Functional inequalities for Feynman-Kac semigroups

Functional inequalities for Feynman-Kac semigroups

James Thompson 15th April 2019

Abstract

Using the tools of stochastic analysis, we prove various gradient estimates and Har- nack inequalities for Feynman-Kac semigroups with possibly unbounded potentials.

One of the main results is a derivative formula which can be used to characterize a lower bound on Ricci curvature using a potential.

Keywords: Feynman-Kac, Gradient, Harnack, Schrödinger AMS MSC 2010: 47D08; 58J65; 58Z05; 60H30

1 Introduction

Suppose M is a geodesically and stochastically complete and connected Riemannian manifold of dimensionn, with Levi-Civita connection∇and Laplace-Beltrami operator

∆. SupposeV ∈L²_loc(M)is bounded below. Then the operatorH :=−¹₂∆ +V is essen- tially self-adjoint [3] and bounded below. Essential self-adjointness means thatH can be uniquely extended from the domainC₀^∞(M)of smooth functions with compact support to a self-adjoint operator H with maximal domainD(H) = {f : f, Hf ∈ L²(M)}. The corresponding semigroup generated by−H consists of bounded self-adjoint operators onL²(M). Under certain conditions onV the semigroup is continuous and given by the Feynman-Kac formula

P_t^Vf(x) =E[Vt^xf(Xt(x))]. HereX(x)is a Brownian motion onM starting atxand

V^xt :=e^{−R0tV(Xs(x))ds}

forx∈ M and t ≥0. A class of potentials that is of particular interest is thoseV for which

lim

t↓0 sup

x∈ME Z t

0

|V(X_s(x))|ds

= 0.

Such V is said to belong to the Kato class, first introduced by Kato in [8] using an equivalent definition. The local Kato class consists of those potentialsV for which1_KV belongs to the Kato class for any compact set K ⊂ M. This is a very large class of potentials, encompassing nearly all physically relevant phenomena, for which one can still expect the Feynman-Kac formula to make sense pointwise for allx∈M. We refer to Aizenman and Simon [1] and Simon [12] for a comprehensive account of all this, and to the more recent work of Güneysu [7] for generalizations to Schrödinger type operators on vector bundles.

Our focus will be on the derivative, or gradient, of the Feynman-Kac semigroupP_t^Vf. For the heat semigroupPtf, a probabilistic formula for the derivativedPtf, not involving the derivative of f, was proved by Bismut in [2] using techniques from Malliavin cal- culus. A transparent generalization of this, using an argument based on martingales,

was later given by Elworthy and Li in [6] and developed further by Thalmaier in [14].

Thalmaier’s formulation is based on local martingales and allows to localize the contri- bution of Ricci curvature. A formula for dP_t^Vf was proved by Elworthy and Li in [6], some applications of which were recently explored in [10].

The present work is a continuation of the author’s recent article [16], in which local- ized versions of the derivative formula were proved, together with an extension to the Hessian. In all cases, these derivative formulae involve the derivative ofV. Therefore, as in [6] and [16], we will assume throughout thatV is continuously differentiable.

In Section 2, we start by using the derivative formula from [16] to obtain explicit gradient estimates for P_t^Vf. These estimates are then used to derive Theorem 3.2, which states that if

sup

x∈ME Z t

0

|dV|(Xs(x))ds

<∞ (1)

with the Ricci curvature ofM bounded below, then (dP_t^Vf)(v) =E

V^xt((df)_Xt(x)(//_tQ_tv)−f(X_t(x)) Z t

0

(dV)_Xs(x)(//_sQ_sv)ds)

for allf ∈C_b¹(M), x∈M, v ∈TxM andt ≥0. The compositionQ//⁻¹ is the damped parallel transport along the paths ofX(x), determined by the solution to equation (3) below, andC_b¹(M) denotes the space of all bounded continuously differentiable func- tions onM with bounded derivative. Note that condition (1) is satisfied if|dV|belongs to the Kato class.

One application of this derivative formula is that it allows to characterize a lower bound on Ricci curvature using the potentialV, as explained in Section 3. For example, supposeV ∈C¹(M)is bounded below with ∇V bounded andK ∈R. Then, according to Theorem 3.3, the following are equivalent:

(1) Ric≥2K;

(2) iff ∈C_b¹(M)then

|∇P_t^Vf| ≤e^−KtP_t^V|∇f|+k∇Vk_∞

1−e^−Kt K

P_t^V|f|

for allt≥0.

In particular, if the above gradient estimate is verified for some suitableV then it must hold for all suchV.

The derivative formula stated above is also applied in Section 4 to prove a Harnack inequality, given by Theorem 4.1. Afterwards, in Section 5, we prove the counterpart to the Bismut formula for differentiation inside the semigroup, namely Theorem 5.3, and use it to derive two further derivative estimates, Propositions 5.4 and 5.5, with corres- ponding shift-Harnack inequalities, Theorems 5.6 and 5.7. Shift-Harnack inequalities were introduced by Wang in [18]. Exploring such inequalities in the present setting was motivated by a question posed by Feng-Yu Wang during a workshop at the University of Swansea in 2017.

2 Uniform gradient estimates

In [11], Priola and Wang derived uniform gradient estimates for semigroups gener- ated by second order elliptic operators with irregular and unbounded coefficients and bounded, measurable potentialsV. They did not require local Hölder continuity of the

coefficients, instead replacing the gradient of the semigroup with the Lipschitz con- stant. Although we will only consider the particular operator−¹₂∆ +V, we do allow for the potential to be unbounded.

Let us first briefly consider the case in which the potential V is bounded. In this case, we have the following gradient estimate, which depends only on the supremum norm ofV:

Proposition 2.1. SupposeV ∈C¹(M)is bounded withRic≥2K. Then for all bounded measurable functionsf we have

kdP_t^Vfk∞≤ r2

πte^K−t 1 + 2tkVk∞e^−tinfV kfk∞

for allt >0.

Proof. According to the variation of constants formula we have P_t^Vf =P_tf−

Z t 0

P_t−s(V P_s^Vf)ds and therefore

|dP_t^Vf| ≤ |dPtf|+ Z t

0

|dP_t−s(V P_s^Vf)|ds. (2) It is well known that

|dP_tf| ≤ r2

πte^K−tkfk_∞

and similarly for the integrand. The result therefore follows by substituting these es- timates into (2), integrating the latter.

We now turn our attention to the case of unbounded V. Let us recall the Bismut formula forP_t^Vf that was proved in [16]. For this, denote by//the stochastic parallel transport along the paths ofX(x)and byBthe anti-development of//toTxM. ThenB is a Brownian motion onT_xM starting at the origin. Denote byQtheEnd(T_xM)-valued solution to the ordinary differential equation

d

dtQt=−1

2//⁻¹_t Ric^]//tQt (3)

withQ0= idTxM. The compositionQ//⁻¹is called the damped parallel transport. Now supposeDis a regular domain inM and denote byτthe first exit time ofX(x)fromD. In [16] it was proved, for a bounded measurable functionf andx∈D, that there is the following local derivative formula:

(dP_t^Vf)_x=−E

V^xtf(X_t(x)) Z t

0

hQ_sh˙_s, dB_si+dV(//_sQ_sh_s)ds

for all t > 0, where h is any bounded adapted process with paths belonging to the Cameron-Martin spaceL^1,2([0, t]; Aut(TxM))such thath0= 1,hs= 0fors≥τ∧twith E[Rτ∧t

0 |h˙s|²ds]<∞. Suchhcan always be constructed, as shown for example in [4]. It follows that if the Ricci curvature is bounded below, in the sense that

inf{Ric(v, v) :v∈T M,|v|= 1}>−∞, and if

κ_V(t) := sup

x∈ME Z t

0

|dV|(X_s(x))ds

<∞

thendP^Vf is uniformly bounded on[, t]×M for all >0andt >0. Arguing as in the proof of [16, Theorem 7], it follows that in this case there is the following non-local, but explicit, version of the derivative formula:

(dP_t^Vf)_x=1 tE

V^xtf(X_t(x)) Z t

0

hQs, dB_si −(t−s)dV(//_sQ_s)ds

(4) for allt >0. Using formula (4), we can quickly derive some simple gradient estimates.

For convenience, and without loss of generality, we will assume thatV is non-negative.

Note also that in the sequel,Kwill always denote a real-valued constant.

Theorem 2.2. Suppose V ∈ C¹(M) is non-negative with κV(t) < ∞ and Ric ≥ 2K. Then for allf ∈L^∞(M)we have

kdP_t^Vfk∞≤e^K−t r2

πt +κ_V(t)

! kfk∞

for allt >0.

Proof. Since for eacht >0the stochastic integralRt

0hQs, dBsiis a centred Gaussian ran- dom variable with varianceRt

0kQsk²ds≤te^2K− and since for centred Gaussian random variable with variance1we haveE[|X|] =q

2

π, we have by formula (4), that

|dP_t^Vf|(x)≤ kfk∞

1 t

E

Z t 0

hQs, dBsi

+e^K−tE Z t

0

(t−s)|dV|(Xs(x))ds

≤ e^K−tkfk_∞ r2

πt +E Z t

0

|dV|(Xs(x))ds !

from which the result follows, by the definition ofκ_V(t).

Forp >1we can similarly obtain an estimate using theL^pnorm (as opposed to the L^∞norm), for which we should assume that

κV,q(t) := sup

x∈ME

"

Z t 0

|dV|(Xs(x))ds ^q#¹_q

<∞

whereqis the conjugate ofp.

Theorem 2.3. SupposeV ∈C¹(M)is non-negative withRic≥2K. Supposep >1, set q=p/(p−1)and assumeκV,q(t)<∞. Then for allf ∈L^p(M)we have

kdP_t^Vfk_p≤e^K−t Cq^1/q

√t +κ_V,q(t)

! kfk_p

for allt >0, whereCq is the constant from the Burkholder-Davis-Gundy inequality.

Proof. By the Burkholder-Davis-Gundy inequality, we see that

E

Z t 0

hQs, dBsi

q

≤E

sup

0≤s≤t

Z s 0

hQr, dBri

q

≤CqE

"

Z t 0

kQsk²ds

q 2#

and so, by formula (4) and Hölder’s inequality, we have

|dP_t^Vf|(x)≤E[|f|^p(X_t(x))]^1/p1 t E

Z t 0

hQ_s, dB_si

q^1/q

+e^K−tE

"

Z t 0

(t−s)|dV|(X_s(x))ds ^q#¹_q



≤e^K−tE[|f|^p(Xt(x))]^1/p

C_q^1/q 1

√t +κV,q(t)

for allt >0andx∈M. Thus

kdP_t^Vfk_p≤e^K−t Cq^1/q

√t +κ_V,q(t)

!Z

M

E[|f|^p(X_t(x))]dx ^1/p

and the result follows by Fubini’s theorem.

Theorem 2.2 implies that if the Ricci curvature is bounded below then dP^Vf is bounded on[,∞)×M for each >0. A similar observation applies to Theorem 2.3. Our next step will be to use Theorems 2.2 and 2.3 to obtain quantitative estimates which are uniform across all space and time. The price we pay is the inclusion of a term involving Hf. These are generalizations of the estimate proved by Cheng, Thalmaier and the author in [4] and [5].

Theorem 2.4. SupposeV ∈ C¹(M) is non-negative with κ_V(δ) < ∞for some δ > 0 andRic≥2K. Supposef ∈ C²(M)withf, Hf bounded. Then the derivativedP^Vf is uniformly bounded on[0,∞)×M and moreover

kdP_t^Vfk∞≤e^K−δ

r 2

πδ+κV(δ)

!

kfk∞+δ r 8

πδ +κV(δ)

!

kHfk∞

!

for allt≥0.

Proof. According the forward Kolmogorov equation we have P_δ+t^V f =P_t^Vf−

Z δ 0

P_s^V(HP_t^Vf)ds and therefore

|dP_t^Vf|(x)≤ |dP_δ+t^V f|(x) + Z δ

0

|dP_s^V(HP_t^Vf)|(x)ds (5) for allx∈M. By Theorem 2.2 we have

|dP_δ+t^V f|(x)≤e^K−δkP_t^Vfk_∞ r 2

πδ+κV(δ)

!

≤e^K−δkfk_∞ r 2

πδ+κV(δ)

!

and similarly

|dP_s^V(HP_t^Vf)|(x)≤e^K−skHfk_∞ r 2

πs+κV(s)

! .

Consequently

|dP_t^Vf|(x)≤e^K−δ

r 2

πδ +κ_V(δ)

!

kfk∞+ r8δ

π + Z δ

0

κ_V(s)ds

!

kHfk∞

!

from which the result follows, sinceκV is non-decreasing.

Theorem 2.5. SupposeV ∈ C¹(M) is non-negative with Ric ≥ 2K. Supposep > 1, set q = p/(p−1) and assumeκ_V,q(δ) <∞ for someδ > 0. Supposef ∈ C²(M)with f, Hf ∈L^p(M). Then the derivativedP^Vf is uniformlyL^p-bounded on [0,∞)×M and moreover

kdP_t^Vfk_p≤e^K−δ

Cq^1/q

√δ +κ_V,q(δ)

!

kfk_p+δ 2Cq^1/q

√δ +κ_V,q(δ)

! kHfk_p

!

whereCq is the constant from the Burkholder-Davis-Gundy inequality.

Proof. By (5) and Minkowski’s inequality we have kdP_t^Vfkp≤ kdP_t+δ^V fkp+

Z δ 0

kdP_s^V(HP_t^Vf)kpds.

By Theorem 2.3 we have

kdP_t+δ^V fkp≤e^K−δ Cq^1/q

√δ +κV,q(δ)

! kfkp

and similarly

kdP_s^V(HP_t^Vf)kp≤e^K−s Cq^1/q

√s +κ_V,q(s)

! kHfkp

from which the result follows, as is the casep=∞.

3 Characterizations of Ricci curvature

It is well known that Ricci curvature can be characterized in terms of the gradient of the heat semigroup; see for example [17, Theorem 2.2.4]. Next we show that this characterization extends to the Feynman-Kac semigroups considered above. This shows a way in which the derivativedV, appearing in the gradient estimates of the previous section, occurs naturally:

Theorem 3.1. SupposeV ∈C¹(M)is bounded below. Letx∈M andX ∈TxM with

|X|= 1. Supposef ∈C₀^∞(M)with∇f =X,Hessf(x) = 0and setα:=f(x). Then for anyp >0we have

limt↓0

P_t^V|∇f|^p(x)− |∇P_t^Vf|^p(x)

pt =1

2Ric(X, X) +

1−1 p

V +αdV(X).

Proof. By Taylor expansion at the pointxwe have

P_t^V|∇f|^p=|∇f|^p+t(¹₂∆−V)|∇f|^p+o(t) and also

|∇P_t^Vf|^p=|∇f|^p+pt|∇f|^p−2h∇(¹₂∆−V)f,∇fi+o(t) for smallt >0. Furthermore at the pointxwe have

(¹₂∆−V)|∇f|^p=p

2|∇f|^p−2 1

2∆−2 pV

|∇f|²+p 2

p 2−1

|∇f|^p−2|∇|∇f|²|²

with|∇|∇f|²|²(x) =|2(Hessf(x))^](∇f)|²= 0, and by the Bochner formula 1

2∆|∇f|²=h∇∆f,∇fi+ Ric(∇f,∇f)

so therefore limt↓0

P_t^V|∇f|^p− |∇P_t^Vf|^p

pt = 1

2Ric(X, X)−1

pV +h∇(V f),∇fi

= 1

2Ric(X, X) +

1−1 p

V +αdV(X) as required.

In the Section 2 we considered uniform estimates ondP^Vf which involved either the supremum norm off and a constant which diverges for small time, or the supremum norms of bothfandHfand a constant which does not. In light of the previous theorem, we would also like to consider derivative estimates for functions belonging toC_b¹(M). To do so we use the following derivative formula:

Theorem 3.2. Suppose V ∈ C¹(M) is bounded below withκ_V(t) <∞. Suppose the Ricci curvature ofM is bounded below withf ∈C_b¹(M). Then

(dP_t^Vf)(v) =E

V^xt((df)_Xt(x)(//tQtv)−f(Xt(x)) Z t

0

(dV)_Xs(x)(//sQsv)ds)

(6)

for allx∈M andv∈TxM.

Proof. First suppose f ∈ C₀^∞(M). Setting f_s := P_t−s^V f and N_s(v) := dP_t−s^V f(//_sQ_sv), using the definition of Qas the solution to equation (3) and by Itô’s formula and the Weitzenböck formula

d∆f = tr∇²df−df(Ric^]) we see

dNs(v)=^m dfs(//s∂sQsv)ds+ (∂sdfs)(//sQsv)dt+1

2tr∇²(dfs)(//sQsv)ds

= V(X_s(x))N_s(v)ds+f_s(X_s(x))dV(//_sQ_sv)ds

where^m=denotes equality modulo the differential of a local martingale. It follows that d(V^xsNs(v))=^mVs^xfs(Xs(x))dV(//sQsv)ds

so that

V^xs(dP_t−s^V f)_Xs(x)(//_sQ_sv)− Z s

0

V^xrP_t−r^V f(X_r(x))(dV)_Xr(x)(//_rQ_rv)dr (7) is a local martingale. To verify that it is a true martingale, we see that

E

"

sup

s∈[0,t]

V^xs(dP_t−s^V f)_Xs(x)(//sQsv)− Z s

0

Vr^xP_t−r^V f(Xr(x))(dV)_Xr(x)(//rQrv)dr #

≤ e^K−t|v| kdP^Vfk_L^∞_([0,t]×M)+kfk∞κV(t)

which is finite, by Theorem 2.4. Formula (6) follows by evaluating the expected value of (7) at times0andt, extending to allf ∈C_b¹(M)by approximation.

Combining Theorems 3.1 and 3.2, we obtain the following equivalence:

Theorem 3.3. SupposeV ∈C¹(M)is bounded below with∇V bounded. LetK ∈R^. Then the following are equivalent:

(1) Ric≥2K;

(2) iff ∈C_b¹(M)then

|∇P_t^Vf| ≤e^−KtP_t^V|∇f|+k∇Vk∞

1−e^−Kt K

P_t^V|f| (8) for allt≥0.

Proof. That (1) implies (2) follows immediately from Theorem 3.2. To prove that (2) implies (1), supposex∈M andX ∈TxM with|X|= 1and choosef ∈C₀^∞(M)so that f(x) = 0,∇f =X andHessf(x) = 0. Then, by applying Theorem 3.1 to the casep= 1, we obtain

1

2Ric(X, X) = lim

t↓0

P_t^V|∇f|(x)− |∇P_t^Vf|(x) t

≥ lim

t↓0

1−e^−Kt Kt

KP_t^V|∇f|(x)− k∇Vk∞P_t^V|f|(x)

= K|∇f|(x)− k∇Vk_∞|f(x)|

= K as required.

For the caseV = 0it is well known thatRic≥2K if and only if (8) holds. So in fact Ric≥2K so long as (8) holds forsome V ∈ C¹(M)which is bounded below with ∇V bounded. Moreover, if (8) holds forsomesuchV then it must hold forallsuchV.

4 Harnack inequalities

Denoting by ρ the Riemannian distance on M, we have also the following Harnack inequality for P_t^Vf, the proof of which is based on that for the V = 0case [17, The- orem 2.3.4]:

Theorem 4.1. SupposeV ∈ C¹(M)is non-negative with∇V bounded andRic≥ 2K. Then for all bounded measurable functionsf ≥0andp >1we have

(P_t^Vf)^p(x)≤(P_t^Vf^p)(y) exp

pρ²(x, y)

2(p−1)C1(t, K)t+tρ(x, y)k∇Vk_∞ 2C2(t, K)

for allx, y∈M andt >0, where C1(t, K) :=e^2Kt−1

2Kt , C2(t, K) := Kt 2 coth

Kt 2

.

Proof. Suppose first thatf ∈C²(M)withf bounded,inff >0andf constant outside a compact set. Givenx6=yandt >0, letγ: [0, t]→M be a geodesic fromxtoyof length ρ(x, y). Letvs:= ˙γs, so that|vs|=ρ(x, y)/t. Let

h(s) :=t

e^2Ks−1 e^2Kt−1

fors∈[0, t], so thath(0) = 0andh(t) =t. Lety_s:=γ_h(s)and define φ(s) := logP_s^V((P_t−s^V f)^p)(ys)

fors∈[0, t]. Then d dsφ(s)

= 1

P_s^V(P_t−s^V f)^p d dsP_s^V

(P_t−s^V f)^p+P_s^V d

ds(P_t−s^V f)^p

+h∇P_s^V(P_t−s^V f)^p,y˙si

(ys)

= 1

P_s^V(P_t−s^V f)^p

(¹₂∆−V)P_s^V(P_t−s^V f)^p−pP_s^V((P_t−s^V f)^p−1(₂¹∆−V)P_t−s^V f) +h∇P_s^V((P_t−s^V f)^p),y˙si

(ys)

= 1

P_s^V(P_t−s^V f)^p

P_s^V(¹₂∆(P_t−s^V f)^p)−pP_s^V((P_t−s^V f)^{p−1 1}₂∆P_t−s^V f) + (p−1)P_s^V(V(P_t−s^V f)^p) +h∇P_s^V(P_t−s^V f)^p,y˙si

(ys).

Since

∆(P_t−s^V f)^p=p(P_t−s^V f)^p−1∆P_t−s^V f+p(p−1)(P_t−s^V f)^p−2|∇P_t−s^V f|², it follows from Theorem 3.3 that

d dsφ(s)

= 1

P_s^V(P_t−s^V f)^p

P_s^V

p(p−1)

2 (P_t−s^V f)^p−2|∇P_t−s^V f|²

+ (p−1)P_s^V(V(P_t−s^V f)^p) +h∇P_s^V((P_t−s^V f)^p),y˙_si

(y_s)

≥ 1

P_s^V(P_t−s^V f)^pP_s^V

(P_t−s^V f)^p

p(p−1)

2 |∇logP_t−s^V f|²+ (p−1)V

−pρ(x, y)

t h⁰(s)e^−Ks|∇logP_t−s^V f|

−ρ(x, y)

t h⁰(s)k∇Vk∞

1−e^−Ks

K (ys)

≥ −pρ²(x, y)h⁰(s)²e^−2Ks

2(p−1)t² −ρ(x, y)

t h⁰(s)k∇Vk∞

1−e^−Ks K

. Since

h⁰(s) = 2Kt

e^2Ks e^2Kt−1

we have d

dsφ(s)≥ −2pρ²(x, y)K²e^2Ks

(p−1)(e^2Kt−1)² −2ρ(x, y)

e^2Ks−e^Ks e^2Kt−1

k∇Vk∞

which integrated between 0 and t yields the inequality. By approximations and the monotone class theorem, as in [17, Theorem 2.3.3], this extends to all non-negative, bounded measurable functions.

5 Shift-Harnack inequalities

In this section we prove two further differentiation formulae, which complement for- mula (4), and use them to deduce shift-Harnack inequalities, first introduced by Wang

in [18] and similar to Theorem 4.1 except that the shift in space variable takes placein- sidethe semigroup. The approach will be similar to that of [15]. In particular, we start by supposing thatαis a bounded continuously differentiable 1-form andαs a solution to the equation

∂

∂sαs= (¹₂∆−V)αs (9)

fors ∈ (0, t] withα0 = α. Here∆ denotes the Hodge Laplacian −(d^?+d)² acting on 1-forms, wered^? denotes the codifferential operator.

Proposition 5.1. Suppose V ∈ C¹(M) is bounded below and that h_s is a bounded adapted process with paths belonging to the Cameron-Martin spaceL^1,2([0, t]; [0,∞)). Then

V^xsd^?α_t−s(Xs(x))hs+V^xsα_t−s

//sQs

Z s 0

h˙rQ⁻¹_r dBr

+ Z s

0

V^xrhdV, αt−ri(Xr(x))h_rdr (10) is a local martingale.

Proof. Sinced^?commutes with∆, with

−d^?(V αs) =−V d^?αs+hdV, αsi, by equation (9) we have

∂

∂sd^?αs= (¹₂∆−V)d^?αs+hdV, αsi.

Consequently, by Itô’s formula, we find that n_s:=Vs^x(d^?α_t−s)(X_s(x)) +

Z s 0

V^xrhdV, α_t−ri(X_r(x))dr is a local martingale and therefore so is

n_sh_s− Z s

0

n_rh˙_rdr (11)

with the latter starting atd^?αth0. Moreover

V^xs(d^?α_t−s)(X_s(x)) ˙h_sds=−V^xs n

X

i=1

(∇//_se_iα_t−s)(//_se_i) ˙h_sds

=−V^xs

DXⁿ

i=1

(∇_//seiα_t−s)(//_sQ_s)dB_sⁱ,h˙_sQ⁻¹_s dB_sE

where{ei}ⁿ_i=1is any orthonormal basis ofTxM. Since

d(V^xsα_t−s(//_sQ_s)) =V^xs n

X

i=1

(∇//_se_iα_t−s)(//_sQ_s)dB_sⁱ (12)

it follows that Z s

0

V^xr(d^?α_t−r)(Xr(x)) ˙hrdr+V^xsα_t−s(//sQs) Z s

0

h˙rQ⁻¹_r dBr

(13)

is also a local martingale. Using the fact that (11) and (13) are local martingales, and since by integration by parts

− Z s

0

Z r 0

V^xuhdV, α_t−ui(Xu(x))duh˙rdr= − Z s

0

Vr^xhdV, α_t−ri(Xr(x))drhs

+ Z s

0

Vr^xhdV, α_t−ri(Xr(x))hrdr, we easily verify that (10) is also a local martingale.

Theorem 5.2. SupposeV ∈C¹(M)is bounded below with∇V andRicbounded. Sup- poseαis a bounded continuously differentiable1-form withd^?αbounded. Then

P_t^V(d^?α)(x) =−1 tE

V^xtα

//_tQ_t

Z t 0

Q⁻¹_s (dB_s+ (t−s)//⁻¹_s ∇V ds)

for allx∈M andt >0.

Proof. According to the assumptions of the theorem, with hs = (t−s)/t, the local martingale (10) is a true martingale. Furthermore, by (12), it follows that

α_t=E[Vtα(//_tQ_t)]

so the result follows by evaluating the martingale (10) at the times0andt, and applying the Markov property to the term involving∇V.

Given a vector field Y, the Bismut formula (4) provides a probabilistic expression for the derivativeY(P_t^Vf)which does not involve derivatives off. By Theorem 5.2, and using the facts thatdivY =−d^?Y^[anddiv(f Y) =Y f+fdivY, we obtain the following theorem which provides a similar expression for the derivativeP_t^V(Y(f)):

Theorem 5.3. SupposeV ∈C¹(M)is bounded below with∇V andRicbounded. Sup- posef is a bounded C¹function andY a boundedC¹ vector field for whichdivY and Y(f)are also bounded. Then

P_t^V(Y(f))(x)

=−E[Vt^xf(Xt(x))(divY)(Xt(x))]

+1 tE

V^xtf(X_t(x))

Y(X_t(x)), //_tQ_t Z t

0

Q⁻¹_s (dB_s+ (t−s)//⁻¹_s ∇V ds)

for allx∈M andt >0.

By Theorem 5.3, we have the following two propositions:

Proposition 5.4. Suppose V ∈ C¹(M) is non-negative with ∇V bounded and 2K ≤ Ric≤2Lfor constantsKandL. Supposef is a boundedC¹ function andY a bounded C¹vector field for whichdivY andY(f)are also bounded. Then

|P_t^V(Y(f))|(x)≤α^V_t (Y)(P_t^Vf²)¹²(x) for allx∈M andt >0, where

α^V_t(Y) := kdivYk_∞+kYk_∞e^−Kt

√t

e^2Lt−1 2Lt

¹2

+kYk∞k∇Vk∞

e^−Kt L

1

Lt

2 coth ^Lt₂

−1−1

! .

Proof. By Theorem 5.3, we have

|P_t^V(Y(f))|(x)

≤ kYk_∞e^−tK t E

"

Z t 0

Q⁻¹_s dBs

²#¹₂

(P_t^Vf²)¹²(x)

+

kdivYk_∞+kYk_∞k∇Vk_∞e^−tK t

Z t 0

e^Ls(t−s)ds

(P_t^Vf²)¹²(x)

= α^V_t(Y)(P_t^Vf²)¹²(x) as required.

Proposition 5.5. Suppose V ∈ C¹(M) is non-negative with ∇V bounded and 2K ≤ Ric ≤ 2L for constants K and L. Supposef > 0 is a bounded C¹ function andY a boundedC¹vector field for whichdivY andY(f)are also bounded. Then

|P_t^V(Y(f))|(x)≤δ(P_t^V(flogf)(x)−P_t^VflogP_t^Vf(x)) +β_t^V(δ, Y)P_t^Vf(x) for allx∈M,t >0andδ >0, where

β_t^V(δ, Y) := kdivYk_∞+kYk²_∞e^−2Kt 2tδ

e^2Lt−1 2Lt

+kYk∞k∇Vk∞

e^−Kt L

1

Lt

2 coth ^Lt₂

−1−1

! .

Proof. By Theorem 5.3 and [13, Lemma 6.45], the latter being essentially Jensen’s in- equality, we have

|P_t^V(Y(f))|(x)

≤ kdivYk∞P_t^Vf(x)

+ 1 tE

V^tf(Xt(x))

Y(Xt(x)), //tQt

Z t 0

Q⁻¹_s (dBs+ (t−s)//⁻¹_s ∇V ds)

≤ δ(P_t^V(flogf)(x)−P_t^Vf(x) logP_t^Vf(x))

+δP_t^Vf(x) logE

exp

kYk∞

e^−Kt tδ

Z t 0

Q⁻¹_s dBs

+

kdivYk_∞+kYk_∞k∇Vk_∞e^−Kt t

Z t 0

e^Ls(t−s)ds

P_t^Vf(x)

≤ δ(P_t^V(flogf)(x)−P_t^VflogP_t^Vf(x)) +β_t^V(δ, Y)P_t^Vf(x) as required.

These estimates can be used to derive shift-Harnack inequalities, as introduced by Wang for Markov operators on a Banach space; [18, Proposition 2.3]. In particular, suppose as in [15] that {Fs:s ∈ [0,1]} is a C¹ family of diffeomorphisms of M with F₀= id_M. For eachs∈[0,1]define a vector fieldY_sonM by

Ys:= (DFs)⁻¹ d dsFs

and assume thatYsanddivYsare uniformly bounded.

Theorem 5.6. SupposeV ∈C¹(M)is non-negative with∇V bounded and2K≤Ric≤ 2Lfor constantsKandL. Supposef ≥0is a bounded measurable function and thatY_s anddivYsare uniformly bounded. Then

P_t^Vf(x)≤P_t^V(f◦F1)(x) + Z 1

0

(α^V_t )²(Ys)ds ¹2

P_t^Vf²¹2(x) for allx∈M andt≥0, whereα^V_t is defined as in Proposition 5.4.

Proof. It suffices to prove forf continuously differentiable. Since d

ds(f◦Fs) =∇V_s(f ◦Fs),

and following the proof of [18, Proposition 2.3], we see for allr >0that d

dsP_t^V f

1 +rsf(F1−s)

=−rP_t^V

f²

(1 +rsf)²(F_1−s)

−P_t^V

Y_1−s f

1 +rsf(F_1−s)

for alls ∈ [0,1]. Applying Proposition 5.4 and proceeding as in the proof of [18, Pro- position 2.3], integrating overs∈[0,1]and minimizing overr >0, we easily obtain the desired inequality.

Theorem 5.7. SupposeV ∈C¹(M)is non-negative with∇V bounded and2K≤Ric≤ 2Lfor constantsKandL. Supposef ≥0is a bounded measurable function and thatY_s anddivYsare uniformly bounded. Then

(P_t^Vf)^p(x)≤P_t^V(f^p◦F1)(x) exp Z 1

0

p

1 + (p−1)sβ_t^V

p−1 1 + (p−1)s, Ys

ds

for allx∈M,t≥0andp >1, whereβtis defined as in Proposition 5.5.

Proof. As for the previous theorem, it suffices to prove forf continuously differentiable.

The result extends to more generalf by approximation. Applying Proposition 5.5, as in the proof of [18, Proposition 2.3] withβ(s) := 1 + (p−1)sfors∈[0,1], we obtain

d dslog

P_t^V

f^β(s)(Fs)

(x)_β(s)^p

≥ − p β(s)β^V_t

p−1 β(s), Ys

fors∈[0,1]. Integrating oversyields the result.

Looking ahead, it would now be desirable to weaken our assumptions ondV, such as to suppose simply thatdV exists in some weak sense and is locally Kato, or indeed to eliminate terms involving dV altogether. To do the latter could however be diffi- cult, sincedV appears natually in Theorem 3.1. Assumptions involvingdV do appear elsewhere in the literature, such as in the celebrated work of Li and Yau [9].

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