First Order Feynman-Kac Formula

arXiv:1608.03856v1 [math.PR] 12 Aug 2016

Xue-Mei Li and James Thompson

Abstract

We study the parabolic integral kernel associated with the weighted Laplacian and the Feynman-Kac kernels. For manifold with a pole we deduce formulas and estimates for them and for their derivatives, given in terms of a Gaussian term and the semi-classical bridge. Assumptions are on the Riemannian data.

AMS Mathematics Subject Classification : 60Gxx, 60Hxx, 58J65, 58J70

1 Introduction

LetM be a complete connected smooth Riemannian manifold of dimension n and ∆ the Laplace-Beltrami operator with the sign convention that∆ is negative definite. Let h be a smooth real valued function on M , define ∆^h = ∆ + 2L_∇h. We study the solutions of the parabolic equation _∂t^∂f = (¹₂∆^h − V )f where t > 0, V a real valued bounded H ¨older continuous function onM , and lim_t↓0f (t, x) = f (x). Without loss of generality we may assume that V ≥ 0. Denote by Pt^h,Vf its solution, which is also denoted by P_t^V ifh = 0, by P_t^h ifV = 0, and by Ptf if both h and V vanish. Their corresponding integral kernels are denoted by the lower case functions: p^h,V_t , p^V_t ,p^h_t, andp_t. Also, if Z is an additional C¹ vector field the notationsp^h,Z,V_t etc. will be used. All stochastic processes are assumed to have infinite life time.

Set V_t = Eh

e^{−R0tV (xs)ds}i

where (x_t) is the canonical h-Brownian motion, by an h- Brownian motion we mean a strong Markov processes with generator ¹₂∆^h. We deduce the following first order Feynman-Kac formula

d(P_T^h,Vf )(v) = 1 tE



V_Tf (x_T)

Z t

0 hWs(v), u_sdB_si − Z t

0

Z r 0

dV (W_s(v)) ds dr



, whereu_sand W_s are respectively the stochastic parallel and stochastic damped parallel translations along its sample paths, under the assumption that Ric−2 Hess(h) ≥ K where K is a constant. This formula can be obtained by filtering out redundant noise from the equation in [19, section 2]. However it is fairly easy to deduce it directly, avoiding some assumptions made in [19]. From this formula, it is clear thatdP_t^h,Vf is uniformly close to

dP_t^hf . We will in fact show that, for explicit constants C₁(t, K) and C₂(t, K) depending ont and K,

|∇Pt^h,Vf |x0 ≤ 1

√t

2C₁E

(f (x_t)V_tlog(f (x_t)V_t))⁺ ¹²

+ t|∇V |∞C₂. Choosingf to be the Feynman-Kac kernel leads to the following estimates:

|∇ log p^h,Vt |^x0 ≤

√2C₁

√t

 sup

y∈M

log p^h_t(y, y₀)

p^h_2t(x₀, y₀) + 2t(sup V − inf V )¹₂

+ t|∇V |∞C2.

Together with estimates onsup_y∈Mlog_p^pt(y,y0)

2t(x0,y0), this leads to the estimate of S.-J. Sheu [41]. Sheu’s estimates and extensions can be found in [28, 44, 20], all for the case V = 0 and h = 0. Feynman-Kac formula is a popular subject see [21, 42, 45, 26, 5, 37, 34, 14]. Differentiating Feynman-Kac semigroups has been studied in connection with Hamiltonian-Jacobian equation, see [15, 12, 11].

IfM is a manifold with a pole y₀, by which we mean that the exponential mapexp_y0is a diffeomorphism, it makes sense to compare the Feynman-Kac kernel and its derivatives with the ‘Gaussian kernel’. We do not make assumptions on uniform ellipticity of the operator; all assumptions will be on Riemmannian data. Let Jy0 denote the Jacobian determinant of the exponential map aty₀andΦ(y) = ¹₂J

1

y20(y)∆J⁻

1

y02(y). The subscript y₀ will be omitted from time to time. For T > 0 fixed, a semi-classical bridge ˜x_sis a time dependent diffusion with generator ¹₂△ + ∇ log kT −s(·, y0) where, ford the Riemannian distance function,

k_t(x₀, y₀):= (2πt)⁻ⁿ²e^−d

2(x0,y0)

2t J⁻¹²(x₀).

On Rⁿ, the semi-classical bridge agrees with the conditioned Brownian motion. Moreover the process r_t = d(˜x_t, y₀) is then-dimensional Bessel bridge. In [15], K. D. Elworthy and A. Truman proved the following formula, whose consideration comes from classical mechanics and semi-classical limits,

p^V_T(x0, y0)= kT(x0, y0)E h

e^{R0T(Φ−V )(˜xs)ds}i

. (1.1)

We give a simple proof for the following formula, see [49] , for allx₀ ∈ M, p^h,V_T (x₀, y₀)= e^{h(y0)−h(x0)}k_T(x₀, y₀)Eβ_T^h,

β_T^h = exp

Z t 0



Φ − V − 1

2|∇h|²−1 2∆h

 (˜x_s)ds

 ,

The methods in [49] are similar to that in [15, 17, 16]. Here we use a different method, which allows us to deduce a first order formula. The functionΦ is bounded on manifolds of constant negative curvature. If−¹₂|∇h|²− ¹₂∆h is bounded, the formula leads easily to a Gaussian upper bound for the integral kernelp^h_t. In this formula, an additional non- gradient type driftZ is also allowed for which we follow a beautiful idea of Watling [49].

The semi-classical bridge will be replaced by the semi-classical Riemannian bridge whose Markov generator is ¹₂△ + ∇ log kT −s(·, y0)+ ∇S, where

S(x) = Z 1

0 h ˙γ(u), Z(γ(u))idu,

forγ : [0, 1] → M the unique geodesic from x to y0, representing the path average of the radial part ofZ.

Letu˜_tdenote the solution to the stochastic differential equation (2.2) on the orthonor- mal frame bundle with u˜₀ ∈ π⁻¹(x₀) and set x˜_t = π(˜u_t). We often need the condi- tion that Ric−2 Hess(h) is bounded from below. Following the notation in [30, 31], set ρ^h= inf_|v|=1{Ric(v, v) − 2 Hess h(v, v)}.

Theorem Assumey0is a pole,V ∈ C^1,α∩ BC¹,Φ and f ∈ L∞andρ^h≥ K. Then dp^h,V_T (·, y⁰)= 1

Te^{h(y0)−h(x0)}kT(x0, y0)E

 β_T^h

Z T

0 h ˜Wr(·), ˜urd ˜Br− (t − r)∇V dri



whered ˜B_r= dB_r+ ˜u⁻¹_r ∇ log(e^−hk_{T −r})(˜x_r)dr and ˜W is the solution to (2.4).

From this theorem we immediately see that, for an explicit constantC(K, h, |∇ log J|∞), depending only onK, h, and |∇ log J|∞, the following estimate holds,

|∇p^h,V_T (·, y0)|x0

≤ Ce^{h(y0)−h(x0)}(2πt)⁻ⁿ²e^−d

2(x0,y0) 2t J⁻

1

y02(x₀)|βT^h|∞

d(x₀, y₀)

T + 1 + |∇h|∞+ 1

√T + T |dV |∞

 .

LettingZ_T = ^βhT

E(β_T^h), we also have:

∇ log p^h,VT (·, y0) 

x0 ≤ C(K, |∇ log J|∞)|ZT|L2(Ω)

d(x₀, y₀)

T + 1 + |∇h|∞+ 1

√T + T |dV |∞

 . There is also a version of the above formula and estimates for H ¨older continuous potential

V , which however involves a ln T |V |∞ term. Note that precise Gaussian estimates for heat kernels and their derivatives using semi-classical bridge were obtained by S. Aida [1] for asymptotically flat Riemannian manifolds with a pole where the derivaties oflog J up to order 4, the Riemmanina curvature and the derivative of the Ricci curvature are assumed to be bounded. Heat kernel formula for Schr¨odinger type operator acting on sections of vector bundles can be found in M. Ndumu [38] and M. Braverman [8]. The study of the probability measure induced by the semi-classical bridge has been followed up in [32].

2 Preliminaries and First Order Feynman-Kac Formula

In this section we introduce the notation and the preliminary results. Denote by BC^r the space of bounded C^rfunctions onM with bounded derivatives, C_K^r its subspace of functions with compact supports andC^1,α the space of functions whose first derivatives are locally H ¨older continuous.

The h-Brownian motion we use will be given by the canonical construction below. Let {Btⁱ} be a family of independent one-dimensional Brownian motions on a filtered prob- ability space {Ω, F, Ft, P}, set Bt = (B_t¹, . . . , B_tⁿ). Let{Hi} be canonical horizontal vector fields on the orthonormal frame bundle OM of M , associated to an orthonormal basis of Rⁿ. The tilde sign over a vector field onM indicates its horizontal lift to OM . If the h-Brownian motion is complete, which holds if Ric −2 Hess(h) is bounded from below, then the following stochastic differential equation (SDE) is complete,

du_t= Xn

i=1

H_i(u_t)◦ dBtⁱ+ f∇h(ut)dt. (2.1)

Furthermore ifπ is the projection from OM to M , x_t:= π(u_t) is a h-Brownian motion on M starting at x₀ := π(u₀). Ifh vanishes it is sufficient to assume that Ric_x ≥ −α(r(x)) where α grows at most quadratically and Ric_x is the Ricci curvature at x ∈ M and Ric= inf_{v∈TxM :|v|=1}{Ricx(v, v)}. The corresponding equation,

d˜u_t= Xn

i=1

H_i(u˜_t)◦ dBtⁱ+ f∇h(˜ut)dt +∇ log k^T −t(u˜_t)dt, t < T, (2.2)

gives rise to the semi-classical bridge in the same way,x˜_t= π(˜u_t).

Let Ric^♯_x : TxM → T^xM be the linear map defined by hRic^♯xu, vi = Ric^x(u, v).

Denote by (W_t) and ( ˜W_t) respectively the solutions to the following two equations, the first, along (x_t), being

D

dtWt= −1

2Ric^#_xt(Wt)+ Hess(h)(Wt), W0 = idT_x0M (2.3) and the second, along (x˜t), being

D

dtW˜_t= −1 2Ric^#_x˜

t( ˜W_t)+ Hess(h)( ˜W_t), W˜₀= id_Tx0M. (2.4) Here^D_dtW_t= u_tdt^du⁻¹_t W_tand_dt^DW˜_t= ˜u_tdt^du˜⁻¹_t W˜_t, and so the first equation, for example, is interpreted as follows:u⁻¹_t W_tsolves the equation

d

dtw_t= −1

2(u⁻¹_t Ric^♯_xtu_t)w_t+ u_t⁻¹Hess(h)(u_tw_t), w₀ = idRⁿ. Throughout this section we assume that theh-Brownian motions do not explode.

Ifh is a smooth real valued function on M then ∆^h = (d + d^∗)² whered^∗ is the L² adjoint of the exterior differential operatord with respect to the measure e^2hdvol and

the initial domain of∆^h consists of smooth and compactly supported differential forms.

Thend + d^∗ and all its powers are essentially self-adjoint, [30, ch.2], and we denote by the same notation their closures. Suppose thatV is bounded, then the operator f → V f is

∆^hbounded and by the Kato-Rellich theorem, ¹₂∆^h− V is self-adjoint on the domain of

∆^h and essentially self-adjoint onC_K^∞. By functional calculuse^{−t(12∆h−V )} is a strongly continuous contraction semi-group on L₂ ∩ Bb, where the L² space is defined by the weighted measuree^2hdx. Furthermore if f ∈ L²(M ; e^2hdx) then e^{−t(12∆h−V )}f belongs to the domain of ∆^h. By direct computation E(f (x_t)V_t), where V_t = e^{−R0tV (xr)dr}, is also a strongly continuous contraction semi-group onL2∩ L∞with generator ¹₂∆^h− V and coreC_K^∞. Consequently

e^{−t(12∆h−V )}f = E (f (x_t)V_t) , f ∈ L2∩ L∞. Furthermore they solve the variation of constant formula:

g_t= P_t^hf + Z t

0

P_t−s^h (V g_s)ds,

and consequently they areC^1,2 functions and solve the parabolic equation. The solution measure has a density p^h,V_t with respect to the volume measure. We need the following formulation for the Feynman-Kac formula.

Lemma 2.1 IfP_t^h,Vf is a C^1,2solution to the parabolic equation, then V_sP_t−s^h,Vf (x_s) = P_t^h,Vf (x₀)+

Z s 0

V_rdP_t−r^h,Vf (u_rdB_r), 0 ≤ s ≤ t. (2.5)

Proof By the assumption on the h-Brownian motion, u_s exists for all time. We may therefore apply Itˆo’s formula to V_sP_t−s^h,Vf (x_s), using dV_s = −V (xs)V_sds and dx_s =

u_s◦ dBs+ ∇h(xs)ds to obtain (2.5). 

Fork ∈ 0, 1, · · · denote by H^kthe completion ofH₀^k, where

H₀^k=



f ∈ C^∞: |f |²Hk = Xk j=0

|∇^(j)f |²L² < ∞

 ,

under the norm | · |H^k. Denote by C_K^∞Hk the closure of C_K^∞ under | · |H^k. Denote also by d^⋆ the dual of d in L²(M ; dx), taking h = 0, the latter having initial domain C_K^∞. Then the Laplace-Beltrami operator on functions is the closure of−d^⋆d and for any complete Riemannian manifold Dom(d) = C_K^∞H

1

= H¹. For higher order derivatives the corresponding statements hold for manifolds with bounded geometry, see [4]. For k = 2, C_K^∞H

2

= H²if the injectivity radius ofM is positive and if the Ricci curvature is bounded below, see E. Hebey [27]. We avoid these assumptions.

Recall thatρ^h(x) = inf_v∈STxM{Ric(v, v) − 2 Hess(h)(v, v)}.

Lemma 2.2 FixT > 0. Assume that V is bounded with V ∈ C^1,α. Iff ∈ L² ∩ BC¹ then for all0 ≤ t < T and v ∈ Tx0M we have

V_t·

dP_{T −t}^h,Vf

(W_t(v)) = dP_T^h,Vf (v) + Z t

0

V_s·

∇dP_{T −s}^h,Vf

(u_sdB_s, W_s(v)) +

Z t 0

V_s· dV (Ws(v)) · P_{T −s}^h,V f (x_s)ds.

(2.6)

If furthermore|dV | is bounded and ρ^h is bounded from below, then for allt ∈ [0, T ), E

hV_t(dP_{T −t}^h,Vf )(Wt(v))i

= d(P_T^h,Vf )(v) + E

Z t 0

V_sdV (Ws(v))P_{T −s}^h,Vf (xs)ds

 . (2.7) Proof SinceV ∈ C^1,α, the solutionP_t^h,Vf is three times differentiable in space and we may differentiate both sides of the parabolic equation. SinceP_t^h,Vf ∈ Dom(d), it follows thatd∆(P_t^h,Vf ) = ∆^1,hd(P_t^h,Vf ), see [30, Chapter 2] or [22] for h = 0 case. Consider the function (t, α, (x, v)) ∈ [0, T ]×R⁺×T M 7→ (α, dP_{T −t}^h,Vf (v)) and apply Itˆo’s formula to it and to the process (t, V_t, W_t(v)) to obtain

V_tdP_{T −t}^h,Vf (W_t(v)) − dP_T^h,Vf (v)

= Z _t

0

V_s∇(dP_{T −s}^h,V f )(u_sdB_s, W_s(v)) + Z _t

0 ∇Vs(dP_{T −s}^h,V f )(∇h(xs), W_s(v))ds +

Z t 0

V_s

 ∂

∂sdP_{T −s}^h,V f



(W_s(v)) ds +1 2

Z t 0

V_str∇²(dP_{T −s}^h,Vf )(W_s(v)) ds +

Z t 0

V_sdP_{T −s}^h,V f

D dsW_s(v)

 ds −

Z t 0

V (x_s)V_sdP_{T −s}^h,Vf (W_s(v)) ds.

(2.8)

Using Bochner’s formula, ∆^1,h = trace ∇² + 2L_∇h − Ric^♯ for the Laplace-Beltrami operator∆¹on differential1-forms, the definition of the Lie derivative and equation (2.3), we thus have

V_tdP_{T −t}^h,Vf (W_t(v)) − dPT^h,Vf (v)

= Z t

0

V_s∇(dP_{T −s}^h,Vf )(u_sdB_s, W_s(v)) − Z t

0

V (x_s)V_sdP_{T −s}^h,Vf (W_s(v)) ds +

Z t 0

V_s

∂

∂sdP_{T −s}^h,Vf



(W_s(v)) ds +1 2

Z t 0

V_s∆^1,h(dP_{T −s}^h,Vf )(W_s(v)) ds.

We can commute the time and space derivatives and also commute∆^hwithd to obtain

∂

∂sd(P_{T −s}^h,V f ) +1

2∆^1,h(dP_{T −s}^h,V f ) = V d(P_{T −s}^h,V f ) + (dV )P_s^h,Vf.

By substituting into equation (2.8) we then obtain, after some cancellation, the required

formula. 

Lemma 2.3 Assume thatρ^h is bounded from below andV is a bounded H¨older continu- ous function. Then for allf ∈ L∞andv ∈ Tx0M ,

(dP_t^h,Vf )(v) =1 tE

 f (x_t)

Z t

0 hWs(v), u_sdB_si



+ E

 f (xt)

Z t 0

e^{−Rt−st V (xu)du}V (x_t−s) t − s

Z _t−s

0 hW^r(v), urdBri

 ds

 .

Proof Letf ∈ BC¹∩ L²and we first assume thatV belongs also to BC¹. We differenti- ate both sides of the variation of constants formulaP_t^h,Vf = P_t^hf +Rt

0P_t−s^h (V P_s^h,Vf ) ds to obtain forv ∈ Tx0M ,

dP_t^h,Vf (v) =dP_t^hf (v) + Z t

0

dP_t−s^h (V P_s^h,Vf )(v) ds

=dP_t^hf (v) + Z t

0

1 t − sE



(V P_s^h,Vf )(x_t−s) Z _t−s

0 hWr(v), u_rdB_ri

 ds.

Since the first two terms in the equation make sense, so does the last term, which by the standard Feynman-Kac formula, Lemma 2.1, and the Markov property, has the following expression:

E



(V P_s^h,Vf )(x_t−s) Z _t−s

0 hWr(·), urdB_ri



= E



V (x_t−s)f (x_t)e^{−Rt−st V (xu)du} Z _t−s

0 hWr(·), urdB_ri

 .

The formula follows from the corresponding formula for P_t^hf which is well known and can be easily seen by multiply both sides of (2.5) by the martingaleRt

0husdB_s, W_s(v)i:

E

Z t

0 husdB_w, W_s(v)iP_{T −t}^h f (x_t)



= E

Z t 0

dP_{T −r}^h f (u_rdB_r) Z t

0 husdB_s, W_s(v)i



= E Z t

0

dP_{T −r}^h f (W_r(v))dr = t dP_T^hf (v),

The last identity follows from the second formula in Lemma 2.2. Then using the nor- malised geodesic σ : [0, 1] → M connecting two points x and y in M so that f (x) − f (y) = R1

0 d

ds[f (σ(s))]ds and by the formula above, and the fact that V is bounded,

|Wt| is bounded to see that |dPt^h,Vf (x) − dPt^h,Vf (y)| ≤ C|f |∞d(x, y). In particular if Q^h,V_t (x, ·) denotes the probability measure associated with DPt^h,Vf , then |Q^h,Vt (x, M ) − Q^h,V_t (y, M )| ≤ Cd(x, y). Thus the total variation norm of |Q^h,Vt (x, ·) − Q^h,Vt (y, ·)| ≤ Cd(x, y) which means that the required formula holds for a bounded measurable func- tion f . For a bounded H¨older continuous V it is sufficient to approximate with regular

functions. 

Proposition 2.4 Suppose that Ric−2 Hess(h) ≥ K and that V ∈ C^1,α∩ BC¹. Then for all0 ≤ t < T , v ∈ Tx0M and f ∈ L∞, (2.7) holds.

Proof We prove the formula using equation (2.6). By the boundedness ofV , dV and f it is clear thatRt

0V_sdV (W_s(v))P_{T −s}^h,V f (x_s)ds is bounded. Since |Wt(v)| is bounded, we use Lemma 2.3 to conclude that|d(P_{T −t}^h,Vf | ∈ L², and so does Vtd(P_{T −t}^h,Vf )(Wt(v)) which means that the stochastic integral appearing in equation (2.6) isL²-bounded and therefore

a true martingale with vanishing expectation. 

Theorem 2.5 Assume thatV ∈ C^1,α∩BC¹andf ∈ L∞. Suppose that Ric−2 Hess(h) ≥ K. Then for all 0 < t ≤ T and v ∈ Tx0M we have

dP_T^h,Vf (v) = 1 tE



V_Tf (x_T) Z t

0 hWs(v), u_sdB_si



−1 tE



V_Tf (x_T) Z t

0

Z r 0

dV (W_s(v)) ds dr

 . Proof By Lemma 2.1 we have

V_tP_{T −t}^h,Vf (x_t)= P_T^h,Vf (x₀)+ Z t

0

V_rd(P_{T −r}^h,V f )(u_rdB_r). (2.9)

Next we multiply the above equation by the L² martingale Rt

0husdB_s, W_s(v)i and use Itˆo’s isometry to obtain

E



V_tP_{T −t}^h,Vf (x_t) Z t

0 husdB_s, W_s(v)i



= E

Z t 0

V_rd(P_{T −r}^h,V f )(W_r(v)) dr



using the fact that the last term in equation (2.9) is an L² martingale. We now apply equation (2.7) to deduce that for any0 < t ≤ T ,

dP_T^h,Vf (v) = 1 tE



V_tP_{T −t}^h,Vf (x_t) Z t

0 hurdB_r, W_r(v)i



−1 tE

Z t 0

Z r 0

V_sdV (Ws(v))P_{T −s}^h,Vf (xs)ds dr

 (2.10)

and the rest follows from the Markov property. 

This extends the formula in [7] for the logarithmic derivative of the heat kernel (on a compact manifold) proved using Malliavin calculus, the latter was extended to non- compact manifolds in [30, 19] by an elementary stochastic calculus. For manifolds whose second fundamental form of an isometric embedding satisfies suitable conditions, the formula can be deduced from that in [19] using the techniques of filtering our redundant noise. Here we do not make any assumptions on the second fundamental form. See also [18] and [33] for reaction-diffusion equation on Rⁿand [46].

For estimations we need the following lemma, which is [43, Lemma 6.45].

Lemma 2.6 Suppose (Ω, F, P) is a probability space and φ ≥ 0 is a measurable function onΩ. If Ψ is a measurable function on Ω such that φΨ is integrable then

E[φΨ] ≤ E [φ log φ] + log E [exp(Ψ)] .

Proposition 2.7 Suppose that Ric−2 Hess(h) ≥ K and V ∈ C^1,α∩ BC¹. Then for a non-negative bounded measurable functionf we have

|∇Pt^h,Vf |x0 ≤ 1

√t

2C₁(t, K)E

(f (x_t)V_tlog(f (x_t)V_t))⁺ ¹²

+ t|∇V |∞C₂(t, K) for allt > 0 where

C1(t, K) := 1 − e^−Kt

Kt , C2(t, K) := 2

Kt 1 + e^−Kt/2− 1 Kt/2

!!

.

Proof Forγ ∈ R set

φ := f (x_t)V_t, ψ_t:= γ Z t

0 hWs(v₀), u_sdB_s− (t − s)∇V (xs)dsi to see, by Theorem 2.5, Fubini’s theorem and Lemma 2.6, that forv₀ ∈ Tx0M ,

γtd(P_t^h,Vf )(v₀)≤ E [φ log φ] + log E

 exp

 γ

Z t

0 hWs(v₀), u_sdB_s− (t − s)∇V (xs)dsi



.

Since ^D_dsW_s= −¹₂Ric^♯(W_s)+ ∇²h(W_s, ·)^♯ we have|Ws| ≤ e^−Ks2 so, log E

 exp

 γ

Z t

0 hWs(v₎, u_sdB_s− (t − s)∇V (xs)dsi



≤ log E

 exp



|γ|

Z t

0 hWs(v₀), u_sdB_si + | γ|

Z t 0

(t − s)h−∇V (ξs), W_s(v₀)ids  



≤ γ² Z t

0

e^−Ks|v0|ds + |γ||∇V |∞

Z t 0

(t − s)e^−Ks2 |v0|ds

≤ γ²tC1(t, K)|v⁰| + |γ||∇V |∞t²C2(t, K)|v⁰|.

Thus we obtain

γtd(P_t^h,Vf )(v₀)≤ E

(φ log φ)⁺

+ γ²tC₁(t, K)|v0| + |γ||∇V |∞t²C₂(t, K) which after minimizing overγ yields

t|∇Pt^h,Vf |x0 ≤ 2tC1(t, K)E

(φ log φ)⁺
¹₂

+ t²|∇V |∞C₂(t, K),

as claimed. 

IfP_t^h,Vf (x₀)> 0 we define the non-negative quantity Ht(f, x₀):= E

"

f (x_t)V_t

P_t^h,Vf (x₀)log f (x_t)V_t P_t^h,Vf (x₀)

!#

,

and replacing the non-negative functionf in Proposition 2.7 by ^f

P_t^h,Vf (x0), we deduce the following corollary.

Corollary 2.8 Suppose that Ric−2 Hess(h) ≥ K and V ∈ C^1,α∩ BC¹. Then for any bounded measurable functionf ≥ 0 we have

|∇ log Pt^h,Vf |x0 ≤ 1

√t



2C₁(t, K)Ht(f, x₀)

¹₂

+ t|∇V |∞C₂(t, K), t > 0.

and

|∇ log p^h,Vt |^x0 ≤ 1

√t

p2C1(t, K) sup

y∈M

log p^h_t(y, y₀)

p^h_2t(x0, y0)+2t(sup V −inf V )¹₂

+t|∇V |∞C2(t, K).

Indeed, choosingf (·) = p^h,Vt (·, y0), the Feynman-Kac kernel associated toP_t^h,V, we have

Ht(f, x₀)≤ sup

y∈M

log

f (y)Eh

e^{−R0tV (xs)ds}|xt= yi P_t^h,Vf (x₀) E

"

f (x_t)V_t P_t^h,Vf (x₀)

#

≤ sup

y∈M

logp^h,V_t (y, y₀)e^{−t inf V} p^h,V_2t (x0, y0)

= sup

y∈M

log

p^h_t(y, y₀)Eh

e^{−R0tV (bt,y,y0s )ds}i

e^{−t inf V} p^h_2t(x₀, y₀)Eh

e^{−R02tV (bt,x0,y0s )ds}i

≤ sup

y∈M

log p^h_t(y, y₀)

p^h_2t(x₀, y₀) + 2t(sup V − inf V )

where b^t,x,y_s is the process obtained by conditioning theh-Brownian motion by x_t = y and x₀ = x. Given suitable heat kernel upper and lower bounds, or more generally a Harnack inequality for p^h_t, we would have an estimate on the logarithmic derivative of p^h,V_t . These two types of assumptions are naturally related. For the first see [10, 29, 9, 25, 13, 39], [29, Corollary 3.1], and [24, Thms 7.4, 7.5, 7.9]. If the Sobolev inequality

|f |²_2n/(n−2) ≤ CR

M|∇f |²dx holds for f ∈ CK^∞ then the heat kernel satisfies the on- diagonal estimate p_t(x, x) ≤ Ct⁻ⁿ² (leading to off-diagonal estimates). In fact, N. Th.

Varopoulos proved that the Sobolev inequality is also necessary for the on-diagonal upper bound, [23, Lemma 5.1, Thm 6.1]. See also [24, 40] for a clear account on the relation between various functional inequalities and the on diagonal Gaussian upper bounds of the type vol(B(x,√

t))⁻¹e^−d(x,y)

2

t . For the case h = 0 and Ric ≥ −K, K ≥ 0, a global

Harnack inequality exists, [29, Thm. 2.2], for positive solutions of the heat equation. For example, [6, Corollary 2],

ft(x) f_t+s(y) ≤

t + s t

ⁿ₂ exp

"

(d(x, y) +√ nKs)² 4s

+

√nK

2 min{(√

2 − 1)d(x, y),

√nK 2 s}

# . From this can be deduced the following theorem,

Theorem 2.9 Suppose that the Ricci curvature is bounded from below andV ∈ C^1,α∩ BC¹. Then for allT > 0 there exists a positive constant C(T ) such that

|∇ log p^Vt (·, y0)(x₀)| ≤ C(T )

1

t +d²(x₀, y₀)

t² + |V |∞

¹₂

+ C(T )t|∇V |∞

for allt ∈ (0, T ] and x0, y₀ ∈ M.

WhenV = 0, this recovers the estimates in J. Sheu [41], see also [28, 35, 44]. For brevity we do not write down the estimate involving h it is however worth noticing that gradient formula forP_t^hf would lead to a parabolic Harnack inequality for p^h_t, see [3] for such an estimate.

3 On Manifolds with a Pole

Letn ≥ 2. Assume that y0is a pole forM . That is, we assume that the exponential map exp_y0is a diffeomorphism betweenTy0M and M . If the Jacobian J : M → R defined by

J(y) ≡ Jy0(y) = | det Dexp⁻¹_y0(y)exp_y0|

is non-singular, then we denote byJ⁻¹the reciprocal ofJ and for t > 0 define Φ(y) = 1

2J¹²(y)∆J⁻¹²(y), k_t(y) = (2πt)⁻ⁿ²e^−r

2(y)

2t J⁻¹²(y)

fory ∈ M, where r denotes the distance to the pole y0. The functionJ¹² is called Ruse’s invariant (see for example A. G. Walker [48]). The objective is to obtain probabilistic representation for the kernelp^V and its derivatives, involving the distance function,J and Φ. On the standard n-dimensional hyperbolic space, the heat kernel p_t(x, y) depends on x and y through r = d(x, y) and is given by iterative formulas:

p_t(x, y) =C(m) 1

√t

 1

sinh r

∂

∂r

m

e^−m2t−r

2

4t, n = 2m + 1,

pt(x, y) =C(m)t⁻³²e^−(2m+1)

2

4 t

 1

sinh r

∂

∂r

mZ _∞

ρ

se^−s

2 4t

cosh s − cosh rds, n = 2m + 2.

If its sectional curvature is−R⁻², then [17],

J =

R r sinh r

R

_n−1

Φ = −(n − 1)²

8R² +(n − 1)(n − 3) 8



r⁻²−

R²sinh² r R

₋₁ . In particularΦ is bounded above.

IfM is a model space, i.e. M is a manifold with a pole p such that for every linear isometry φ : T_pM → TpM there exists an isometry Φ : M → M such that φ(p) = p and dΦ_p = φ. In the geodesic polar coordinates, (r, θ) ∈ (0, ∞) × Sⁿ⁻¹, the pull back metric in (0, ∞) × Sⁿ⁻¹ can be written asdr²+ f (r)²dθ². The function f is C^∞ and satisfies f (0) = 0, f^′(0) = 1, f (r) > 0 for r > 0 and f^′′(r) = −R(r)f (r) where R(r) is the sectional curvature in a plane containing the radial direction ∂_r at a point x with d(x, p) = r. Then log J_p(x) = (n − 1) log^{f (r)}_r . If R(r) = R, a constant, then f (r) = ^sinh(√^√Rr)

R andlog f has bounded derivatives of all order. In general,

|∇ log J| = (n − 1) 

(log f )^′(r) −1 r 

 , ∆r = (n − 1)(log f )^′(r),

∆(log J) = (n − 1)²(log f )^′(r)



(log f )^′(r) − 1 r



+ (n − 1)



(log f )^′′(r) + 1 r²

 , Φ = 1

4(n − 1)

n − 2

r² −n − 1

r (log f )^′(r) − (log f )^′′(r)

 .

For general manifolds, volume comparison theorem has seen studied in [50]. But we are not aware of any comparison theorems for∇ log J and Φ.

3.1 Girsanov Transform and Zeroth Order Formula

LetT be a positive number and x₀, y₀ ∈ M. The semi-classical bridge process, between x₀andy₀in timeT , is the diffusion process starting at x₀with generator¹₂∆+∇ log kT −s

wherek_t(x₀, y₀) := (2πt)⁻ⁿ²e^−d

2(x0,y0)

2t J⁻¹²(x₀). Ifu˜_ssatisfies d˜us=X

Hi(u˜s)◦ dBsⁱ+∇ log k^T −s(˜us)ds

withu˜0= u0and ˜Asthe horizontal lift ofAsthenx˜s= π(˜us) is a semi-classical bridge.

Since|∇r| = 1 away from y0, Itˆo’s formula implies for0 ≤ t < T that r(˜x_t)− r(x0) = β_t+

Z t 0

1

2∆r(˜x_s)ds − Z t

0

r(˜x_s)

T − sds −1 2

Z t

0 dr (∇ log J(˜xs)) ds where β_t is a standard one-dimensional Brownian motion. It is clear from this and the formula

∆r = n − 1

r + dr(∇ log J) (3.1)

that r(˜x_t) is distributed as a Bessel bridge, starting at r(x₀) and ending at 0 at time T , from which it follows thatlim_t↑Tx˜_t= y₀, almost surely.

We follow a beautiful idea of Watling [49] and use a modification of the construction by Elworthy-Truman to define the semi-classical Riemannian bridge for a given vector fieldZ. For γ : [0, 1] → M the unique geodesic from x to y0,

S(x) = Z 1

0 h ˙γ(u), Z(γ(u))idu, and naturallyS(y₀)= 0.

Lemma 3.1 [Watling] Letx_tbe a ¹₂∆ + Z + ∇S + ∇(log(kT −s)) diffusion (the semi- classical Riemannian bridge). Thend(y0, ˜xt) is then-dimensional Bessel bridge process.

In the formula for rt := d(˜xt, y0), the additional driftZ + ∇S appears in the form of h∇d, Z+∇Si and this vanishes. Indeed, S(γx(t)) =R1

th ˙γ(u), Z(γ(u))idu so_dt^d|t=0S(γ_x(t)) =

−h ˙γx(0), Z(x)i on one hand, and _dt^d|t=0S(γ_x(t)) = h∇S, ˙γx(0)i on the other hand.

The following basic lemma will be used repeatedly, in which we denote by P the probability distribution of the Brownian motion with drift∇h + Z and by Q the proba- bility distribution of a semi-classical Riemannian bridge. Note thatΦ = ¹₂J¹²∆(J⁻¹²) =

1

2|∇ log J|²−¹₄∆(log J). Define Φ^h = −1

2|∇h|²− 1

2∆h + Φ, Ψ = 1

2|∇S|²+1

2∆S + hZ, ∇S + ∇hi.

Lemma 3.2 Suppose that h ∈ C²(M ; R) and Z is a C¹ vector field. Suppose that the

1

2∆^h+ Z diffusion x_t is complete. Fixt ∈ [0, T ). Then P and Q are equivalent on Ft

with Radon-Nikodym derivative

M_t:= k_T(x₀)e^(h−S)(x0) k_{T −t}(x˜_t)e^{(h−S)(˜xt)}exp

Z t 0

(Φ^h+ Ψ)(˜x_s)ds



. (3.2)

Proof For anyu₀ ∈ OM with π(u0)= x₀, letu˜_sbe the solution of the following equation with initial valueu0:

du_s=X

H_i(u_s)◦ dBsⁱ + ( eZ + f∇h)(us)ds. (3.3) Then (ut) exists for all time andxt= π(ut). Letu˜tbe the solution to

d˜u_s=X

H_i(˜u_s)◦ dBsⁱ+ (∇ log k^T −s+ eZ + g∇S)(˜us)ds, u˜₀ = u₀ (3.4) Thenx˜_s= π(˜u_s) is a semi-classical Riemannian bridge. It has generator ¹₂∆ + Z + ∇S +

∇(log(kT −s)). One crucial observation is that 1

2∆^h+ Z + ∇ log(kT −se^Se^−h)= 1

2∆ + ∇S + Z + ∇ log kT −s(·, y⁰),

and it is possible to treatx˜_tas the (x_t) process by adding the additional drift∇ log(kT −se^S−h).

Since both processes are well defined before timeT , they are equivalent on Ftfor any t < T .

Let{ei}ⁿi=1be an orthonormal basis of Rⁿand let B˜_tⁱ := B_tⁱ+

Z t 0

d log(k_{T −s}e^S−h)(˜u_se_i)ds.

By the Girsanov-Cameron-Martin theorem we obtain,

M_t= exp

"

− Xm

i=1

Z t

0 h∇ log(kT −se^S−h)(˜x_s), ˜u_se_iidBsⁱ− 1 2

Z t

0 |∇ log(kT −se^S−h)|²(˜xs)ds

# .

Now Itˆo’s formula implies

log (e^S−hk_{T −t})(˜x_t)= log (e^S−hk_T)(x₀)+ Xm

i=1

Z t

0 h∇ log(kT −se^S−h)(˜x_s), ˜u_se_iidBsⁱ

+ Z t

0 |∇ log(kT −se^S−h)(˜xs)|²ds + Z t

0

 ∂

∂s +1

2∆^h+ Z



log(k_{T −s}e^S−h)(˜xs)ds so the stochastic integral appearing in the formula forM_tcan be eliminated. Then, by the

relation (3.1) we see that

∂

∂slog k_{T −s}= n

2(T − s)− r² 2(T − s)²,

|∇ log(kT −se^−h)|² = r²

(T − s)² +1

4|∇ log J|²+rdr(∇ log J)

T − s − 2h∇h, ∇ log(kT −s)i + |∇h|²,

∆ log(k_{T −s}e^−h)= − n

T − s−rdr(∇ log J) T − s −1

2∆(log J) − ∆h.

ForS = 0, we have, 1

2|∇ log(kT −se^−h)|²+

∂

∂s +1

2∆^h+ Z



log(k_{T −s}e^−h)

= 1

8|∇ log J|²−1

4∆(log J) + 1

2|∇h|²−1

2∆h − |∇h|²

= 1

8|∇ log J|²−1

4∆(log J) − 1

2∆h − 1 2|∇h|², from which we deduce the formula with non-vanishingS ,

1

2|∇ log(kT −se^−h+S)|²+

∂

∂s +1

2∆^h+ Z



log(k_{T −s}e^−h+S)

=1

2|∇S|²+ h∇S, ∇ log(kT −se^−h)i +1

2∆^hS + hZ, ∇Si + hZ, ∇ log(kT −se^−h)i +1

8|∇ log J|²−1

4∆(log J) − 1

2∆h −1 2|∇h|².

Since∇S + Z vanishes on radial directions, see the proof for Lemma 3.1, the first line on the right hand side of the equation is

1

2|∇S|²− h∇S + Z, ∇hi + 1

2∆S + h∇h, ∇Si + hZ, ∇Si

= 1

2|∇S|²+1

2∆S + hZ, ∇S + ∇hi.

Finally we see that

log (e^S−hk_{T −t})(˜xt)= log (e^S−hkT)(x0)+ Xm

i=1

Z t

0 h∇ log(kT −se^S−h)(˜xs), ˜useiidBsⁱ

+1 2

Z t

0 |∇ log(kT −se^S−h)(˜x_s)|²ds + Z t

0

(Φ^h+ Ψ)(˜x_s)ds.

and the required identity follows. 

In particular, iff ∈ Bb then Lemma 3.2 implies that for0 ≤ t < T , Z

M

f (y)p^h,Z,V_t (x₀, y) dy = E[V_tf (x_t)]= k_T(x₀, y₀)e^(S−h)(x0)E

f (˜x_t)e^{(h−S)(˜xt)} k_{T −t}(˜x_t) β_t^h,Z

 , (3.5) where

β_t^h,Z = exp

Z t 0

(Φ^h+ Ψ − V )(˜xs)ds



. (3.6)

Elworthy and Truman’s proof of the following theorem, for the case h = 0, was inspired by semiclassical mechanics. They used a semiclassical bridge which arrives aty₀ at timeT + λ and took the limit as λ ↓ 0. We give a slightly modified proof, generalising their result for∆^h, the method of which will later be used to derive a derivative formula.

The following generalises a formula by Elworthy-Truman, [17]. Watling [49] treated Brownian motion with a general driftZ.

Theorem 3.3 [15, 49] LetV ∈ C^0,α∩ L∞and suppose that the Brownian motion with drift ∇h + Z does not explode. Suppose that Φ^h + Ψ − V is bounded above, or more generally the following convergencelim_t→T

β_t^h− β^h_T

 = 0. Then p^h,Z,V_T (x₀, y₀)= e^{(h−S)(y0)−(h−S)(x0)}k_T(x₀, y₀)E

 exp

Z T 0

(Φ^h+ Ψ − V )(˜xs)ds



. (3.7) Proof Letφ be a smooth function with compact support with φ(y0) = 1. Denote by Et

the standard Gaussian kernel in the tangent spaceT_y0M . Then, by a change of variables, we see that

limt↑T lim

r↑tp^h,Z,V_t (φk_{T −r})(x₀)= lim

t↑T lim

r↑t

Z

M

p^h,Z,V_t (x₀, y)φ(y)k_{T −r}(y) dy

= lim

t↑T lim

r↑t

Z

T_y0M

(p^h,Z,V_t (x₀, ·) · φ · J^1/2)(exp_y0v)E_{T −r}(v) dv

= (p^h,Z,V_T (x₀, ·) · φ · J^1/2)(exp_y00_y0)= p^h,Z,V_T (x₀, y₀)