (x)^{^} t; here (x) is the rst exit time of X(x) from an (arbitrarily chosen) relatively compact neighbourhood D of x.

Proof 1) We rst assume f ² C¹(M). In this case ^kd(Psf)_x^k is bounded for (s;x) ² [0;t]^D. Now, let N⁰ be the local martingale (1.8) stopped at ⁰ =

(x) ^{^}t, i.e. N_r⁰ := N^{0^}r, r ^ 0. It suces to show that N⁰ is already a martingale. Namely, then

d(Ptf)xv=^EN⁰ =^?Eh(Pt^?0f)^?X⁰(x)

Z ⁰

0

(TxXs) _hs;A^?Xs(x)
dBs

i; and (2.5) follows from the Markov property (Pt^?0f)^?X⁰(x)
=^EF0[f^Xt(x)].

To check the martingale property ofN⁰, we rst note that sup^0_s^0kTxXs^k2L^p for any 1 ^ p < ¹: for this we may assume that M is already compact, since otherwise M can be modied outside of D without changing T_xX_s for s^{ 0}; on compact manifolds the above integrability of the derivative process is well-known, e.g. [14]. Using this integrability of the derivative process the stochastic integral in (1.8) can be estimated by means of Burkholder-Davis-Gundy and Holder's in- equality:

E 



 Z 

0

A^?Xs(x)
^(TxXs) _hs;dBs







c^{EZ 0}

0

(TxXs) _hs ²ds^1=2

c^hE sup

0s^0kT_xX_s^{k1+""i1+""
hEZ 0}

0

kh__s^k2ds^{1+2"i1+1" }const<¹ for any stopping time  ^0. This veries that N⁰ is indeed a martingale.

{6{

2) In case f ²B(M) only, we use that ^kd(P_sf)_x^k is bounded for (s;x)²[";t]^D if " > 0. However, depending on ", the process h may be modied such that h_"s = h_s for s ^ (x)^{^}(t^?") and h_"s = 0 for s ^ (x)^{^}(t^?"=2), and cutting o appropriately between. Then the arguments used in 1) carry over to give (2.5) with h replaced by h^". Finally, the claimed formula follows by" ^!0.

Note that in Theorem 2.3 the condition ^?R0tkh__s^k2ds
^{1=2 2} L^1+" guarantees that

R⁽x^{)^}r

0

(TxXs) _hs;A^?Xs(x)
dBs, 0 ^ r ^ t, is a martingale, i.e. it assures the uniform integrability of

R

0

(TxXs) _hs;A^?Xs(x)
dBs : 0^ ^(x)^{^}t;  stopping time : The same strategy as above can be applied to get derivative formulae for the heat semigroup in cases when (1.1) is explosive. More precisely, we have the following result.

Theorem 2.4 Let (P_tf)(x) =^E?f^X_t(x)
1^f_t<⁽_x^)gbe the minimal semigroup associated to (1.1) acting on bounded measurable functions f: M ^!R. Then

d(P_tf)_xv =^?Eh?f ^X_t(x)
1^f_t<⁽_x^{)gZ (x)^t}

0

(T_xX_s) _h_s;A^?X_s(x)
dB_sⁱ (2.6) for any bounded adapted process h with sample paths in ^H(^R+;TxM) such that

?R⁽x^{)^}t

0

kh_s^k2ds
^{1=2 2}L^1+" for some" >0, and the property that h⁰ =v, hs = 0 for all s ^ (x) ^{^}t; here (x) is again the rst exit time of X(x) from some relatively compact neighbourhood D of x.

Proof If ⁰ = (x)^{^}t, then (P_t^?0f)^?X⁰(x)
= ^EF0?f ^X_t(x)
1^f_t<⁽_x^)g by the Markov property. The rest of the proof of Theorem 2.3 carries over verbatim to give (2.6).

From (2.6) a Bismut type formula can be derived for the transition kernel associ- ated to (1.1).

Corollary 2.5 Letp(t;

.

_;

.

_{): M} ^_M ^!R+_{,t >}0, be the (smooth) heat kernel (with respect to the Riemannian volume) associated to (1.1) such that

(Ptf)(x) =^E?f ^Xt(x)
1^ft<⁽x^)g=

Z

Mp(t;x;y)f(y)vol(dy) for any f ²B(M). Then

d^?logp(t;

.

_;y)
_xv=^{?EhZ (x)^t}

0

(TxXs) _hs;A^?Xs(x)
dBs





Xt(x) =yⁱ (2.7) with h and (x) as in Theorem 2.4.

Proof Let f ² C(M) of compact support. By the smoothness of p(t;

.

_;

.

_{) for} t >0, we can dierentiate under the integral to obtain

d(Ptf)xv =

Z

dp(t;

.

_{;y)xv f(y)vol(dy):}

{7{

On the other hand, (2.6) may be rewritten as d(Ptf)_xv =^?Z p(t;x;y)f(y)^{EhZ 0}

0

(TxXs) _hs;A^?Xs(x)
dBs





Xt(x)=yⁱvol(dy) with ⁰ =(x)^{^}t. Comparing the last two equations proves formula (2.7).

We conclude this section with some remarks on dierentiation under the expecta- tion, more precisely, for instance, on the question under which conditions

dPtf =P_t⁽¹⁾(df) (2.8)

holds for f ²bC¹(M). As above, let (Ptf)(

.

_{) =} ^E?_f ^_Xt(

.

_)
1^f_t<⁽

.

^)g be the minimal semigroup associated to our (possibly explosive) system (1.1), whereas P_t⁽¹⁾(df) is given by (1.11). To make P_t⁽¹⁾(df) well-dened we assume that

(df)_Xt⁽_x⁾(TxXt)v1^f_t<⁽_x^{)g 2}L¹(^P):

Fixing a bounded adapted process h such that h

.

^{(!) 2 H([0;t];TxM}) for almost all !, we know that

Nr ^N_r^(h)=d(Pt^?rf)_Xr⁽_x⁾(TxXr)hr

?

?Pt^?rf
^?Xr(x)

Z r

0

(TxXs) _hs;A^?Xs(x)
dBs (2.9) denes a local martingale on the stochastic interval [0;t^{^}(x)[. Crucial for formula (2.8) to hold are basically two things: rstly, N^(h) is required to be a uniformly integrable martingale for certain choices of h, and secondly, we need to know that d(Ptf)x ^!0 suciently fast, asx^!1in the one-point-compacticationM^[f1g of M.

Theorem 2.6 Let h be a bounded adapted process with paths in ^H0([0;t];TxM) such that ht =v. Given the above setting, suppose that a.s.

d(Pt^?rf)_Xr⁽_x⁾(TxXr)v^!0 on ^f(x)^t^g asr ^%(x). (2.10) If N^(h) denes a martingale, then

P_t⁽¹⁾(df)v=^Eh?f ^Xt(x)
1^f_t<⁽_x^)g

Z t

0

(TxXs) _hs;A^?Xs(x)
dBs

i: If N^(h0) denes a martingale where h⁰_s =v^?hs, then

d(Ptf)xv =^Eh?f ^Xt(x)
1^ft<⁽x^)g

Z t

0

(TxXs) _hs;A^?Xs(x)
dBs

i:

Proof The assertions follow from evaluating ^E[N⁰] = ^E[lim_r^%_t^{^}_⁽_x⁾Nr], rst for N =N^(h), and then for N =N^(h0).

Remark 2.7 Keeping the notations of Theorem 2.6, we get the following criterion for d(Ptf) = P_t⁽¹⁾(df). Suppose that assumption (2.10) holds (which is void for conservative systems). Moreover, suppose that

Nr ^N_r^(v) =d(Pt^?rf)_Xr⁽_x⁾(TxXr)v ; 0^r ^t^{^}(x); is already a martingale. Note that N^(v)N^(h) for h ^v. Then

d(Ptf)xv=P_t⁽¹⁾(df)v ; which is seen again by taking expectations of N^(v).

{8{

We want to stress that for explosive systems the domain of P_t⁽¹⁾ on 1-forms, i.e.

(P_t⁽¹⁾)xv =^E_Xt⁽_x⁾(TxXt)v1^f_t<⁽_x^)g; (2.11) generally does not include bounded forms²?(T^M), since ^k(TxXt)v^k1^f_t<⁽_x^)g will not be integrable in this case [13]. More precisely, we have the following.

Remark 2.8 Suppose that on a complete Riemannian manifold M relation (2.8) holds for all f ² C¹(M) with compact support. If the derivative process has rst moments, e.g.,^Ek(T_xX_t)1^f_t<⁽_x^)gk<¹ for all x in some open setU and all t ^ t⁰ where t⁰ > 0, then the system (1.1) is already non-explosive [13]. Indeed, on any (geodesically) complete Riemannian manifold M one can construct an increasing sequence (fn) of nonnegative smooth functions of compact support such that fn ^% 1 and ^kdfn^{k1 } 1=n for each n. Then d(Ptfn) ^! d(Pt1) on M, by standard Schauder type estimates. However, for x²U and t ^t⁰,

d(Ptfn)x = P_t⁽¹⁾(dfn)x ^ dfn ^1E (TxXt)1^f_t<⁽_x^{)g !}0:

Thus, if u(t;x) = (Pt1)(x) = ^Pft < (x)^g then u ^ 1 on [0;t⁰]^U, and nally Pt1^1.

3. The dierentiation of Poisson integrals

We consider again a non-degenerate SDE of the type (1.1). Let D ^ M be a nonvoid relatively compact open subset with D⁶=M, and

(x) = inf^ft ^0 :Xt(x)⁶²D^g

the rst exit time of X from D when started at x ² D. For ' ² C(@D) let u(x) =^E['^X⁽_x⁾(x)]. Then Lu= 0 on D.

Theorem 3.1 Assume that (1.1) is non-degenerate. Let u(x) =^E'^X⁽_x⁾(x)^. Then

(du)xv=^?Eh?'^X⁽x⁾(x)

Z ⁽x⁾

0

(TxXs) _hs;A^?Xs(x)
dBs

i (3.1) for any bounded adapted process h such that h

.

^{2 H(R+;TxM), h0 = v, and} hs ^ 0 for s ^ (x), almost surely, provided ^{R0(x)^r}(TxXs) _hs;A^?Xs(x)
dBs, r^0, is a uniformly integrable martingale.

Proof Note thatu^X(x) is a martingale on [0;(x)], in particular a local martin- gale on [0;(x)[. By Lemma 2.1, also

Nr = (du)_Xr⁽_x⁾(TxXr)hr^?u^Xr(x)

Z r

0

(TxXs) _hs;A^?Xs(x)
dBs

(0 ^ r < (x)) is a local martingale. Since D is compact, both u and du are bounded; moreover sup^0_s^(_x^)kTxXs^{k 2} L^p for any 1 ^ p < ¹. Using these properties it is easily checked that N is already a martingale on [0;(x)]. The assertion follows then by taking expectations.

Note that, in the situation of Theorem 3.1, the process

R⁽x^{)^}t

0

(T_xX_s) _h_s;A^?X_s(x)
dB_s; t ^0; (3.2) is a uniformly martingale if for instance^?R0(x)kh_s^k2ds
^{1=2 2}L^1+"for some" > 0.

Using sup^0_s^(_x^)kTxXs^k2L^p for any 1^p <¹, the stochastic integrals in (3.2) {9{

can be estimated by means of Burkholder-Davis-Gundy and Holder's inequality, as in the proof of Theorem 2.3.

Corollary 3.2 Assume that D ^ M is a nonvoid relatively compact open do- main with D ⁶= M and with smooth boundary. Let p: D^@D ^{! R+} be the (smooth) Poisson kernel (with respect to the induced surface measure  on @D) so that

PfX⁽x⁾(x)² dz^g=p(x;z)(dz): Then

d^?logp(

.

_{;z)
xv =}^{?EhZ (x)}

0

(T_xX_s) _h_s;A^?X_s(x)
dB_sX⁽_x⁾(x) =zⁱ (3.3) where h with sample paths in the Cameron-Martin space is as in Theorem 3.1.

Proof For '² C(@M), let u(x) = ^E'^X⁽_x⁾(x)^. We dierentiate u under the integral to obtain

(du)xv=

Z

@Mdp(

.

_{;z)xv '(z)(dz):}

On the other hand, by rewriting (3.1) we get (du)xv=^?

Z

p(x;z)'(z)^Eh

Z ⁽x⁾

0

(TxXs) _hs;A^?Xs(x)
dBs





X⁽_x⁾(x) =zⁱ(dz): Comparing the last two equations gives formula (3.3).

4. Remarks on the choice of the process

h

Let Dbe an open (relatively compact) domain in M. Givenx²D and v²T_xM, most of our formulae require the choice of a bounded adapted process h with sample paths in ^H(^R+;T_xM) such that, e.g.,h⁰ = 0 and h_s=v for s^(x), and the property that ^?R0(x)kh_s^k2ds
^{1=2 2}L^1+" for some " >0; here (x) is the rst exit time of X from D when starting atx. We describe a method of constructing such processes; see [15] for more details.

Suppose that D has smooth boundary. We take f ² C²( D) with f^j@D = 0 and f >0 inD. Further xx²D and write instead of(x). Consider the increasing process

T(t) =^{Z t}

0

f^?2?X_s(x)
ds; t^ ; and (t) = inf^fs ^0 :T(s)> t^g; t ^T():

Obviously T^?(t)
=t for t ^T(), and ^?T(t)
=t for t ^. Since X

.

^{(x) is an} L-diusion with generatorL = ¹²
+Z, the time-changed process ~Xt(x) =X_⁽_t⁾(x) is an ~L-diusion where ~L = f²L. The following lemma shows that ~L-diusions on D have innite lifetime. As a consequence, we get T() =¹ a.s.

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Lemma 4.1 Let ~X be an ~L-diusion on D with ~X⁰ =x²D. Then

~^inf^fs^0 : ~X_s ²@D^g=¹; a.s.

Proof Recall that ~X is a ~L-diusion if, for any '²C²(D), '^X^~t ^?'(x)^?

Z t

0

L'~ ( ~Xs)ds; 0^t^ ;~

is local martingale. For n ^ 1, let n = inf^fs ^ 0 : f( ~Xs) ^ 1=n^g, and choose n^{0 } 1 such that f(x)^ 1=n⁰. Note that ~Lf^?1 = ^?Lf +f^?1kgradf^{k2 } cf^?1 for some constant c=c(f). Thus

E^xf^?1( ~Xt^{^}n)^f^?1(x)e^ct; t^0; n^n⁰: But ^Ef^?1( ~Xt^{^}n)^n^Pfn < t^g, hence

Pfn < t^gn^?1f^?1(x)e^ct:

Therefore, ^Pf < t~ ^g= 0 for any t^0. This proves the Lemma.

Now, for xed t⁰ >0, let hs =vt¹⁰

Z s

0

f^?2?Xr(x)
1^f_r<⁽_t^0)gdr : (4.1) Then, for s^ (t⁰),

h_s=h^?(t⁰)
=vt¹⁰

Z ⁽t⁰⁾

0

f^?2?X_r(x)
dr =v :

It remains to verify that^?R0(t0)kh__s^k2ds
^{1=2 2}L^1+" for some" >0. For instance, we may take "= 1. Obviously,

Z ⁽t⁰⁾

0

kh_s^k2ds=^kv^k2t¹²⁰

Z ⁽t⁰⁾

0

f^?4?Xs(x)
ds=^kv^k2t¹²⁰

Z t⁰

0

f^?2?X_⁽_s⁾(x)
ds:

Recall that X_⁽_s⁾(x) = ~Xs(x) and df^?2( ~Xs) =dNs+ ~Lf^?2( ~Xs)ds where (Ns) is a local martingale. But ~Lf^?2 =f²Lf^?2 =f^23f^?4kgradf^k2?2f^?3Lf^cf^?2 for some constant c=c(f), hence

E

f^?2?X^~s(x)
^f^?2(x)e^cs; and thus, as claimed,

E h

Z ⁽t⁰⁾

0

f^?4?Xs(x)
dsⁱ =^Eh

Z t⁰

0

f^?2?X^~s(x)
ds^{i }f^?2(x)e^{ct0 ?}1

t⁰ <¹: Note that the process h, as dened in (4.1), depends on t⁰ and f. For any t⁰ >0 and any f ²C²( D) with f^j@D = 0 and f >0 in D, formula (4.1) gives a process with the required properties.

5. Extensions to closed dierential 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 toh-Brownian motion (see [6])

{11{