Derivatives of Feynman-Kac Semigroups

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Derivatives of Feynman-Kac semigroups

James Thompson

^*

9th March 2020

Abstract

We prove Bismut-type formulae for the first and second derivatives of a Feynman-Kac semigroup on a complete Riemannian manifold. We derive local estimates and give bounds on the logarithmic derivatives of the integral kernel. Stationary solutions are also considered. The arguments are based on local martingales, although the assumptions are purely geometric.

Keywords: Brownian motion ; Feynman-Kac ; Bismut AMS MSC 2010: 47D08 ; 53B20; 58J65 ; 60J65

1 Introduction

SupposeM is a complete Riemannian manifold of dimensionnwith Levi-Civita connec- tion∇. Denote by∆the Laplace-Beltrami operator, supposeZ is a smooth vector field and set L := ¹₂∆ + Z. Any elliptic diffusion operator on a smooth manifold induces, via its principle symbol, a Riemannian metric with respect to which it takes this form.

Denote byx_ta diffusion onM starting atx₀∈ M with generatorLand explosion time ζ(x₀). The explosion time is the random time at which the process leaves all compact subsets of M. Suppose V : [0, ∞) × M → R is a smooth function which is bounded below and denote byP_t^Vf the associated Feynman-Kac semigroup, acting on bounded measurable functionsf. For T > 0fixed, P_t^Vf is smooth and bounded on(0, T ] × M, satisfies the parabolic equation

∂tφt= (L − Vt)φt (1)

on(0, T ] × Mwithφ0= f and for

V^t:= e⁻^R⁰^t^V^{T −s}^(x^s^)ds (2) is represented probabilistically by the Feynman-Kac formula

P_T^Vf (x0) = EVTf (xT)1_{{T <ζ(x}₀_)} . (3) In the self-adjoint case, equation (1) corresponds (via Wick rotation) to the Schrödinger equation for a single non-relativistic particle moving in an electric field in curved space.

In this sense, the derivative dP_T^Vf corresponds to the momentum of the particle and LP_T^Vf the kinetic energy.

In this article, we prove probabilistic formulae and estimates for dP_T^Vf, LP_T^Vf and

∇dP_T^Vf. In doing so, we extend results in [18] (by includingV) and in [1] (by including Z andV). In each case, we allow for unbounded and time-dependentV. Our approach

*University of Luxembourg, Email: james.thompson@uni.lu

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is more concise than that of [1], since we avoid the extrinsic argument in favour of the differential Bianchi identity. Our results imply new Bismut-type formulae for the derivatives of the heat kernel in the forward variable (see, for example, Corollary 2.3).

Our formula fordP_T^Vf is given by Theorem 2.2. Forv ∈ Tx₀M it states

(dP_T^Vf )(v) = −E

"

V^Tf (xT)1_{{T <ζ(x}₀_)}

Z T 0

hWs( ˙ksv), //sdBsi + dVT −s(Ws(ksv))ds

#

where//_tand W_tare the usual parallel and damped parallel transports, respectively, and Bt the martingale part of the antidevelopment of xt toTx₀M. The process kt is chosen so that it vanishes oncextexits a regular domain (an open connected subset with compact closure and smooth boundary). Imposing this condition onktobviates the need for any assumptions onRicZ. Conversely, if we assumeRicZ is bounded below then we can choosek_t= (T − t)/T and our formula fordP_t^Vf reduces to that of [5, Theorem 5.2].

Formulae in [5] are derived from the assumption that one can differentiate under the expectation, and thus require global assumptions. Our approach, on the other hand, follows that of [18] and [1] in using local martingales to obtain local formula for which no assumptions are needed.

Our formula forLP_T^Vf is given by Theorem 2.6. It states L(P_T^Vf )(x0)

= E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

h ˙ksZ, //_sdB_si

#

+1 2E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

hW_s˙l_s, //_sdB_si + dV_{T −s}(W_sl_s)ds

!Z T 0

˙k_sW_s⁻¹//_sdB_s

#

where the processes k and l are assumed to vanish outside of a regular domain. A formula for∆PT (acting on differential forms) was previously given in [6], for the case of a compact manifold withZ = 0andV = 0.

Our formula for∇dPtf is given by Theorem 2.8. Forv, w ∈ Tx₀M it states (∇dP_T^Vf )(v, w)

= − E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

hW_s⁰( ˙k_sv, w), //_sdB_si

#

− E

"

V^Tf (xT)1_{{T <ζ(x}₀_)}

Z T 0

((∇dVT −s)(Ws(ksv), Ws(w)) + (dVT −s)(W_s⁰(ksv, w)))ds

#

+ E

"

V^Tf (xT)1_{{T <ζ(x}₀_)}

Z T 0

hWs( ˙lsw), //sdBsi + dVT −s(Ws(lsw))ds

!

· Z T

0

!#

whereW_t⁰ solves a covariant Itô equation determined by the curvature tensor and its derivative. This extends [1, Theorem 2.1] while avoiding the use of a stochastic differential equation.

The formulae mentioned above are derived in Section 2. Solutions to the time inde- pendent equation

(L − V )φ = −Eφ

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withE ∈ Rare subject to a similar analysis, as outlined in Section 3. In Section 4 we derive local estimates, using the formulae of Section 2 and local assumptions on curvature and the derivative of the potential function. We do so by choosing the processeskand lappropriately, as in [18] and [1], and applying the Cauchy-Schwarz inequality. These local estimates are given by Theorems 4.1, 4.3 and 4.5; global estimates are then given as corollaries. The global estimates are derived under appropriate global assumptions and imply the boundedness ofdP_t^Vf, LP_t^Vf and∇dP_t^Vf on[, T ] × M. These bounds lead to the non-local formulae of Section 5, in which the processeskandl are chosen deterministically. For the case in which Z is a gradient, estimates on the logarithmic derivatives of the integral kernel can then be derived, using Jensen’s inequality. They are given in Section 6 and extend those of [8] and [17].

2 Local Formulae

For the remainder of this article, we fixT > 0and setft:= P_{T −t}^V f.

2.1 Gradient

Denote by Ric^]_Z := Ric^]− 2∇Z the Bakry-Emery tensor (see [3]). Then the damped parallel transport Wt : Tx₀M → Tx_tM is the solution, along the paths of xt, to the covariant ordinary differential equation

DWt= −1

2Ric^]_ZWt (4)

withW0 = idT_x0M. SupposeD is a regular domain inM withx0 ∈ Dand denote byτ the first exit time ofxtfromD.

Lemma 2.1. Supposev ∈ Tx₀M and thatkis a bounded adapted process with paths in the Cameron-Martin spaceL^1,2([0, T ]; Aut(Tx₀M ))such thatkt= 0fort ≥ T − . Then

V^tdft(Wt(ktv)) − V^tft(xt) Z t

0

hWs( ˙ksv), //sdBsi − Z t

0

V^sfs(xs)dVT −s(Ws(ksv))ds (5) is a local martingale on[0, τ ∧ T ).

Proof. SettingN_t(v) := df_t(W_t(v))we see by Itô’s formula and the relations d∆f = tr ∇²df − df (Ric^])

dZf =∇_Zdf + df (∇Z) dVtf =f dVt+ Vtdf (the first one is the Weitzenböck formula) that

dNt(v)^m= dft(DWt(v))dt + (∂tdft)(Wt(v))dt + 1

2tr ∇²+ ∇Z

(dft)(Wt(v))dt

= VT −tNt(v)dt + ft(xt)dVT −t(Wt(v))dt

where^m=denotes equality modulo the differential of a local martingale. Recalling the definition ofVtgiven by equation (2), it follows that

d(V^tNt(ktv))^m= V^tNt( ˙ktv)dt + V^tft(xt)dVT −t(Wt(ktv))dt so that

V^tNt(ktv) − Z t

0 V^sdfs(Ws( ˙ksv))ds − Z t

0 V^sfs(xs)dVT −s(Ws(ksv))ds

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is a local martingale. By the formula

Vtf_t(x_t) = f₀(x₀) + Z t

0

Vsdf_s(//_sdB_s)

and integration by parts we see that Z t

0

Vsdf_s(W_s( ˙k_sv))ds − Vtf_t(x_t) Z t

0

hW_s( ˙k_sv), //_sdB_si

is also a local martingale and so the lemma is proved.

Theorem 2.2. Supposex₀ ∈ D withv ∈ T_x₀M,f ∈ B_b, V bounded below andT > 0. Supposekis a bounded adapted process with paths belonging to the Cameron-Martin spaceL^1,2([0, T ]; Aut(Tx₀M )), such thatk0= 1,kt= 0fort ≥ τ ∧TandRτ ∧T

0 | ˙ks|²ds ∈ L¹. Then

(dP_T^Vf )(v) = −E

"

V^Tf (xT)1_{{T <ζ(x}₀_)}

Z T 0

# . (6)

Proof. As in the proof of [18, Theorem 2.3], the processktcan be modified tok_tso that k_t = kt fort ≤ τ ∧ (T − 2) and k_t = 0 fort ≥ τ ∧ (T − ), cutting off appropriately in between. Since(dft)x is smooth and therefore bounded for(t, x) ∈ [0, T − ] × D, it follows from Lemma 2.1 and the strong Markov property that formula (6) holds withk_t in place ofk_t. The result follows by taking ↓ 0.

Denoting byp^Z_T(x, y)the transition density of the diffusion with generatorL, using The- orem 2.2 we can easily obtain the following Bismut formula, for the derivative ofp^Z_T(x, y) the in the forward variabley.

Corollary 2.3. Suppose x₀ ∈ D with div Z bounded below and k as in Theorem 2.2.

Then

d log p^Z_T(y, ·)x₀ = −Eh

e⁻^R⁰^T^{div Z(x}^s^)dsRT

0 hWs˙ks, //sdBsi + d(div Z)(Wsks)ds

xT = yi Eh

e⁻^R⁰^T^{div Z(x}^s^)ds

x_T = yi where herex_tis a diffusion onM with generator ¹₂∆ − Zstarting atx₀. Proof. According to the Fokker-Planck equation, we have

p^Z_T(x, y) = p^{−Z,− div Z}_T (y, x)

where p^{−Z,− div Z}_T (y, x) denotes the minimal integral kernel for the semigroup gener- ated by the operatorL^∗ = ¹₂∆ − Z − div Z. The result is therefore obtained simply by conditioning in Theorem 2.2, having replacedZ with−ZandV withdiv Z.

2.2 Generator

Now supposeD₁andD₂ are regular domains withx₀ ∈ D1andD₁⊂ D2. Denote byσ andτ the first exit times ofx_tfromD₁andD₂, respectively.

Lemma 2.4. Suppose x0 ∈ D1 and 0 < S < T and that k, l are bounded adapted processes with paths in the Cameron-Martin spaceL^1,2([0, T ]; R) ^{such that}ks = 0 for

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s ≥ σ ∧ S,l_s= 1fors ≤ σ ∧ Sandl_s= 0fors ≥ τ ∧ (T − ). Then

Vt(Lf_t)(x_t)k_t−1 2Vtdf_t

W_tl_t

Z t 0

+1

2Vtf_t(x_t) Z t

0

hWs˙ls, //_sdB_si Z t

0

˙ksW_s⁻¹//_sdB_s

+1 2

Z t 0

V^sfs(xs)dVT −s(Wsls)ds Z t

0

˙ksW_s⁻¹//sdBs

+ V^tft(xt) Z t

0

h ˙ksZ − ks∇VT −s, //sdBsi

− Z t

0

Vsf_sk_sLV_{T −s}ds is a local martingale on[0, τ ∧ T ).

Proof. Defining

nt:= (Lft) (xt) we have, by Itô’s formula, that

dn_t= d(Lf_t)_x_t//_tdB_t+ ∂_t(Lf_t)(x_t)dt + L(Lf_t)(x_t)dt

= d(Lft)x_t//tdBt+ L(VT −tft)dt

= d(Lft)x_t//tdBt+ (LVT −t)ftdt + VT −tntdt + hdft, dVT −tidt.

It follows that

d(V^tntkt)= V^m ^tnt˙kt+ ktV^t(ftLVT −t+ hdft, dVT −ti)dt and so

Vt(Lf_t)(x_t)k_t− Z t

0

Vs(Lf_s)(x_s) ˙k_sds − Z t

0

Vsk_s(f_sLV_{T −s}+ hdf_s, dV_{T −s}i)ds is a local martingale, with

−(Lft)(xt) ˙ktdt = 1

2d^∗d − Z

ft(xt) ˙ktdt = 1

2d^∗(dft) − (dft)(Z)

˙ktdt.

By the Weitzenbock formula

d((df_t)(W_t)) = (∇_//_t_dB_tdf_t)(W_t) − V_{T −t}(df_t)(W_t)dt + f_t(x_t)dV_{T −t}(W_t)dt from which it follows that

d(V^t(dft)(Wt)) = V^t(∇_//_t_dB_tdft)(Wt) + V^tft(xt)dVT −t(Wt)dt.

Consequently, for an orthonormal basis{ei}ⁿ_i=1ofT_x₀M, by integration by parts we have

V^td^∗(dft) ˙ktdt = −

n

X

i=1

V^t(∇_//_t_e_idft)(//tei) ˙ktdt

= − Vm ^t(∇//_tdB_tdft)(Wt˙ktW_t⁻¹//tdBt)

= − d(Vtdf_t(W_t Z t

0

˙k_sW_s⁻¹//_sdB_s))

+ d

Z t 0

V^sfs(xs)dVT −s(Ws)ds Z t

0

˙ksW_s⁻¹//sdBs

.

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Furthermore

Z t 0

Vsdf_s(Z) ˙k_sds − Vtf_t(x_t) Z t

0

h ˙ksZ, //_sdB_si is a local martingale and therefore

Z t 0

Vs(Lf_s)(x_s) ˙k_sds

−1

2V^tdft(Wt

Z t 0

˙ksW_s⁻¹//sdBs) +1 2

Z t 0

V^sfs(xs)dVT −s(Ws)ds Z t

0

˙ksW_s⁻¹//sdBs

+ V^tft(xt) Z t

0

h ˙ksZ, //sdBsi

is also a local martingale. By the assumptions onkandlit follows from Lemma 2.1 that

O_t¹ = V^tdft(Wt((lt− 1))) − Vtft(xt) Z t

0

hWs( ˙ls), //sdBsi

− Z t

0 V^sfs(xs)dVT −s(Ws((ls− 1)))ds, O_t² =

Z t 0

are two local martingales. So the productO_t¹O²_t is also a local martingale, sinceO¹= 0 on[0, σ ∧ S]withO²constant on[σ ∧ S, τ ∧ (T − )). Consequently

− Vtdft

Wt

Z t 0

˙ksW_s⁻¹//sdBs

+ V^tdft

Wtlt

Z t 0

˙ksW_s⁻¹//sdBs

− Vtft(xt) Z t

0

hWs˙ls, //sdBsi Z t

0

˙ksW_s⁻¹//sdBs

− Z t

0 V^sfs(xs)dVT −s(Ws((ls− 1)))ds Z t

0

˙ksW_s⁻¹//sdBs

is a local martingale and therefore so is

V^t(Lft)(xt)kt−1 2V^tdft

Wtlt

Z t 0

˙ksW_s⁻¹//sdBs

+1

2V^tft(xt) Z t

0

hWs˙ls, //sdBsi Z t

0

˙ksW_s⁻¹//sdBs

+1 2

Z t 0

Vsf_s(x_s)dV_{T −s}(W_sl_s)ds Z t

0

˙ksW_s⁻¹//_sdB_s

+ Vtf_t(x_t) Z t

0

h ˙k_sZ, //_sdB_si

− Z t

0

V^sks(fsLVT −s+ hdfs, dVT −si)ds.

Since

Z t 0

V^sdfs(∇VT −s)ksds − V^tft(xt) Z t

0

hks∇VT −s, //sdBsi is a local martingale, the result follows.

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Lemma 2.5. Supposex₀∈ D1,f ∈ B_b,V bounded below and0 < S < T. Supposekis a bounded adapted process with paths in the Cameron-Martin spaceL^1,2([0, T ]; R)^such thatks= 0fors ≥ σ ∧ S,k0= 1andRσ∧S

0 | ˙ks|²ds ∈ L¹. Then

V_T(x₀)P_T^Vf (x₀) = E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

(k_sV˙_{T −s}(x_s) − ˙k_sV_{T −s}(x_s))ds

# .

Proof. By Itô’s formula, we have

d(V^tVT −tftkt)^m= −ktV^tV˙T −tft+ ˙ktV^tVT −tft

which implies

VtV_{T −t}f_tk_t− Z t

0

( ˙k_sVsV_{T −s}f_s− k_sVsV˙_{T −s}f_s)ds

is a local martingale on[0, τ ∧ T ). The assumptions onf andV imply it is a martingale on[0, τ ∧ T ], so result follows by taking expectations and applying the strong Markov property.

Theorem 2.6. Supposex₀∈ D1,f ∈ B_b,V bounded below and0 < S < T. Supposek, l are bounded adapted processes with paths in the Cameron-Martin spaceL^1,2([0, T ]; R) such that ks = 0 for s ≥ σ ∧ S, k0 = 1, ls = 1 fors ≤ σ ∧ S, ls = 0 fors ≥ τ ∧ T, Rσ∧S

0 | ˙ks|²ds ∈ L¹andRτ ∧T

σ∧S | ˙ls|²ds ∈ L¹. Then L(P_T^Vf )(x0) =

E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

h ˙ksZ, //_sdB_si

#

+1 2E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

!Z T 0

# .

Proof. Modifying the process lt to l_t as in the proof of Theorem 2.2, it follows from Lemma 2.4, the strong Markov property, the boundedness ofP_t^Vf on[0, T ] × D2and the boundedness ofdP_t^Vf andLP_t^Vf on[, T ] × D2that the formula

L(P_T^Vf )(x0)

= E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

h ˙ksZ, //_sdB_si − Z T

0

k_s(dV_{T −s}(//_sdB_s) + LV_{T −s}ds)

!#

+1 2E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

!Z T 0

#

holds withl_t in place oflt. The formula also holds as stated, in terms oflt, by taking

↓ 0. Applying the Itô formula yields Z T

0

ks(dVT −s(//sdBs) + LVT −sds) = −VT(x0) + Z T

0

(ksV˙T −s(xs) − ˙ksVT −s(xs))ds

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and therefore

(L − V_T(x₀))(P_T^Vf )(x₀) = E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

h ˙k_sZ, //_sdB_si − Z T

0

(k_sV˙_{T −s}(x_s) − ˙k_sV_{T −s}(x_s))ds

!#

+1 2E

"

V^Tf (xT)1_{{T <ζ(x}₀_)}

Z T 0

hWs˙ls, //sdBsi + dVT −s(Wsls)ds

!Z T 0

˙ksW_s⁻¹//sdBs

# .

The result follows from this by Lemma 2.5.

2.3 Hessian

For eachw ∈ T_x₀M define an operator-valued processW_t⁰(·, w) : T_x₀M → T_x_tM by W_s⁰(·, w) := Ws

Z s 0

W_r⁻¹R(//rdBr, Wr(·))Wr(w)

−1 2W_s

Z s 0

W_r⁻¹(∇Ric^]_Z+ d^?R − 2R(Z))(W_r(·), W_r(w))dr.

Here the operatorR(Z)is defined byR(Z)(v1, v2) := R(Z, v1)v2and the operatord^?Ris defined byd^?R(v1)v2:= − tr ∇·R(·, v1)v2and satisfies

hd^?R(v₁)v₂, v₃i = h(∇_v₃Ric^])(v₁), v₂i − h(∇_v₂Ric^])(v₃), v₁i

for allv1, v2, v3 ∈ TxM andx ∈ M. The processW_t⁰(·, w)is the solution to the covariant Itô equation

DW_t⁰(·, w) = R(//_tdB_t, W_t(·))W_t(w)

−1 2

d^?R − 2R(Z) + ∇Ric^]_Z

(Wt(·), Wt(w))dt

−1

2Ric^]_Z(W_t⁰(·, w))dt

withW₀⁰(·, w) = 0. As in the previous section, supposeD₁andD₂are regular domains withx₀∈ D₁andD₁⊂ D₂. Denote byσandτthe first exit times ofx_tfromD₁andD₂, respectively.

Lemma 2.7. Suppose v, w ∈ Tx₀M, 0 < S < T and that k, l are bounded adapted processes with paths in the Cameron-Martin space L^1,2([0, T ]; Aut(Tx₀M )) such that ks= 0fors ≥ σ ∧ S,ls= 1fors ≤ σ ∧ T andls= 0fors ≥ τ ∧ (T − ). Then

V^t(∇dft)(Wt(ktv), Wt(w)) + V^t(dft)(W_t⁰(ktv, w)) − V^tft(xt) Z t

0

hW_s⁰( ˙ksv, w), //sdBsi

− Z t

0

Vsf_s(x_s)((∇dV_{T −s})(W_s(k_sv), W_s(w)) + (dV_{T −s})(W_s⁰(k_sv, w)))ds

+ Vtf_t(x_t) Z t

0

hW_s( ˙l_sw), //_sdB_si Z t

0

hW_s( ˙k_sv), //_sdB_si

− Vtdft(Wt(ltw)) Z t

0

hWs( ˙ksv), //sdBsi

+ Z t

0 V^sfs(xs)dVT −s(Ws((ls− 1)w))ds Z t

0

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+ Z t

0 V^sfs(xs)dVT −s(Ws(w)) Z s

0

hWr( ˙krv), //rdBrids

− 2 Z t

0

Vs(df_s dV_{T −s})(W_s(k_sv), W_s(w))ds (2) is a local martingale on[0, τ ∧ T ).

Proof. Setting

N_t⁰(v, w) := (∇dft)(Wt(v), Wt(w)) + (dft)(W_t⁰(v, w)) and

R^],]_x (v1, v2) := Rx(·, v1, v2, ·)^]∈ TxM ⊗ TxM we see by Itô’s formula and the relations

d∆f = tr ∇²df − df (Ric^]) dZf =∇Zdf + df (∇Z) dV f =f dV + V df

∇d(∆f ) = tr ∇²(∇df ) − 2(∇df )(Ric^] id −R^],]) − df (d^?R + ∇Ric^])

∇d(Zf ) =∇Z(∇df ) + 2(∇df )(∇Z id) + df (∇∇Z + R(Z))

∇d(Vtf ) =f ∇dVt+ 2df dVt+ Vt∇df

(the fourth one is a consequence of the differential Bianchi identity; see [4, p. 219], and the fifth one a consequence of the Ricci identity) that

dN_t⁰(v, w)

= (∇_//_t_dB_t∇dft)(Wt(v), Wt(w)) + (∇dft) D

dtWt(v), Wt(w)

dt + (∇dft)

Wt(v),D dtWt(w)

dt + ∂_t(∇df_t)(W_t(v), W_t(w))dt + 1

2tr ∇²+ ∇_Z

(∇df_t)(W_t(v), W_t(w))dt + (∇_//_t_dB_tdft)(W_t⁰(v, w)) + (dft) (DW_t⁰(v, w)) + hd(dft), DW_t⁰(v, w)i + ∂t(dft)(W_t⁰(v, w))dt + 1

2tr ∇²+ ∇Z

(dft)(W_t⁰(v, w))dt

= fm t(xt)(∇dVT −t)(Wt(v), Wt(w))dt + ft(xt)(dVT −t)(W_t⁰(v, w))dt + 2(df_t dVT −t)(W_t(v), W_t(w))dt + V_{T −t}N_t⁰(v, w)dt

for which we calculated

[d(df ), DW⁰(v, w)]_t= (∇df_t)(R^],](W_t(v), W_t(w)))dt.

It follows that

d(V^tN_t⁰(ktv, w))^m= V^tft(xt)((∇dVT −t)(Wt(ktv), Wt(w)) + (dVT −t)(W_t⁰(ktv, w)))dt + VtN_t( ˙k_tv, w)dt + 2Vt(df_t dVT −t)(W_t(k_tv), W_t(w))dt so that

VtN_t⁰(k_tv, w) − Z t

0

Vs(∇df_s)(W_s( ˙k_sv), W_s(w))ds − Z t

0

Vs(df_s)(W_s⁰( ˙k_sv, w))ds

− 2 Z t

0

V^s(dfs dVT −s)(Ws(ksv), Ws(w))ds

− Z t

0 V^sfs(xs)((∇dVT −s)(Ws(ksv), Wsw)) + (dVT −s)(W_s⁰(ksv, w)))ds

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is a local martingale. By the formula

V^tft(xt) = f0(x0) + Z t

0

V^sdfs(//sdBs)

and integration by parts we see that Z t

0

V^s(dfs)(W_s⁰( ˙ksv, w))ds − V^tft(xt) Z t

0

is a local martingale. Similarly, by the formula

V^tdft(Wt) = df0+ Z t

0 V^s(∇dfs)(//sdBs, Ws) + Z t

0 V^sfs(xs)dVT −s(Ws)ds and integration by parts we see that

Z t

0 V^s(∇dfs)(Ws( ˙ksv), Ws(w))ds − V^tdft(Wt(w)) Z t

0

+ Z t

0

Vsf_s(x_s)dV_{T −s}(W_s(w)) Z s

0

hW_r( ˙k_rv), //_rdB_rids

is yet another local martingale. Therefore

V^t(∇dft)(Wt(ktv), Wt(w)) + V^t(dft)(W_t⁰(ktv, w))

− Z t

0

V^sfs(xs)((∇dVT −s)(Ws(ksv), Ws(w)) + (dVT −s)(ksW_s⁰(v, w)))ds

− Vtft(xt) Z t

0

hW_s⁰( ˙ksv, w), //sdBsi − 2 Z t

0 V^s(dfs dVT −s)(Ws(ksv), Ws(w))ds +

Z t 0

Vsf_s(x_s)dV_{T −s}(W_s(w)) Z s

0

hW_r( ˙k_rv), //_rdB_rids

− Vtdft(Wt(w)) Z t

0

is a local martingale. By Lemma 2.1 it follows that

O_t¹ = V^tdft(Wt((lt− 1)w)) − Vtft(xt) Z t

0

hWs( ˙lsw), //sdBsi

− Z t

0

Vsf_s(x_s)dV_{T −s}(W_s((l_s− 1)w))ds,

O_t² = Z t

0

are two local martingales. So the productO_t¹O²_t is also a local martingale, sinceO¹= 0 on[0, σ ∧ S]withO²constant on[σ ∧ S, τ ∧ (T − )). Applying this fact to the previous equation completes the proof.

Theorem 2.8. Suppose x0 ∈ D1 with v, w ∈ Tx₀M, f ∈ Bb, V bounded below and 0 < S < T. Assumek, l are bounded adapted processes with paths in the Cameron- Martin spaceL^1,2([0, T ]; Aut(Tx₀M ))such that ks = 0fors ≥ σ ∧ S, k0 = 1, ls = 1for

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s ≤ σ ∧ S,ls= 0fors ≥ τ ∧ T,Rσ∧S

0 | ˙ks|²ds ∈ L¹andRτ ∧T

σ∧S | ˙ls|²ds ∈ L¹. Then (∇dP_T^Vf )(v, w)

= − E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

#

− E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

((∇dV_{T −s})(W_s(k_sv), W_s(w)) + (dV_{T −s})(W_s⁰(k_sv, w)))ds

#

+ E

"

V^Tf (xT)1_{{T <ζ(x}₀_)}

Z T 0

hWs( ˙lsw), //sdBsi + dVT −s(Ws(lsw))ds

!

· Z T

0

!#

.

Proof. Modifying the process l_t to l_t as in the proof of Theorem 2.2, it follows from Lemma 2.7, the strong Markov property, the boundedness ofP_t^Vf on[0, T ] × D2and the boundedness ofdP_t^Vf and∇dP_t^Vf on[, T ] × D2that the formula

(∇dP_T^Vf )(v, w)

= − E

"

V^Tf (xT)1_{{T <ζ(x}₀)}

Z T 0

#

+ E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

hWs( ˙l_sw), //_sdB_si Z T

0

hWs( ˙k_sv), //_sdB_si

#

− E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

#

+ E

"

V^Tf (xT)1_{{T <ζ(x}₀_)}

Z T 0

dVT −s(Ws(lsw))ds Z T

0

#

− E

"

V^Tf (xT)1_{{T <ζ(x}₀_)}

Z T 0

Z s 0

dV (Wr(w))dr

#

− E

"

Z T

0 V^sdfs(Ws(ksv))dVT −s(Ws(w))ds

#

− E

"

Z T

0 V^sdfs(Ws(w))dVT −s(Ws(ksv))ds

#

holds with l_t in place oflt, and therefore in terms of lt by taking ↓ 0. Paying close attention to the assumptions on l andk, it follows from this, by Theorem 2.2 and the strong Markov property, that

− E

"

Z T 0

V^sdfs(Ws(w))dVT −s(Ws(ksv))ds

#

= + E

"

V^Tf (xT)1_{{T <ζ(x}₀)}

Z T 0

Z T r

hWs( ˙lsw), //sdBsidVT −r(Wr(krv))dr

#

− E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

Z r 0

dV_{T −u}(W_u(w))du

dV_{T −r}(W_r(k_rv))dr

#

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+ E

"

V^Tf (xT)1_{{T <ζ(x}₀)}

Z T 0

dVT −r(Wr(krv))dr Z T

0

dVT −s(Ws(lsw))ds

#

= + E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

0

#

− E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

Z r 0

dV_{T −u}(W_u(w))du

#

+ E

"

V^Tf (xT)1_{{T <ζ(x}₀_)}

Z T 0

0

dVT −s(Ws(lsw))ds

#

from which it follows that (∇dP_T^Vf )(v, w)

= − E

"

V^Tf (xT)1_{{T <ζ(x}₀)}

Z T 0

#

+ E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

0

hWs( ˙k_sv), //_sdB_si

#

− E

"

VTf (x_T)1_{{T <ζ(x}₀_)}

Z T 0

#

+ E

"

V^Tf (xT)1_{{T <ζ(x}₀_)}

Z T 0

dVT −s(Ws(lsw))ds Z T

0

#

+ E

"

V^Tf (xT)1{T <ζ(x)}

Z T 0

hWs˙lsw, //sdBsi Z T

0

dVT −r(Wr(krv))dr

#

+ E

"

Z T 0

0

dVT −s(Ws(lsw))ds

#

− E

"

Z T 0

Z s 0

dVT −r(Wr(w))dr

#

− E

"

VTf (x_T)1{T <ζ(x)}

Z T 0

Z s 0

dV_{T −r}(W_r(w))dr

dV_{T −s}(W_s(k_sv))ds

#

− E

"

Z T 0

Vsdf_s(W_s(k_sv))dV_{T −s}(W_s(w))ds

# .

Finally, by the stochastic Fubini theorem [20, Theorem 2.2] we have E

V^Tf (xT)1{T <ζ(x₀)}

Z T 0

Z s 0

dVT −r(Wr(w))dr(hWs( ˙ksv), //sdBsi + dVT −s(Wsksv)ds)

= E

VTf (xT)1{T <ζ(x₀)}

Z T 0

Z T s

hWr( ˙krv), //rdBri + dVT −r(Wrkrv)dr

dVT −s(Ws(w))ds

which cancels the final three terms in the previous equation, by the strong Markov property, Theorem 2.2 and the assumptions onk.

For the caseZ = 0andV = 0, Theorem 2.8 reduces to [1, Theorem 2.1].

Remark 2.9. We have assumed thatV is bounded below and smooth. However, so long asV is bounded below and continuous withVt∈ C¹ for eacht ∈ [0, T ]andP^Vf ∈

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C^1,3([, T ] × M )then the results of Subsection 2.1 evidently remain valid. Similarly, the results of Subsection 2.2 evidently remain valid ifV is bounded below,C¹withV_t∈ C² for eacht ∈ [0, T ]and P^Vf ∈ C^1,4([, T ] × M ). Similarly, the results of Subsection 2.3 evidently remain valid if V is bounded below and continuous with Vt ∈ C² for each t ∈ [0, T ]andP^Vf ∈ C^1,4([, T ] × M ).

3 Stationary Solutions

Now supposeφ ∈ C²(D) ∩ C(D)solves the eigenvalue equation (L − V )φ = −Eφ

on the regular domain D, for some E ∈ R and a function V ∈ C² which does not depend on time and which is bounded below. Denoting byτ the first exit time fromD of the diffusionxt with generatorLand assuming x0 ∈ D, one has, in analogy to the Feynman-Kac formula (3), the formula

φ(x0) = EVτφ(xτ)e^Eτ .

Furthermore, the methods of the previous section can easily be adapted to find formulae for the derivatives ofφ. In particular, one simply setsf_t= φ, replacesV_{T −t}withV − E and the calculations carry over almost verbatim (although there is no application of the strong Markov property; in this case the local martingale property is enough). In particular, for the derivativedφ, supposingkis a bounded adapted process with paths in the Cameron-Martin spaceL^1,2([0, ∞), Aut(Tx₀M ))withk0 = 1,kt= 0fort ≥ τ and Rτ

0 | ˙ks|²ds ∈ L¹, one obtains (dφ)(v) = −E

V^τφ(xτ)e^Eτ Z τ

0

hWs( ˙ksv), //sdBsi + dV (Ws(ksv))ds

for each v ∈ Tx₀M. When V = 0 and E = 0 this formulae reduces to the one given in [18]. Similarly, denoting byD1 a regular domain withx0 ∈ D1 and D1 ⊂ Dand by σthe first exit time ofxtfrom D1, supposingk, l are bounded adapted processes with paths in the Cameron-Martin spaceL^1,2([0, ∞); Aut(Tx0M ))such thatks= 0fors ≥ σ, k₀ = 1, l_s = 1 fors ≤ σ, l_s = 0fors ≥ τ, Rσ

0 | ˙ks|²ds ∈ L¹ and Rτ

σ | ˙ls|²ds ∈ L¹, for the Hessian ofφone obtains

(∇dφ)(v, w)

= − E

Vσφ(x_σ)e^Eσ Z σ

0

− E

V^σφ(xσ)e^Eσ Z σ

0

((∇dV )(Ws(ksv), Ws(w)) + (dV )(W_s⁰(ksv, w)))ds

+ E

V^τφ(xτ)e^Eτ

Z τ 0

hWs( ˙lsw), //sdBsi + dV (Ws(lsw))ds

·

Z σ 0

hWs( ˙ksv), //sdBsi + dV (Ws(ksv))ds

for allv, w ∈ Tx₀M.

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4 Local and Global Estimates

4.1 Gradient

Theorem 4.1. SupposeD0, Dare regular domains withx0∈ D0⊂ D,V bounded below andT > 0. Set

κ_D:= inf{RicZ(v, v) : v ∈ TyM, y ∈ D, |v| = 1};

v_D:= sup{|(dV_t)_y(v)| : v ∈ T_yM, y ∈ D, |v| = 1, t ∈ [0, T ]}.

Then there exists a positive constant

C ≡ C(n, T, inf V, κ_D, v_D, d(∂D₀, ∂D)) such that

|dP_t^Vf_x₀| ≤ C

√t|f |_∞ (13)

for all0 < t ≤ T,x0∈ D0andf ∈ Bb.

Proof. According to [1], the processk_tappearing in Theorem 2.2 can be chosen so that

|ks| ≤ c(T )for alls ∈ [0, T ], almost surely, with

E

"

Z T 0

| ˙ks|²ds

#¹₂

≤ C˜

p1 − e^{− ˜}^C²^T

for a positive constant C˜ which depends continuously on κ, n and d(∂D0, ∂D). The details of this can be found in [19]. By Theorem 2.2 and the Cauchy-Schwarz inequality, using equation (4) and the parameter κ_D to control the size of the damped parallel transport, we have

|dP_T^Vf | ≤ |f |∞e^{− inf V}



E

"

1_{{T <ζ(x}₀)}

Z T 0

|Ws|²| ˙ks|²ds

#¹₂

+E

"

1_{{T <ζ(x}₀_)}

Z T 0

|dVT −s||Ws||ks|ds

#!

≤ |f |_∞e^{(− inf V −}¹²^(κ^D^∧0))T



E

"

Z T 0

| ˙ks|²ds

#¹₂ + vDE

"

Z T 0

|ks|ds

#



so the estimate (13) follows by substituting the bounds onkand ˙k.

Note that in the above theorem, the dependence of the constantCond(D0, D)is such that if one tries to shrink D onto D0, so as to reduce the information needed about RicZanddV, then the constant blows up at rate1/d(D0, D). There is therefore a trade- off between the size of the domain and the size of the constant. This behaviour is unavoidable and also occurs with respect to the domainsD₀, D₁ andD₂ which appear in Theorems 4.3 and 4.5 below.

Corollary 4.2. Suppose Ric_Z is bounded below with |dV | bounded and V bounded below. Then for allT > 0there exists a positive constantC ≡ C(n, T )such that

|dP_t^Vfx| ≤ C

√t|f |_∞ for all0 < t ≤ T,x ∈ M andf ∈ Bb.