Dimension-Free Harnack Inequality and ItsApplications ∗

Feng-Yu WangSchool of Mathematical Science, Beijing Normal University, Beijing 100875, China

Email: wangfy@bnu.edu.cn

January 13, 2006

Abstract

This paper presents a self-contained account concerning a dimension-free Harnackinequality and its applications. This new type of inequality not only implies heat kernelbounds as the classical Li-Yau’s Harnack inequality did, but also provides a direct wayto describe various dimension-free properties of finite- and infinite-dimensional diffusionsemigroups. We will start from a standard weighted Laplace operator on a Riemannianmanifold with curvature bounded from below, then move further to the unboundedbelow curvature case and infinite-dimensional settings.

AMS subject Classification: 58G32, 60J60

Keywords: Harnack inequality, diffusion semigroup, Riemannian manifold, heat kernel, hy-percontractivity, supercontractivity, ultracontractivity.

1 Introduction

The purpose of this paper is to survey some representative contributions of the probabilitygroup at Beijing Normal University. In the past a few years, virtual progress has been made bythe group in the following three directions: Markov processes and spectral theory, stochasticanalysis and geometry, and interacting particle systems and superprocesses. Since the studyin the first direction has been already well summarized in monographs and survey articles (see[?, ?, ?] and references therein), and that in the third direction will be introduced by Professor

∗Supported in part by Creative Research Group Fund of the National Natural Science Foundation of China(No. 10121101) and RFDP(20040027009).

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Zenghu Li in the same issue, we focus this paper on a main topic in the second direction. Moreprecisely, we intend to present a self-contained account concerning a dimension-free Harnackinequality and applications for some finite- and infinite-dimensional diffusion semigroups,which are important in stochastic analysis, functional analysis, and geometric analysis.

The dimension-free Harnack inequality was first established by the author in 1997 ([?])for diffusion semigroups on manifolds with curvature bounded from below. Since then thisinequality has been applied and further developed intensively to the study of functional in-equalities (see [?, ?, ?, ?, ?]), heat kernel estimates (see [?]), high order eigenvalues (see[?, ?, ?]), transportation cost inequalities (see [?]), and short time behavior of transitionprobabilities (see [?, ?, ?]). In particular, the aspect of functional inequalities are relevant tothe ergodicity theory and spectral theory for Markov semigroups, which have been introducedin details in [?, ?] and references therein.

Before moving on, let us briefly recall the classical Li-Yau Harnack inequality and explainthe reason why a new type of dimension-free inequality is needed in applications.

Let M be a d-dimensional compact connected Riemannian manifold such that

(1.1) Ric(X,X) ≥ −K|X|2, X ∈ TM

holds for some K ≥ 0, where Ric is the Ricci curvature. Let Pt := et∆(t ≥ 0) be the heatsemigroup, one has the probability representation

Ptf(x) = Exf(xt), f ∈ Cb(M), x ∈ M, t ≥ 0,

where xt is the diffusion process generated by ∆.

Theorem 1.1. (Li-Yau [?]) For any s, t > 0, α > 1, and f ∈ Cb(M),

Ptf(x) ≤ (Pt+sf(y))(t + s

t

)αd/2

exp

[αρ(x, y)2

4s+

αdKs

4(α− 1)

], x, y ∈ M,

where ρ(x, y) is the Riemannian distance between x and y.

This inequality has been widely applied in geometric analysis, for instances, the heatkernel estimates, estimates of the first eigenvalue and log-Sobolev constant, boundedness ofRiesz transformations (Meyer’s inequality) (cf. [?, ?, ?]). Moreover, this parabolic Harnackinequality has been extended and improved by Bakry and Qian [?] for a more general ellipticdifferential operator L satisfying the following curvature-dimension condition due to Bakryand Emery:

(1.2) Γ2(f, f) :=1

2L|∇f |2 − 〈∇f,∇Lf〉 ≥ −K|∇f |2 +

(Lf)2

n, f ∈ C∞(M)

for some K ∈ R and n > 0. According to the Bochner-Weizenbock formula, when L = ∆,(1.1) is equivalent to (??) with n := d.

The condition (??), however, excludes many important models including the Ornstein-Uhlenbeck operator L := ∆ − x · ∇ on the Euclidean space. Indeed, according to [?] if the

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semigroup generated by L := ∆ +∇V on a complete Riemannian manifold satisfies Li-Yau’sinequality and the distance function is exponential integrable w.r.t. the measure eV dx, wheredx is the Riemannian volume measure, then the manifold has to be compact. On the otherhand, the Ornstein-Uhlenbeck process is very important in stochastic analysis. In particular,it plays the role of Brownian motion in infinite-dimensional analysis. So, it is useful to findout a new type Harnack inequality which works also for operators without the dimension-curvature condition.

It is well-known that although the Ornstein-Uhlenbeck operator does not satisfy thedimension-curvature condition, but its curvature in the sense of Bakry-Emery is positive.More precisely, (2.1) below holds for K = −1. In the next section we first establish thedimension-free Harnack inequality for diffusion semigroups with curvature bounded from be-low, which in particular applies to (finite- or infinite-dimensional) Ornstein-Uhlenbeck semi-groups. As applications of this new type inequality, estimates of heat kernels, various contrac-tivity properties of semigroups, and transportation-cost inequalities are derived. In section3 we extend the dimension-free Harnack inequality for operators with curvature unboundedbelow, while in section 4 we study this inequality and its applications for a class of non-linearstochastic partial differential equations without any curvature condition.

2 Diffusions with Curvature Bounded Below

Consider L := ∆ + Z for some C1-vector field Z such that

(2.1) Ric(X,X)− 〈∇XZ, X〉 ≥ −K|X|2, X ∈ TM

holds for some K ∈ R. Then it is well-known that the diffusion process generated by L isnon-explosive and the corresponding semigroup Pt satisfies the gradient estimate (see e.g. [?])

(2.2) |∇Ptf | ≤ eKtPt|∇f |, t > 0, f ∈ C1b (M).

Starting from this gradient estimate, the author [?] obtained the following Harnack inequality(2.3). This inequality is optimal since according to [?], it is indeed equivalent to the curvaturecondition (2.1).

Theorem 2.1. ([?, ?]) The curvature condition (2.1) holds if and only if for any α > 1 andnonnegative f ∈ Cb(M),

(2.3) (Ptf(x))α ≤ (Ptfα(y)) exp

[Kαρ(x, y)2

2(α− 1)(1− e−2Kt)

], x, y ∈ M.

We remark that comparing with the above Li-Yau’s inequality a remarkable feature of(2.3) is its independence of the dimension, so that the inequality applies also to operatorswith infinite dimensions. Another striking difference between these two inequalities is that

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in (2.3) we commutate the power α > 1 and the semigroup rather than to shift the timeas in Li-Yau’s inequality. To explain this point, we first note from the Bochner-Weizenbockformula that the curvature condition (2.1) is indeed equivalent to (??) for n = ∞. So, merelywith (2.1) we are not able to follow Li-Yau’s argument involving in the leading term (Lf)2,which is crucial in proving the Harack inequality and is exactly corresponding to the shift oftime according to the heat equation. Alternatively, we introduce a power α > 1 such thatthe commutation of this power and the semigroup provides a new leading term |∇Ptf |2, seethe proof below for details. Finally, as explained in the introduction, the Ornstein-Uhlenbecksemigroup does not satisfy Li-Yau’s inequality but does satisfy the present inequality (2.3)for K = −1.

Sketch of the proof. (a) Necessity. Assume 0 < ε ≤ f ≤ c < ∞. Let γ : [0, t] → M be asmooth curve linking x and y, that is γ0 = x and γt = y. Let

φ(s) := log Ps(Pt−sf)α(γs), s ∈ [0, t].

Then

(2.4) logPtf

α(y)

(Ptf(x))α= φ(t)− φ(0).

On the other hand, by the chain rule we obtain

φ′(s) :=Ps

L(Pt−sf)α − α(Pt−sf)α−1LPt−sf

+ 〈∇Ps(Pt−sf)α, γs〉

Ps(Pt−sf)α(γs)

≥ α

Ps(Pt−sf)α(γs)Ps

(Pt−sf)α

[(α− 1)|∇Pt−sf |2(Pt−sf)2

− |γs|eKs|∇Pt−sf |Pt−sf

](γs)

≥ α infr>0(α− 1)r2 − |γs|eKsr = −α|γs|2e2Ks

4(α− 1).

Combining this with (2.4) we arrive at

logPtf

α(y)

(Ptf(x))α≥ − α

4(α− 1)

∫ t

0

|γs|2eKsds.

Maximizing the lower bound over the smooth curve γ, we complete the proof.(b) Sufficiency. According to [?], the curvature condition is equivalent to

(2.5) (Ptf)Pt

f log

f

Ptf

≥ 1− e−2Kt

2K|∇Ptf |2, t ≥ 0, f ∈ C1

b (M), f ≥ 0.

Then it suffices to prove that (2.3) implies (??). Let f ∈ C∞b (M) be positive and equal to

constant outside a compact set, and let x ∈ M be fixed. Let v ∈ TxM with |v| = 1 such that

∂vPtf(x) := 〈v,∇Ptf(x)〉 = −|∇Ptf(x)|.

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Next, for fixed δ > 0, let xs := expx[sv] and αs := 1+δs. Applying (??) to y = xs and α = αs

we obtain

[Ptf(x)

]1+δs ≤ (Ptf1+δs)(xs) exp

[K(1 + δs)s

2δ(1− e−2Kt)

].

Since the equality holds for s = 0, we are able to take derivative for both sides w.r.t. s ats = 0 to obtain

δ[Ptf(x)] log f(x) ≤ δPt(f log f)(x) + ∂vPtf(x) +KPtf(x)

2δ(1− e−2Kt).

Thus,

∂vPtf(x) ≥ −

δ[Pt(f log f)(x)− Ptf(x) log Ptf(x)] +KPtf(x)

2δ(1− e−2Kt)

.

Maximizing the right-hand side w.r.t. δ > 0, we obtain (??).

We now apply (2.3) to the following three typical contractivity properties of Pt (cf. [?]).

Definition 2.1. A semigroup Pt on L2(µ) is called

(I) hypercontractive if there exists t > 0 such that ‖Pt‖L2(µ)→L4(µ) ≤ 1;

(II) supercontractive if ‖Pt‖L2(µ)→L4(µ) < ∞ for any t > 0;

(III) ultracontractive if ‖Pt‖L2(µ)→L∞(µ) < ∞ for any t > 0.

It is well-known that these three contractivity properties are corresponding to Gross’ log-Sobolev inequalities [?]. More precisely, letting (E , D(E )) be the Dirichlet form associatedto a symmetric Markov semigroup Pt on L2(µ), one has (see e.g. [?, ?, ?, ?] and referencestherein):

(i) Pt is hypercontractive if and only if there exists a constant C > 0 such that

Entµ(f 2) := µ(f 2 log

f 2

µ(f 2)

)≤ CE (f, f), f ∈ D(E ).

This inequality implies the exponential decay of Pt in entropy.

(ii) Pt is supercontractive if and only if there exists β : (0,∞) → (0,∞) such that

Entµ(f 2) ≤ rE (f, f) + β(r)µ(f 2), r > 0, f ∈ D(E ).

If this inequality holds for good enough function β, then Pt is ultracontractive with anexplicit upper bound of ‖Pt‖L2(µ)→L∞(µ) determined by β.

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The above two inequalities belong to a general family of functional inequalities, called superPoincare inequalities, which is introduced in [?] to describe the essential spectrum. See also[?] for more applications of these inequalalities including estimates of high order eigenvalues.Moreover, the log-Sobolev inequality can be described as a Poincare type inequality by usingOrlicz norms (cf. [?]). We refer to [?, ?] and references therein for the study of generalPoincare type inequality in Orlicz or Banach norms.

We first apply the Harnack inequality (2.3) to describe the hypercontractivity of Pt.

Theorem 2.2. ([?]) Assume (2.1) and that Pt has an invariant probability measure µ. Letρ(x) := ρ(x, o) for a fixed point o ∈ M.

(1) If there exists λ > K/2 such that µ(eλρ2) :=

∫eλρ2

dµ < ∞, then Pt is hypercontractive.Consequently, if K ≤ 0 (nonnegative curvature) then Pt is hypercontractive if and onlyif µ(eλρ2

) < ∞ for some λ > 0.

(2) For any K > 0 and λ < K/4, there exists an example of M and L := ∆ + Z such that(2.1) holds and µ(eλρ2

) < ∞ but Pt is not hypercontractive.

Sketch of the proof. We only prove (1) since the construction of the counterexample for(2) is very technical. Let f ∈ Cb(M) with µ(|f |α) = 1. By (2.3) we have

1 ≥ |Psf(x)|α∫

exp[− αKρ(x, y)2

2(α− 1)(1− exp[−2Ks])

]µ(dy)

≥ |Psf(x)|αµ(B(o, 1)) exp[− αK(ρ(x) + 1)2

2(α− 1)(1− exp[−2Ks])

],

where and in the sequel B(o, r) is the geodesic ball with center o and radius r. Then

(2.6) |Psf(x)|α ≤ 1

µ(B(o, 1))exp

[ αK(ρ(x) + 1)2

2(α− 1)(1− exp[−2Ks])

], s > 0, x ∈ M.

From (??) it is easy to see that if there exists λ > K/2 such that µ(exp[λρ2]) < ∞, thenthere exists t > 0, α > 1 and q > α such that ‖Pt‖Lα(µ)→Lq(µ) < ∞. This will imply thehypercontractivity in the present setting by using the semigroup log-Sobolev inequality, theinterpolation theorem, and the uniform positivity improving property of Pt, see [?] for details.

Open Problem. Given K ≥ 0, let λ(K) be the smallest possible constant such that for anyconnected complete Riemannian manifold M and any operator L := ∆ + Z on M satisfying(2.1) and having a unique invariant probability measure µ, if µ(eλρ2

) < ∞ for some λ > λ(K)then Pt := eLt is hypercontractive in L2(µ). By Theorem ?? we have λ(K) ∈ [K/4, K/2]. Butthe exact value of λ(K) is still open except for K = 0.

Next, by (??) we have the following criteria for the supercontractivity and ultracontrac-tivity.

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Theorem 2.3. ([?]) In the situation of Theorem ?? we have:

(1) Pt is supercontractive if and only if µ(eλρ2) < ∞ for any λ > 0.

(2) Pt is ultracontractive if and only if ‖Pteλρ2‖L∞(µ) < ∞ for any λ, t > 0.

Sketch of the proof. The sufficiency follows from (??) with α = 2 while the necessity isdue to the concentration property of µ led by log-Sobolev inequalities, see [?] for details.

Moreover, the Harnack inequality provides the following Li-Yau type estimate of the tran-sition density pt(x, y) of Pt.

Theorem 2.4. ([?]) Assume (2.1) and that Pt is symmetric w.r.t. µ (not necessarily finite).Then for any ε > 0 there exists a constant C(ε) > 0 such that

pt(x, y) ≤ C(ε) exp[−ρ(x, y)2/(4 + ε)t]√µ(B(x,

√t))µ(B(y,

√t))

, x, y ∈ M, t > 0.

Sketch of the proof. Given t > 0 and x ∈ M , applying (??) for α = 2 and f :=ps(x, ·)/

√pt(x, x) and s := t/2 we arrive at

pt(x, x) ≤ ∫

M

exp[− Kρ(x, y)2

1− exp[−Kt])

]µ(dy)

−1

≤ c

µ(B(x,√

t)

for some constant c > 0 independent of t and x. Combining this and an argument ofGrigor’yan [?], we complete the proof (see [?] for details).

Finally, we introduce an application of the dimension-free Harnack inequality made byBobokov, Gentile and Ledoux [?] to the so-called HWI inequality, which links the relativeentrop, the Wasserstein distance (transport cost), and the Dirichlet form (energy). We referto [?, ?] and references therein for additional results in this direction. For any two probabilitymeasures µ and ν on M , define the L2-Wasserstein distance

W2(ν, µ) := infπ∈C (ν,µ)

∫

M×M

ρ2dπ

1/2

,

where C (ν, µ) is the set of all couplings of ν and µ.

Theorem 2.5. ([?]) Assume (2.1) for Z := ∇V such that dµ := eV (x)dx is a probabilitymeasure. Then

µ(f 2 log f 2) ≤ 2√

2µ(|∇f |2)W2(f2µ, µ) + KW2(f

2µ, µ)2, µ(f 2) = 1.

7

Sketch of the proof. Let f > 0 such that µ(f) = 1. Repeating the proof of Theorem 2.1with

φ(s) := Pt(log P2t−sf)(γs), s ∈ [0, t]

and taking the optimal choice of γ, we arrive at

Pt(log Ptf)(x) ≤ log P2tf(y) +Ke2Kt

2(e2Kt − 1)ρ(x, y)2.

Combining this with the Monge-Kantorovitch representation of the Wasserstein distance andthe fact that µ(log P2tf) ≤ 0, we complete the proof.

To conclude this section, we introduce some additional references concerning the dimension-free Harnack inequality and applications in infinite dimensional diffusion semigroups satisfyingthe gradient estimate as in (2.2).

(1) Aida and Kawabi [?]: short times behavior of transition probability for infinite-dimensionalO-U processes.

(2) Aida and Zhang [?]: short times behavior of transition probability for diffusions on pathgroups (path spaces over Lie groups).

(3) Kawabi [?]: short times behavior of transition probability for diffusions on Hilbertspaces.

(4) Rockner and Wang [?]: Functional inequalities, transition density estimates for Levytype diffusion-jump processes on Hilbert spaces.

3 Diffusions with Curvature Unbounded Below

It is well-known that the gradient estimate (2.2) used in Section 2 is equivalent to the curvaturecondition (2.1), so that the argument in the proof of Theorem 2.1 is no longer available ifthe curvature of the generator is unbounded below. So, in this section we introduce a newapproach by using coupling and the Girsanov theorem. This approach will be also appliedin Section 4 to a class of non-linear SPDEs. In order to ensure the integrability of theexponential term involved in the underlying Girsanov transformation, we need the followingpointwise condition of the Ricci curvature and the drift: there exists a constant c > 0 suchthat

Ricx := infRic(X,X) : X ∈ TxM, |X| = 1 ≥ −c(1 + ρ(x)2),

hZ(x) := sup〈∇XZ, X〉, X ∈ TxM, |X| = 1 ≤ c(1 + ρ(x)),

〈Z,∇ρ(x)〉 ≤ c(1 + ρ(x)).

(3.1)

The main idea works as follows. Given x0 6= y0 on M , let (xt, yt) be the coupling of theL-diffusion process by parallel transformation. To force the two marginal processes to meet

8

before a given time T , we make a Girsanov transformation of yt, denoted by yt, which is equalto xt at t = T and is generated by L under a weighted probability Q := RP for a density Rinduced by the Girsanov transformation. Then for every bounded measurable function f onM , one has

|PT f(y0)|α = |EQf(yT )|α = (ER|f(xT ))|α ≤ (PT |f |α(x0))(ERα/(α−1))α−1, α > 1.

Therefore, to derive a Harnack inequality, it suffices to prove that ERp < ∞ for p > 1 andto estimate this quantity. We will be able to realize this idea to obtain the following resultunder (??).

Theorem 3.1. ([?]) If (??) holds, then there exists a constant c′ > 0 such that

|Ptf(y)|α ≤ (Pt|f |α(x)) exp

α(α + 1)ρ(x, y)2

2(α− 1)t+

c′α2(α + 1)2

(α− 1)2(1 + ρ(x, y)2)ρ(x, y)2

+α− 1

2(1 + ρo(x)2)

holds for all α > 1, t > 0, x, y ∈ M and bounded measurable function f on M .

As an application of the above Harnack inequality, we are able to present the followingheat kernel estimate as in Theorem ??.

Corollary 3.2. ([?]) Assume (??). For any δ > 2 there exists a constant c(δ) > 0 such that

pt(x, y) ≤ exp[−ρ(x, y)2/2δt + c(δ)(1 + t + t2 + ρo(x)2 + ρo(y)2)]√µ(B(x,

√2t))µ(B(y,

√2t))

, x, y ∈ M, t > 0,

where B(x, r) is the geodesic ball at x in M with radius r.

We remark that as in Theorem ??, the heat kernel upper bound presented above is sharpin short time according to Varadhan’s asymptotic formula. On the other hand, however, thelong time behaviors for the heat kernel upper bounds included in Theorem ?? and Corollary?? are qualitatively different. This difference is due to the fact that the long time behaviorof the L-diffusion process is highly related to the curvature according to the Ito formula andthe second variational formula of the distance.

Sketch of the proof of Theorem ??. We simply assume that M does not have cut-locus,so that the parallel transformation is smoothly defined between the tangent spaces at any twopoints. We first introduce the construction of coupling by parallel transformation. Let Bt bea d-dimensional Brownian motion. Then the L-diffusion process starting at x0 ∈ M can beconstructed by solving to the following SDE:

9

(3.2) dxt =√

2Φt dBt + Z(xt)dt, x0 ∈ M,

where Φt the horizontal lift of xt; that is, letting H : TM → TO(M) be the lift operator,

dΦt = HΦt dxt, Φ0 ∈ Ox(M).

For given y 6= x, let e(x, y) : [0, ρ(x, y)] → M be the unique geodesic from x to y. LetPx,y : TxM → TyM be the parallel transformation along the geodesic e(x, y). In particularPx,x = I, the identity operator. Consider

(3.3) dyt =√

2Pxt,ytΦt dBt + Z(yt)dt, y0 ∈ M.

Since Px,y is smooth, yt is a well-defined L-diffusion process starting at y0. We call (xt, yt)the coupling by parallel transformation of the L-diffusion process.

To calculate the radial process ρ(xt, yt), let Mx,y be the mirror reflection operator alongthe geodesic e(x, y); that is, Mx,yX = Px,yX if X ⊥ e while Mx,yX = −Px,yX if X ‖ eat point x. Let uid−1

i=0 be an orthonormal basis in Rd such that Φtu0 = e at xt. Define

vi := (Ψ−1t Pxt,ytΦt)u

i, i = 0, · · · , d− 1. Since 〈Φtui, e〉(xt) = 0 for all i 6= 0, we have

v0 = −(ΨtMxt,ytΦt)u0, vi = ΨtMxt,ytΦt)u

i, i 6= 0.

Then [?, Theorem 2 and (2.5)] implies

(3.4) dρ(xt, yt) ≤ IZ(xt, yt)dt, t ≤ τ,

where τ := inft ≥ 0 : xt = yt is the coupling time and

IZ(x, y) =d−1∑i=1

∫ ρ(x,y)

0

(|∇e(x,y)Ji|2 − 〈R(e(x, y), Ji)e(x, y), Ji〉)sds + Zρ(·, y)(x) + Zρ(x, ·)(y)

for R the curvature operator, e(x, y) the tangent vector of the geodesic e(x, y), and Jid−1i=1

the Jacobi fields along e(x, y) which, together with e(x, y), consist of an orthonormal basis ofthe tangent space at x and y. Let

K(x, y) := supz∈e(x,y)

(−Ricz)+, δ(x, y) := sup〈∇XZ, X〉z : z ∈ e(x, y), X ∈ TzM, |X| = 1.

We have

Zρ(·, y)(x) + Zρ(x, ·)(y) =

∫ ρ(x,y)

0

〈∇e(x,y)Z, e(x, y)〉sds ≤ δ(x, y)ρ(x, y).

10

Thus, by [?, Theorem 2.1.4] (see also [?] and [?]) we obtain

(3.5) IZ(x, y) ≤ 2√

K(x, y)(d− 1) tanh(ρ(x, y)

2

√K(x, y)/(d− 1)

)+ δ(x, y)ρ(x, y).

To construct a coupling so that the coupling time is less than a given time T > 0, let usconsider

dyt =√

2Pxt,ytΦt dBt + Z(yt)dt + N(yt)dLt(y)

−(IZ(xt, yt) +

ρ(x0, y0)

T

)n(yt, xt)dt, y0 = y0,

(3.6)

where n(y, x) := e(y, x)|y ∈ TyM for any x 6= y. Since n(x, y) is smooth outside the diagonalD := (x, x) : x ∈ M, the solution yt exists uniquely until the coupling time τ := inft ≥ 0 :xt = yt. We let yt = xt for t ≥ τ . As in (??) we have

dρ(xt, yt) ≤ −ρ(x0, y0)

Tdt, t ≤ τ

so that τ ≤ T. Let

Nt :=1

2

∫ t∧τ

0

〈dMs,(IZ(xs, ys) +

ρ(x0, y0)

T

)n(ys, xs)〉, Rt := eNt− 1

2[Nt],

wheredMs :=

√2(xs, ys)Pxs,ysΦs dBs.

By Girsanov theorem yt is an L-diffusion process under the weighted probability measureQ := RTP. Therefore,

PT f(y) = EQf(yT ) = ERT f(xT ) ≤ (Efα(xT ))1/α(ERβT )1/β, α−1 + β−1 = 1.

Finally, to complete the proof one only has to estimate ERβT by using exponential martingale

argument.

4 Stochastic Generalized Porous Media and Fast Diffu-

sion Equations

In this section we consider an infinite-dimensional model where no any curvature condition isavailable.

Let (E, M ,m) be a separable probability space and (L, D(L)) a negative definite self-adjoint linear operator on L2(m) having discrete spectrum. Let

(0 <)λ1 ≤ λ2 ≤ · · ·be all eigenvalues of −L with unit eigenfunctions ei belonging to Lr+1(m), where r > 0 isa fixed number throughout this paper.

11

To state our equation, we first introduce the state space of the solutions. Let H be thecompletion of L2(m) under the inner product

〈x, y〉H :=∞∑i=1

1

λi

〈x, ei〉〈y, ei〉,

where 〈 , 〉 is the inner product in L2(m). It is well-known that H is the dual space of theSobolev space H1 := D((−L)1/2) and hence, is often denoted by H−1 in the literature. LetLHS denote the space of all Hilbert-Schmidt operators from L2(m) to H. Let Wt be thecylindrical Brownian motion on L2(m) w.r.t. a complete filtered probability space (Ω, Ft,P);that is, Wt :=

∑∞i=1 Bi

tei for a sequence of independent one-dimensional Ft-Brownian motionsBi

t. Let

Ψ, Φ : [0,∞)× R× Ω → Rbe progressively measurable and continuous in the second variable, and let

Q : [0,∞)× Ω → LHS

be progressively measurable such that

E∫ T

0

‖Qt‖2LHS

dt < ∞, T > 0.

We consider the equation

(4.1) dXt =LΨ(t,Xt) + Φ(t,Xt)

dt + QtdWt.

To ensure the existence and uniqueness of the solution to (??), we assume that the embed-ding Lr+1(m) ⊂ H is continuous and there exist functions δ, η, γ, σ ∈ C([0,∞)) with δ > 0such that

|Ψ(t, s)|+ |Φ(t, s)− σts| ≤ ηt(1 + |s|r), s ∈ R, t ≥ 0,

2〈Ψ(t, x)−Ψ(t, y), y − x〉 − 2〈Φ(t, x)− Φ(t, y), L−1(x− y)〉≤ −δ2

t ‖x− y‖r+1r+1 + γt‖x− y‖2

H , x, y ∈ Lr+1(m), t ≥ 0,

(4.2)

where and in the sequel, ‖ · ‖p denotes the norm in Lp(m) for p ≥ 1 and L−1 is assumed tobe bounded in Lr+1(m) if Φ 6= 0. We note that this extra assumption holds automatically ifL is a Dirichlet operator. A very simple example satisfying (??) is that Ψ(t, s) := |s|r−1s andΦ(t, s) := γts. If, in particular, Φ = 0, Q = 0 and Ψ(t, s) := |s|r−1s then (??) reduces back tothe classical porous medium (for r > 1) and fast diffusion (for r < 1) equation (see e.g. [?]).Thus, (??) is called the stochastic generalized porous medium and fast diffusion equation.

Definition 4.1. An adapted continuous process Xt is called a solution to (??), if

E∫ T

0

‖Xt‖r+1r+1dt < ∞, T > 0

12

and

〈Xt, ei〉 = 〈X0, ei〉+

∫ t

0

m(Ψ(s,Xs)Lei + Φ(s,Xs)ei

)ds +

∫ t

0

〈Q(s,Xs)dWs, ei〉, i ≥ 1, t > 0.

According to [?], (??) has a unique solution for any X0 ∈ L2(H; F0;P). Moreover, it iseasy to prove that the solution has distribution uniqueness, i.e. its distribution does notdepend on the choice of cylindrical Brownian motions.

For any x ∈ H, let Xt(x) be the unique solution to (??) with X0 = x. Define

PtF (x) := EF (Xt(x)), x ∈ H

for any bounded measurable function F on H. To establish the Harnack inequality for Pt, weassume that Qt is non-degenerate, that is, Qtx = 0 implies x = 0. Let

‖x‖Qt :=

‖y‖2, if y ∈ L2(m), Qty = x,

∞, otherwise.

We call ‖ · ‖Qt the intrinsic distance induced by Qt.Unlike known Harnack inequalities for infinite-dimensional models where the intrinsic

distance (normally equals to infinity almost surely) is involved, the inequality presented belowonly includes the usual norm on the state space H. This will imply stronger regularityproperties of the semigroup such as the strong Feller property of Pt and estimates of itstransition density pt(x, y). Moreover, as have been shown in Section 2, this inequality canalso be applied to derive the hypercontractivity and ultracontractivity of the semigroup.

Theorem 4.1. ([?]) Assume (??) and let r ∈ (0, 3). If

(4.3) ‖x‖r+1r+1 ≥ ξ2

t ‖x‖2Qt‖x‖r−1

H , x ∈ Lr+1(m), t ≥ 0

holds for some strictly positive function ξ ∈ C([0,∞)), then for any t > 0, Pt is strong Fellerand for any positive bounded measurable function F and α > 1,

(4.4)(PtF )α(y)

PtFα(x)≤ exp

[27α‖x− y‖3−r

H

(α− 1)(3− r)3(∫ t

0δsξs exp[ r−3

4

∫ s

0γudu]ds)2

], x, y ∈ H.

Sketch of the proof. (a) For any x, y ∈ H, let (Xt, Yt) be the unique solution to theequations

dXt = A(t,Xt)dt + QtdWt, X0 = x,

dYt = A(t, Yt)dt +αt(Xt − Yt)

‖Xt − Yt‖εH

1t≤τdt + QtdWt, Y0 = y,

where τ := inft ≥ 0 : Xt = Yt. By the pathwise uniqueness, we have Xt = Yt for t ≥ τ.

13

Our first aim is to choose αt such that τ ≤ T . To this end, by (??) and Ito’s formula dueto [?, Theorem I.3.2], we have

d‖Xt − Yt‖2H ≤ (− δ2

t ‖Xt − Yt‖r+1r+1 + γt‖Xt − Yt‖2

H − αt‖Xt − Yt‖2−εH

)dt, t ≤ T.

Then

(4.5) d‖Xt − Yt‖2

He−R t0 γsds

≤ −(δ2t ‖Xt − Yt‖r+1

r+1 + αt‖Xt − Yt‖2−εH

)e−R t0 γsdsdt, t ≤ T.

From this one can prove τ ≤ T provided α satisfies

(4.6)

∫ T

0

exp

[− ε

2

∫ t

0

γsds

]αtdt ≥ 2

ε‖x− y‖ε

H .

(b) Let ε = (3− r)/4. By (??) and (??) we obtain

d‖Xt − Yt‖2

He−R t0 γsds

ε ≤ −εδ2t ‖Xt − Yt‖2(ε−1)

H e−εR t0 γsds‖Xt − Yt‖r+1

r+1dt

≤ −εδ2t ξ

2t ‖Xt − Yt‖2

Qte−ε

R t0 γsds‖Xt − Yt‖2(ε−1)+r−1

H dt

= −εδ2t ξ

2t e−εR t0 γsds

‖Xt − Yt‖2Qt

‖Xt − Yt‖2εH

dt,

(4.7)

where the last step follows from the fact that ε = (3− r)/4. Let

α2t := c2δ2

t ξ2t e−εR t0 γsds,

where

c :=2‖x− y‖ε

H

ε∫ T

0δtξt exp[−ε

∫ t

0γs]ds

.

Then (??) holds and (??) implies

(4.8)ε

c2

∫ T

0

α2t‖Xt − Yt‖2

Qt

‖Xt − Yt‖2εH

dt ≤ ‖x− y‖2εH .

Letting

ξt :=αtQ

−1t (Xt − Yt)

‖Xt − Yy‖εH

1t<τ,

we have

dYt = A(t, Yt)dt + Qt(dWt + ξtdt), Y0 = y.

14

By (??) and Girsanov theorem,

Wt := Wt +

∫ t

0

ξsds, t ≤ T

is a cylindrical Brownian motion on L2(m) under the probability measure RP, where

R := exp

[ ∫ T

0

〈dWt, ξt〉 − 1

2

∫ T

0

‖ξt‖22dt

].

Therefore, by (a),

(4.9) (PT F (y))α =(ERF (YT )

)α=

(ERF (XT )

)α ≤ (ERα/(α−1)

)α−1PT Fα(x).

Then the desired Harnack inequality follows by estimating ERα/(α−1) using exponential mar-tingales.

(c) Finally, since

PT F (y) = ERF (YT ) = ERF (XT ),

we have

(4.10) |PT F (y)− PT F (x)| = |E(R− 1)F (XT )| ≤ ‖F‖∞E |R− 1|.It is easy to see from (??) that limy→x E|R−1| = 0. Then (??) implies that PT F ∈ Cb(H).

To apply Theorem ?? to contractivity properties of Pt, we consider the following time-homogenous case where the semigroup has a unique invariant probability measure.

Theorem 4.2. ([?]) In the situation of Theorem ?? and let Ψ, Φ and Q be deterministic andtime-homogenous such that ξ, δ are constant and γ ≤ 0.

(1) The Markov semigroup Pt has a unique invariant probability measure µ with full support

on H and µ(eε0‖·‖r+1H ) < ∞ for some ε0 > 0.

(2) For any x ∈ H, t > 0 and p > 1, the transition density pt(x, y) of Pt w.r.t. µ satisfies

(4.11) ‖pt(x, ·)‖Lp(µ) ≤ ∫

H

exp

[− 27p‖x− y‖3−r

H

(3− r)3(δξt)2

]µ(dy)

−(p−1)/p

.

(3) If r = 1 then Pt is hyperbounded (i.e. ‖Pt‖L2(µ)→L4(µ) < ∞) and compact on L2(µ) forsome t > 0; while if r ∈ (1, 3) then Pt is ultracontractive and there exists c > 0 suchthat

(4.12) ‖Pt‖L2(µ)→L∞(µ) ≤ exp[c(1 + t−(1+r)/(r−1)

)], t > 0.

Consequently, when r ∈ (1, 3), Pt is compact for any t > 0 and hence, the essential spectrumof its generator is empty.

15

Sketch of the proof. (1) The existence and uniqueness of µ can be proved by using Ito’sformula and a standard tightness argument. By (??) with γ ≤ 0 and the continuity of theembedding Lr+1(m) ⊂ H, for any x ∈ H and Xt(x) the solution to (??) with X0 = x, thereexists δ, θ > 0 such that

d‖Xt(x)−Xt(0)‖2H ≤ −δ2‖Xt(x)−Xt(0)‖r+1

r+1dt ≤ −θ‖Xt(x)−Xt(0)‖r+1H dt.

Thus, limt→∞ ‖Xt(x) − Xt(0)‖H = 0 for any x ∈ H. This implies that Ptf(x) → µ(f) ast → ∞ for any x ∈ H and any f ∈ Cb(H). Therefore, µ is the unique invariant probabilitymeasure.

Moreover, if suppµ 6= H, then there exists x0 ∈ H and r > 0 such that B(x0, r) := y ∈H : ‖x0− y‖H ≤ r is a null set of µ. Since for any t > 0, Pt(x0, dy) is absolutely continuousw.r.t. µ, we have Pt(x0, B(x0, r)) = 0. Thus, letting Xt be the solution to (??) with X0 = x0,we obtain

P(‖Xt − x0‖H ≤ r) = 0, t > 0.

Since Xt is a continuous process on H, this implies P(‖X0−x0‖H ≤ r) = 0 which is impossible.So, µ has full support on H.

Finally, by Ito’s formula for the square of the norm, for some c, θ > 0 and c′ := (r+1)ε0/2we have

(4.13) deε0‖Xt‖r+1H ≤ (

c− θ‖Xt‖r+1r+1 + 2c′‖Q‖2

LHS‖Xt‖r+1

H

)c′‖Xt‖r−1

H eε0‖Xt‖r+1H dt + dMt

for some martingale Mt. Since ‖ · ‖r+1 ≥ c0‖ · ‖H for some constant c0 > 0, when ε0 > 0 issmall enough there exist c1, θ1 > 0 such that

deε0‖Xt‖r+1H ≤ (

c1 − θ1‖Xt‖r+1r+1e

ε0‖Xt‖r+1H

)dt + dMt.

This implies

µn(eε0‖·‖r+1H ) ≤ 1

θ1n+

c1

θ1

, n ≥ 1.

Hence, µ(eε0‖·‖r+1H ) < ∞ since µ is the weak limit of a subsequence of µn.

(2) Let t > 0 and x ∈ H be fixed. Since µ is the invariant probability measure of Pt, by(??) there exists a constant ct > 0 such that

(Pt1A(x))2

∫

H

e−ct‖x−y‖3−rH µ(dy) ≤

∫

H

Pt1A(y)µ(dy) = µ(A), A ∈ M .

Thus, the transition kernel Pt(x, dy) is absolutely continuous w.r.t. µ so that it has a densitypt(x, y).

Next, for any p > 1 and any nonnegative measurable function f with µ(fp/(p−1)) ≤ 1, itfollows from (??) that

(Ptf(x)

)p/(p−1) ≤ (Ptf

p/(p−1)(y))exp

[ct‖x− y‖3−r

H

], x, y ∈ H

16

holds for

ct :=27p

(3− r)3(∫ t

0δsξs exp[−3−r

4

∫ s

0γudu]ds)2

.

Thus,

(Ptf(x)

)p/(p−1)∫

H

e−ct‖x−y‖3−rH µ(dy) ≤ µ(fp/(p−1)) ≤ 1.

Therefore,

〈pt(x, ·), f〉 = Ptf(x) ≤( ∫

H

e−ct‖x−y‖3−rH µ(dy)

)−(p−1)/p

.

This implies (??).(3) Let f ∈ L2(µ) with µ(f 2) = 1. By (??) with γ = 0 and constants ξ, δ > 0, there exists

a constant c = c(r) > 0 such that

(Ptf)2(x) exp

[− c‖x− y‖3−r

H

t2

]≤ Ptf

2(y), x, y ∈ H, t > 0.

Taking integration for both sides w.r.t. µ(dy), we obtain

(4.14) (Ptf)2(x) ≤ 1

µ(B(0, 1))exp

[c(‖x‖H + 1)3−r

t2

], x ∈ H, t > 0,

where B(0, 1) := y ∈ H : ‖y‖H ≤ 1 has positive mass of µ.(a) If r = 1 then by (??) and Theorem ??(1) we have

∫

H

(Ptf)4(x)µ(dx) ≤ 1

µ(B(0, 1))

∫

H

exp[c(‖x‖H + 1)3−r

t2

]µ(dx) < ∞

for sufficiently big t > 0. Thus, Pt is hyperbounded. Since Pt has transition density w.r.t. µ,according to e.g. [?] it is compact in L2(µ) for large t > 0.

(b) If r > 1, then (??) implies

deε0‖Xt‖r+1H ≤ c2 − θ2‖Xt‖2r

H eε0‖Xt‖r+1H dt + dMt

for some small ε0 > 0 and some c2, θ2 > 0. Thus, letting h(t) solve the equation

(4.15) h′(t) = c2 − θ2ε−2r/(1+r)0 h(t)

log h(t)

2r/(r+1), h(0) = eε0‖x‖r+1

H ,

we have

(4.16) E eε0‖Xt(x)‖r+1H ≤ h(t).

Since 2r/(r + 1) > 1, (??) and (??) imply

17

(4.17) E eε0‖Xt(x)‖r+1H ≤ exp

[c3

(1 + t−(r+1)/(r−1)

)], t > 0, x ∈ H

for some constant c3 > 0. Next, by (??) we have

(4.18) ‖Ptf‖∞ = ‖Pt/2Pt/2f‖∞ ≤ c4 supx∈H

E exp[c4

t2‖X t

2(x)‖3−r

H

], t > 0

for some c4 > 0. Since there exists c5 > 0 such that

c4

t2u3−r ≤ ε0u

r+1 + c5t−(r+1)/(r−1), u, t > 0,

(??) follows immediately from (??) and (??). Finally, the compactness of Pt follows since Pt

has transition density w.r.t. µ.

To apply Theorems ?? and ??, one has to check condition (??) which is however lessexplicit. To this end, we present below some simple sufficient conditions for (??) to hold. Forthe case where r ∈ (0, 1) we shall need following the Nash (or Sobolev) inequality. Recall thatL satisfies the Nash inequality with dimension d > 0, if there exists a constant R > 0 suchthat

(4.19) ‖f‖2+4/d2 ≤ R〈f,−Lf〉, f ∈ D(L), µ(|f |) = 1.

Corollary 4.3. In the situation of Theorem ??. Let Qei := qiei for i ≥ 1 such that∑∞

i=1 q2i /λi <

∞. Then Q is Hilbert-Schmidt from L2(m) to H and Theorems ?? and ?? apply with someconstant ξ2 > 0 in each of the following cases.

(1) For r ≥ 1 and infi q2i > 0.

(2) r ∈ (0, 1), L is a Dirichlet operator, and there exist c1, ε > 0 such that

(4.20) q4i ≥ c1λ

ε(1+r)+1−ri , i ≥ 1

and (??) holds for some d < 4(r + 1)ε/(1− r).

Proof. If r ≥ 1 and c0 := infi≥1 q2i , then

‖x‖2Q =

∞∑i=1

〈x, ei〉q−2i ≤ 1

c0

‖x‖22

and ‖x‖2r+1 ≥ ‖x‖2

2 ≥ λ1‖x‖2H . Hence, (??) holds for ξ2 := λ

(r−1)/21 c0.

On the other hand, for r ∈ (0, 1) it follows from the Holder inequality that

18

‖x‖2Q =

∞∑i=1

〈x, ei〉2q−2i =

∞∑i=1

〈x, ei〉2λi

λiq−2i

≤( ∞∑

i=1

〈x, ei〉2λi

)(1−r)/2( ∞∑i=1

〈x, ei〉2λi

(λiq−2i )2/(r+1)

)(r+1)/2

= ‖x‖1−rH

( ∞∑i=1

〈x, ei〉2q−4/(r+1)i λ

(1−r)/(1+r)i

)(r+1)/2

.

(4.21)

Next, by (??) and [?, Theorem 1.3], there exists a constant Cε > 0 such that

‖f‖2+4ε/d2 ≤ Cε〈f, (−L)εf〉, f ∈ D((−L)ε).

Then by the uniform heat kernel upper bound induced by the Sobolev inequality (see e.g. [?])and that d/ε < 4(1 + r)/(1− r), there exists a constant c2 > 0 such that

‖x‖2r+1 ≥ c2‖(−L)−ε/2x‖2

2 = c2

∞∑i=1

〈x, ei〉2λ−εi .

According to (??), this implies

‖x‖2r+1 ≥ c3

∞∑i=1

〈x, ei〉2q−4/(r+1)i λ

(1−r)/(1+r)i

for some c3 > 0. Then the proof is finished by combining this with (??).

Finally, to illustrate our results we introduce below a concrete example.

Example 4.1. Let Qtei := qiei, Ψ(t, x) := |x|r−1x and Φ(t, x) := cx for some constants c ≤ 0and r ∈ (0, 3). Let L := ∆ on a bounded domain in R with Dirichlet boundary conditions.

(i) Let r ≥ 1. If there exist constants c1, c2 > 0 and ε ∈ [(r − 1)/(r + 1), r/(r + 1)) suchthat

c1 ≤ q2i ≤ c2i

(r+1)ε+1−r, i ≥ 1,

then all assertions in Theorem ?? hold.

(ii) Let r ∈ (1/5, 1). Assertions in Theorem ??(1) and (2) hold if there exist c1, c2 > 0 andε1, ε ∈ ((1− r)/4(1 + r), r/(r + 1)) such that

c1i(r+1)ε1+1−r ≤ q2

i ≤ c2i(r+1)ε+1−r, i ≥ 1.

19

Proof. It is well-known that (??) holds for d = 1 and λi = O(i2) for large i ≥ 1. Moreover, itis easy to see that (??) holds for γ := c and some constant δ > 0. Finally, since ε < r/(1+ r)in both cases on has

∞∑i=1

q2i

λi

≤ c3

∞∑i=1

i(r+1)ε−1−r < ∞.

(i) For r ≥ 1, by the condition that q2i ≥ c1 > 0 we obtain the desired assertions from

Corollary ??(1).(ii) For r ∈ (0, 1), the condition ε1 > (1− r)/4(r + 1) implies d := 1 < 4(r + 1)ε1/(1− r).

Then the desired assertions follow from Corollary ??(2) and the fact that λi = O(i2).

Acknowledgement. The author would like to thank Professor Mu-Fa Chen for encourage-ment and useful suggestions during the preparation of this note.

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