Evolution systems of measures and
semigroup properties on evolving manifolds
Li-Juan Cheng
*†Anton Thalmaier
‡Abstract
An evolving Riemannian manifold (M, gt)t∈I consists of a smooth d-dimensional
manifoldM, equipped with a geometric flowgt of complete Riemannian metrics,
parametrized byI = (−∞, T ). Given an additionalC1,1family of vector fields(Zt)t∈I
onM. We study the family of operatorsLt= ∆t+ Ztwhere∆tdenotes the Laplacian
with respect to the metricgt. We first give sufficient conditions, in terms of space-time
Lyapunov functions, for non-explosion of the diffusion generated byLt, and for
exis-tence of evolution systems of probability measures associated to it. Coupling methods are used to establish uniqueness of the evolution systems under suitable curvature conditions. Adopting such a unique system of probability measures as reference mea-sures, we characterize supercontractivity, hypercontractivity and ultraboundedness of the corresponding time-inhomogeneous semigroup. To this end, gradient estimates and a family of (super-)logarithmic Sobolev inequalities are established.
Keywords: Evolution system of measures; geometric flow; inhomogeneous diffusion; semigroup;
supercontractivity; hypercontractivity; ultraboundedness.
AMS MSC 2010: 60J60; 58J65; 53C44.
Submitted to EJP on August 16, 2017, final version accepted on February 2, 2018. Supersedes arXiv:1708.04951v2.
1
Introduction
LetM be ad-dimensional differentiable manifold equipped with a family of complete Riemannian metrics(gt)t∈I which isC1intand evolves according to
∂
∂tgt= 2ht, t ∈ I,
whereI = (−∞, T )for someT ∈ (−∞, +∞], andhta time-dependent 2-tensor onT M.
Denote by∇t,∆
tthe Levi-Civita connection, resp. Laplacian onM, both with respect
to the metricgt. For a given C1,1 family (Zt)t∈I of vector fields onM, we study the
time-dependent second order differential elliptic operatorLt= ∆t+ Zt.
In this paper, we develop the basis for a general theory of the following backward Cauchy problem: ( ∂su(·, x)(s) = −Lsu(s, ·)(x) u(t, x) = φ(x) ) , (s, t) ∈ Λ, x ∈ M, (1.1)
*Mathematics Research Unit, FSTC, University of Luxembourg, 4364 Esch-sur-Alzette, Luxembourg.
E-mail: [email protected]
†Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China.
E-mail: [email protected]
‡Mathematics Research Unit, FSTC, University of Luxembourg, 4364 Esch-sur-Alzette, Luxembourg.
whereφ ∈ C2(M ) ∩ C
b(M )andΛ := {(s, t) : s ≤ tands, t ∈ I}.
We investigate this problem from a probabilistic point of view. LetXtbe the diffusion
process generated by Lt (called Lt-diffusion) which is assumed to be non-explosive
before timeT (see [2, 8, 14] for details). As in the time-homogeneous case, we construct
Lt-diffusionsXtvia horizontal diffusionsutaboveXt.
Let F(M )be the frame bundle overM and Ot(M )the orthonormal frame bundle with
respect to the metricgt. We denote byπ : F(M ) → Mthe projection from F(M )ontoM.
For a frameu ∈Ot(M ), denote byHYt(u)the∇t-horizontal lift ofY ∈ TπuM. This allows
one to determine standard-horizontal vector fieldsHt
i on Ot(M ), via the formula Hit(u) = Huet
i(u), i = 1, 2, . . . , d, where(ei)di=1 denotes the canonical orthonormal basis ofR
d. Furthermore, we denote
by(Vα, β)dα, β=1the standard-vertical fields on F(M ). Then givens ≥ 0, the diffusionutis
constructed fort ≥ sas the solution to the following Stratonovich SDE:
dut= √ 2 d X i=1 Hit(ut) ◦ dBti+ H t Zt(ut) dt − 1 2 d X α,β=1 (∂tgt)(uteα, uteβ)Vα, β(ut) dt, us∈Os(M ), πus= x, (1.2)
whereBtis a standard Brownian motion onRd. The projectionXt:= πutofutontoM
then gives the wanted Lt-diffusion process on M, see [2]. In the next section we
complement existing results on non-explosion ofXtwhich is a subject already studied
in [14].
The backward Cauchy problem (1.1) is the Kolmogorov equation to the following non-autonomous SDE onM:
dXt= ut◦ dBt+ Zt(Xt) dt, Xs= x. (1.3)
Denote byXt(s,x)the solution to Eq. (1.3) which is assumed to be non-explosive before timeT. Then function
u(s, x) := E[φ(Xt(s,x))]
satisfies Eq. (1.1) and gives rise to a family of inhomogeneous Markov evolution operators
(Ps,t)(s,t)∈ΛonM: Ps,tφ(x) := E[φ(X (s,x) t )] = E (s,x)[φ(X t)].
This is completely standard in the case of a fixed metric and a time-independent operatorLt= LwherePs,t= Pt−s= e(t−s)LandLp-spaces are taken with respect to an
invariant measureµ, i.e., a Borel probability measureµonM such that
Z M Ptφ dµ = Z M φ dµ, t > 0, φ ∈ Bb(M ).
Under suitable conditions, see [5, 6], existence and uniqueness of the invariant measure can be shown. In this case,Ptextends to a contraction semigroup onLp(M, µ)for every p ∈ [1, ∞), see e.g. [3, 11, 18, 21, 22].
Let us start by reviewing the notion of an evolution system of measures. A family of Borel probability measures(µt)t∈I onM is called an evolution system of measures
(see [9]) if Z M Ps,tφ dµs= Z M φ dµt, φ ∈ Bb(M ), (s, t) ∈ Λ. (1.4)
Recently, Angiuli, Lorenzi, Lunardi et al. investigated evolution system of measures and related topics for non-autonomous parabolic Kolmogorov equations with unbounded coefficients onRd (see [1, 12, 15, 16]). For instance, in [12] sufficient conditions for existence and uniqueness of evolution systems of measures are given; in [1], using a unique tight evolution system of measures as reference measures, hypercontractivity and the asymptotic behavior are studied; the asymptotics in time-periodic parabolic problems with unbounded coefficients is addressed in [16]. All this work motivates us to study evolution systems of measures on evolving manifolds and to investigate contractivity properties of the semigroup. Our probabilistic approach simplifies and extends in particular earlier results obtained by analytic methods.
We start by formulating some hypotheses which will be needed later on. Letρt(x, y)
be the Riemannian distance fromxtoywith respect to the metricgt. Fixingo ∈ M, we
writeρt(x) := ρt(o, x)for simplicity. LetCuttbe the set of the cut-locus of(M, gt). Let Cut := {(x, t) : x ∈ Cutt}.
At different places in the paper, some of the hypotheses listed below will be put in force.
(H1) There exists an increasing functionϕ ∈ C2(R+)such that
lim
r→+∞ϕ(r) = +∞ and
(Lt+ ∂t)(ϕ ◦ ρt)(x) ≤ m(t), (x, t) ∈ M × I \ Cut,
for some continuous functionmonI.
(H2) There exists an increasing functionϕ ∈ C2
(R+)such thatϕ(0) = 0, lim
r→+∞ϕ(r) = +∞ and
(Lt+ ∂t)(ϕ ◦ ρt)(x) ≤ a(t) − c(t)(ϕ ◦ ρt)(x), (x, t) ∈ M × I \ Cut,
for some non-negative functionaand a functionconIsuch that
H(t) := Z t −∞ exp − Z t r c(u) du a(r) dr < ∞.
(H3) There exists a functionkonIsuch that
RZ
t := Rict− ht− ∇tZt≥ k(t), t ∈ I.
For any > 0, positive function`onIandt ∈ I, set
A1= 2k − `, B1(t) = 2d + 1 4 3(d − 1) −1+ 3k (t) + 2|Zt|t(o) 2 `−1(t), where k(t) = sup |Rict| (x) : ρt(x) ≤ . (1.5)
There exists a positive constantand a positive function`onIsuch that
Remark 1.1. In (H1) and (H2) the Lyapunov function ϕ ◦ ρt is by definition
time-dependent.
(a) From condition (1.6) it can be seen that the functionkin (H3) must satisfy
Z t −∞ exp −2 Z t r k(s) ds dr < ∞ and Z t −∞ k(s) ds = +∞, t ∈ I. (1.7)
(b) Hypothesis (H1) gives a sufficient condition for non-explosion ofLt-diffusions.
Hypo-thesis (H2) ensures existence of an evolution system of measures(µt)t∈I, whereas
(H3) guarantees uniqueness of the evolution system of measures(µt)t∈I.
(c) As indicated, the Lyapunov functionϕ ◦ ρtis time-dependent. Comparatively, in [12]
the Euclidean distance is used as reference distance and then a space only Lyapunov condition is sufficient for existence and uniqueness of an evolution system of mea-sures. In [1, 12] the coefficients in the Lyapunov condition are uniformly bounded, and as consequence a time-homogeneous process can be used for comparison with the original process. In our setting, the coefficients in the Lyapunov conditions need to be time-dependent to preserve the information about the varying space.
In general, evolution systems of measures are far from being unique. If there is a unique system it plays an important role. Indeed, it is related to the asymptotic behavior ofPs,tass → −∞. We shall prove that if Hypothesis (H3) holds, then forx ∈ M and (s, t) ∈ Λ,
lim
s→−∞kPs,tf (x) − µt(f )kL2(M,µs)= 0,
whereµt(f )denotes the average off with respect to the measureµt.
In Sections 3-5 we use Hypothesis (H3) as standing assumption. Taking the unique evolution system of measures(µs)s∈I as reference measures, we study contractivity
properties of the time-inhomogeneous semigroup Ps,t. For the sake of brevity, we
introduce the following notations:
kPs,tk(p,t)→(q,s):= kPs,tkLp(M, µt)→Lq(M, µs),
kPs,tk(p,t)→∞:= kPs,tf kLp(M, µt)→L∞(M ).
Definition 1.2. The evolution operatorPs,tis called
(i) hypercontractive if it mapsLp(M, µt)intoLq(M, µs)for some1 < p < q < +∞and (s, t) ∈ Λsuch that
kPs,tk(p,t)→(q,s)≤ 1;
(ii) supercontractive if it mapsLp(M, µ
t)intoLq(M, µs)for any1 < p < q < +∞and (s, t) ∈ Λ, and if there exists a positive functionCp,q : Λ → (0, +∞)such that
kPs,tk(p,t)→(q,s)≤ Cp,q(s, t);
(iii) ultrabounded if it mapsLp(M, µ
t)intoL∞(M )for everyp > 1and(s, t) ∈ Λ, and if
there exists a functionCp,∞: Λ → (0, +∞)such that
kPs,tf k(p,t)→∞≤ Cp,∞(s, t). (1.8)
Remark 1.3. It is easy to see that due to contractivity of the semigroup, the function
(i) For fixed s ∈ I, the function Cp,q(s, s + ·) : (0, ∞) → (0, ∞) is a non-increasing
function;
(ii) for fixedt ∈ I, the functionCp,q(t− ·, t) : (0, ∞) → (0, ∞)is a non-increasing function.
Note that the functionCp,q takes into account both the position and the length of the
interval[s, t].
In what follows, we use the abbreviation
k·kp, s:= k·kLp(M, µs).
In Section 4, we extend the arguments of [18] to consider hypercontractivity and su-percontractivity via logarithmic Sobolev inequalities (in short log-Sobolev inequalities). In fact, under the assumption thatRZ
t ≥ k(t)fort ∈ I, there is a family of log-Sobolev
inequalities with respect toPs,t:
Ps,t(f2log f2) ≤ 4 Z t s exp −2 Z t r k(u) du dr Ps,t|∇tf |2t+ Ps,tf2log Ps,tf2, f ∈ Cb1(M ), (s, t) ∈ Λ.
Hypercontractivity ofPs,tinLpspace, related to the unique evolution system of measures,
is then obtained as a consequence of the log-Sobolev inequalities.
In Section 5 we then prove that supercontractivity of the evolution operatorsPs,tis
equivalent to the validity of the following family of super-log-Sobolev inequalities
Z M f2log |f | kf k2,s dµs≤ r |∇sf |s 2 2,s+ βs(r) kf k 2 2,s, r > 0, for everys ∈ I,f ∈ H1(M, µ
s)and some positive decreasing functionβs. Note that the
functionβsmay depend on the current timeswhich generalizes the notion of
super-log-Sobolev inequalities for non-autonomous systems onRdin [1]. Moreover, combining the super-log-Sobolev inequalities and dimension-free Harnack inequalities, we prove that the exponential integrability of radial function with respect to(µt)t∈I or(Ps,t)(s,t)∈Λ is
equivalent to supercontractivity or ultraboundedness of the corresponding semigroup. The paper is organized as follows. In Section 2 we first give sufficient conditions for existence and uniqueness of evolution systems of measures. Then in Section 3, by means of Bismut type formulas, gradient estimates inLp(M, µ
s)are established for p ∈ (1, +∞], which are used in Sections 4-5 to study hypercontractivity, supercontractivity and ultraboundedness for the corresponding semigroup.
2
Diffusion processes and evolution system of measures
2.1 Non-explosion
Recall thatρt(x)denotes the distance functionρt(o, x)with respect to a fixed
refer-ence pointo ∈ M. A sufficient condition for non-explosion ofLt-diffusions can be given
as follows.
Theorem 2.1. Suppose that Hypothesis (H1) holds. Then Lt-diffusion process Xt is
non-explosive before timeT.
Proof. Without loss of generality, we suppose that theLt-process Xtstarts fromxat
times. For fixedt∗∈ (s, T ], there existsc := sup
t∈[s,t∗]m(t) > 0such that
Then, by the Itô formula for the radial part ofXt(see [14, Theorem 2]), we obtain dϕ ◦ ρt(Xt) ≤ √ 2u−1 t ∇ tϕ ◦ ρ t(Xt), dBt + (Lt+ ∂t) ϕ ◦ ρt(Xt) dt ≤√2u−1 t ∇ tϕ ◦ ρ t(Xt), dBt + c dt
up to the lifetimeζ ∧ t∗whereζ := limn→∞ζnwith
ζn:= inf{t ∈ (s, T ) : ρt(x, Xt) ≥ n}.
In particular, ifXs= x ∈ M, then
ϕ(n)Px{ζn ≤ t} ≤ Ex[ϕ ◦ ρt∧ζn(Xt∧ζn)] ≤ ϕ(ρs(x)) + ct, t ∈ [s, t
∗].
According to Hypothesis (H1), ϕis an increasing function such that ϕ(r) → +∞as
r → ∞. Thus, there existsm ∈ N+ such thatϕ(n) > 0for alln ≥ mand
Px{ζ ≤ t} ≤ lim n→∞P x{ζ n≤ t} ≤ lim n→∞ ϕ(ρs(x)) + ct ϕ(n) = 0, t ∈ [s, t ∗].
Therefore we haveP{ζ ≥ t∗} = 1. Sincet∗ is arbitrary, we obtain
P{ζ ≥ T } = 1
which completes the proof.
From Theorem 2.1 we get the following corollary which has been proved in [14] in the case of a Lyapunov condition with constant coefficients.
Corollary 2.2. Letψ ∈ C(R+)andh ∈ C(I)be non-negative such that for anyt ∈ I, (Lt+ ∂t)ρt(x) ≤ h(t)ψ(ρt(x)) (2.1)
holds outsideCutt(o), the cut-locus ofoassociated with the metricgt. If Z ∞ 1 dt Z t 1 exp − Z t r ψ(s) ds dr = ∞, (2.2)
then theLt-diffusion process is non-explosive.
Proof. Suppose that the processXtgenerated byLt starts fromxat times ∈ I. For
fixedt∗∈ (s, T ], letc = supt∈[s,t∗]h(t)and
ϕ(s) = Z s 1 dt Z t 1 exp −c Z t r ψ(u) du dr.
It is easy to see from condition (2.2) thatϕis an increasing function onR+withϕ(r) → ∞
asr → ∞, satisfying
(Lt+ ∂t)ϕ(ρt(x)) ≤ 1, t ∈ [s, t∗].
2.2 Evolution systems of measures
Fort ∈ I consider the linear second order differential operatorLtgiven on a smooth
functionf by
Ltf = (∆t+ Zt)f.
As indicated, Hypothesis (H1) guarantees the existence of a unique Markov semigroup
Ps,tgenerated byLt. Indeed, for fixedt ∈ I andf ∈ Cb(M ), the function(s, x) 7→ Ps,tf (x)
is the unique bounded classical solution inCb((−∞, t] × M ) ∩ C1,2((−∞, t] × M )to the
backward Cauchy problem:
(
∂su(·, x)(s) = −Lsu(s, ·)(x), (s, x) ∈ (−∞, t) × M,
u(t, x) = f (x), x ∈ M. (2.3)
According to the uniqueness of solutions to Eq. (2.3), we obtain
Ps,rPr,t = Ps,t, s ≤ r ≤ t < T.
Moreover, for any(s, t) ∈ Λ, x ∈ M and f ∈ C2(M )with kL
tf k∞ < ∞, the forward
Kolmogorov equation reads as
∂
∂tPs,tf (x) = Ps,tLtf (x),
and for anyf ∈ Bb(M ),(s, t) ∈ Λandx ∈ M, the backward Kolmogorov equation is given
by
∂
∂sPs,tf (x) = −LsPs,tf (x).
Based on Hypothesis (H2) or (H3), one can prove existence and uniqueness of an evolution system.
Theorem 2.3. Suppose that Hypothesis (H2) holds, then there exists an evolution
system of measures(µt)t∈Ifor(Ps,t)(s,t)∈Λsuch that sup
s∈(−∞,t] Z
M
(ϕ ◦ ρs)(y) µs(dy) ≤ H(t). (2.4)
Suppose that Hypothesis (H3) holds, then there exists a unique evolution system and
sup s∈(−∞,t]
Z
M
ρs(y)2µs(dy) < H1(t). (2.5)
Proof. (a) We first show existence. Givent ∈ I, a family of measures can be constructed as follows (see e.g. [6] for details). ForA ∈ B(M )and(s, t) ∈ Λ, let
µs,t(A) := 1 t − s Z t s Pr,t(o, A) dr.
We claim that under Hypothesis (H2), the family of measures(µs,t)s∈(−∞,t]is compact.
Suppose thatXtstarts fromoat times. Under Hypothesis (H2), applying the Itô formula
to the radial processρt(Xt), we get
i.e., E " exp Z t∧ζn s c(r) dr ! ϕ ◦ ρt∧ζn(Xt∧ζn) # ≤ ϕ(0) + E " Z t∧ζn s exp Z r s c(u) du a(r) dr # .
Using the conditionϕ(0) = 0and lettingn → ∞, we arrive at
E [ϕ ◦ ρt(Xt)] ≤ Z t s exp Z r s c(u) du a(r) dr exp − Z t s c(r) dr ≤ Z t s exp − Z t r c(u) du a(r) dr ≤ H(t).
Therefore, according to the monotonicity ofH, we have
sup s∈(−∞,t]
(Ps,tϕ ◦ ρt)(o) ≤ H(t), (2.6)
from which it follows that
µs,t(ϕ ◦ ρt) = 1 t − s Z t s (Pr,tϕ ◦ ρt)(o) dr ≤ H(t).
In addition, since ϕ is a compact and increasing function such that ϕ(r) → +∞ as
r → +∞, we know that (µs,t)s∈(−∞,t] is a family of compact measures, i.e., for each n ∈ Z, there exists a sequence(tnk),tnk→ +∞ask → +∞such that
µtnk,n*∗µn.
Let µs := Pn,s∗ µn. It is easy to check that the family µs satisfies Eq. (1.4), i.e., for φ ∈Bb(M ),
µs(Ps,tφ) = Pn,s∗ µn(Ps,tφ) = µn(Pn,tφ) = µt(φ).
By this and the bound (2.6), we get the existence of an evolution system(µs)s∈I.
More-over, we have the estimate
µs(ϕ ◦ ρs) = µn(Pn,sϕ ◦ ρs) ≤ lim tnk→∞ 1 n − tnk Z n tnk sup r∈(−∞,s] (Pr,sϕ ◦ ρs)(o) dr ≤ H(s)
which completes the proof of Eq. (2.4).
(b) If Hypothesis (H3) holds, we claim that there exists a unique evolution system of probability measures(µt)t∈I such that
sup s∈(−∞,t]
µs(ρ2s) < H1(t).
First recall the formula (see [17, Lemma 5 and Remark 6])
∂tρt(x) = 1 2
Z
∂tgt( ˙r(s), ˙r(s)) ds (2.7)
where r : [0, ρt(x)] → M is agt-geodesic connectingo andx. By this formula and the
index lemma, we have
whereGis the solution to the equation G00(s) = −Rict( ˙r(s), ˙r(s)) d − 1 G(s), G(0) = 0, G0(0) = 1.
Under Hypothesis (H3), by [14, Lemma 9], we have
(Lt+ ∂t)ρt≤ (d − 1)G0(ρt) G(ρt) − k(t)ρt+ Z ρt 0
Rict( ˙r(s), ˙r(s)) ds + hZt, ˙r(0)it(o) ≤(d − 1)G 0(ρ t) G(ρt) − k(t)ρt− Z ρt 0 (d − 1)G00(s) G(s) ds + |Zt|t(o) ≤ Ft(ρt) − k(t)ρt+ |Zt|t(o) whereFt(s) =pk(t)(d − 1) coth pk(t)/(d − 1) (s ∧ ) + k(t) (s ∧ )and k(t) := sup{|Rict| : ρt(x) ≤ }.
It is easy to see thatFt(s)is non-increasing insandlimr→0rFt(r) < ∞. Hence, by means
of the positive function`in Hypothesis (H3), we obtain
(Lt+ ∂t)ρ2t= 2ρt(Lt+ ∂t)ρt+ 2 ≤ 2ρt(Ft(ρt) − k(t)ρt+ |Zt|t(o)) + 2 ≤ 2d + 2nk(t) + (d − 1)−1+ p (d − 1)k(t) + |Zt|t(o) o ρt− 2k(t)ρ2t ≤ 2d +3 k(t) + (d − 1)−1 + 2|Zt|t(o) ρt− 2k(t)ρ2t ≤ 2d +3k(t) + 3(d − 1) −1+ 2|Z t|t(o) 2 4`(t) − (2k(t) − `(t))ρ 2 t.
By a similar argument as in part (a), we obtain an evolution system of measures such that
sup t∈(−∞,s]
µt(ρ2t) ≤ H1(s).
We now use a coupling method to prove uniqueness of the evolution system. Let
(Xt, Yt)be a parallel coupling starting from(x, y)at times. Then, by [7] or [13], we know
that ifRZ t ≥ k(t),t ∈ I, then E(s,(x,y))[ρt(Xt, Yt)] ≤ exp − Z t s k(r) dr ρs(x, y).
Let(µt)t∈I be an evolution system of measures. Then, we have the estimate: |Ps,tf (o) − µt(f )| = Z
In addition, from Eq. (1.7), we know that exp − Z t −∞ k(r) dr = 0 and sup s∈(−∞,t] µs(ρ2s) < ∞.
Now lettings → −∞, we conclude that
lim
s→−∞|Ps,tf (o) − µt(f )| = 0.
If there exists another evolution system of probability measures(νt)t∈I, thenνt(f )is also
the limit ofPs,tf (o)ass → −∞, and henceνt= µt.
Directly from Eq. (2.9) we have the following asymptotic results.
Corollary 2.4. Suppose that Hypothesis (H3) holds. Then we have the following
con-vergence result: for anyf ∈ C1(M )being constant outside a compact set, there exists a functioncinC(I)such that
kPs,tf − µt(f )k2,s≤ c(t) exp − Z t s k(r) dr |∇tf |t ∞, (s, t) ∈ Λ.
Proof. Let(Xt, Yt)be parallel coupling process associated toLt. For anyf ∈ C1(M )
being constant outside a compact set, we have
|Ps,tf (x) − µt(f )| = |Ps,tf (x) − Ps,tf (o) + Ps,tf (o) − µt(f )| ≤ E (s,(x,o))[f (X t) − f (Yt)] + e −Rt sk(r) dr |∇tf |t ∞ µs(ρ 2 s) 1/2 ≤ E (s,(x,o)) f (Xt) − f (Yt) ρt(Xt, Yt) ρt(Xt, Yt) + e−Rstk(r) dr |∇tf |t ∞ µs(ρ 2 s) 1/2 ≤ |∇tf | t ∞E(s,(x,o))[ρt(Xt, Yt)] + e− Rt sk(r) dr |∇tf |t ∞ µs(ρ 2 s) 1/2 ≤ e−Rstk(r) dr |∇tf |t ∞ ρs(x) + µs(ρ2s) 1/2 (2.10) which implies that
kPs,tf − µt(f )k2,s≤ 2 exp − Z t s k(r) dr |∇tf | t ∞ µs(ρ2s) 1/2 .
Now using Theorem 2.3 and
sup s∈(−∞,t]
µs(ρ2s) < ∞,
we obtain the result directly.
Corollary 2.5. Suppose that Hypothesis (H3) holds andsups∈(−∞,t]ρs(x) < ∞for any x ∈ M andt ∈ I. Then we have the following convergence result: for anyf ∈ C1
b(M ),
there exists a functionCinC(I)such that
|Ps,tf − µt(f )| ≤ C(t) exp − Z t s k(r) dr |∇tf | t ∞, (s, t) ∈ Λ.
Proof. Ifsups∈(−∞,t]ρs(x) < ∞for anyx ∈ Mandt ∈ I, then the result can be directly
derived from the inequality (2.10).
Remark 2.6. Actually, our results can be applied to the following forward Cauchy
problem via a time reversal: fors ∈ [T, +∞),
(
∂tu(·, x)(t) = Ltu(t, ·)(x), (t, x) ∈ (s, +∞) × M ;
2.3 Some examples
We now investigate some non-autonomous systems on evolving manifolds to illustrate the results of Subsection 2.2 above.
Example 2.7. The manifoldM is the Euclidean spaceRd and the geometric flowg
tis
given by
(gt)ij = g(t)δij
for some positive functiong ∈ C1(I). Consider the operatorLt= ∆t+ Zt= g(t)−1(∆ + Z).
It is easy to see thatkε= 0and|Zt|t(o) = g(t)−1/2|Z|(o)wherekεis defined as in (1.5).
Moreover, assume that there exists constantCsuch that∇Z ≤ C. Then the curvature satisfies RZ t = − 1 2g 0(t)h·, ·i − g−2 (t)h∇·Z, ·i ≥ −12g0(t) − g−2(t)C =: A(t).
Hence, by Theorem 2.3, if we can choose` > 0such that
Z ∞ t exp − Z t r (2A − `)(s) ds 1 + (g(r)`(r))−1 dr < ∞,
then the non-autonomous system has an unique evolution system of measures.
Example 2.8. The evolving manifoldM carries a backward Ricci flow(gt): ∂tgt= 2Rict, t ∈ (−∞, T ].
Consider the operatorLt= ∆t+ ∇tV for someV ∈ Cb2(M ). Assume thatHess t
V ≤ −k(t), t ∈ (−∞, T ]and that there existso ∈ M such that for smallε > 0,
kε(t) ≤ K and |Zt|t(o) ≤ C
for some constantsKandCwherekεis defined as in (1.5). Hence, by Theorem 2.3, if Z t −∞ exp − Z t r (2k(s) − `(s)) ds (1 + `(r)−1) dr < ∞ (2.11) for some positive function`onI, the non-autonomous diffusion system has an unique evolution system of measures. Here, for instance, ifk(t) = |t|−αwith0 < α < 1, choosing
`(t) = |t|−α, it is easy to check that (2.11) holds.
Example 2.9. Suppose thatM is a hypersurface parameterized locally byX = {xi}in
Rd and evolving by its backward mean curvature flow,t ∈ (−∞, T ]. Let{H
ij} be the
second fundamental form ofM andH = gijHij its mean curvature. It is well known that ∂
∂tgij = 2HHij;
Ricij = HijH − gmrHirHmj.
Consider the process(Xt)generated byLt= ∆t. Then
(RZt)ij := (Rict− ht)ij = −gmrHirHmj.
Assume thatRZ
t ≥ k(t)and that there existso ∈ Msuch thatkε≤ Kfor some constantK.
Hence, by Theorem 2.3, if Z t −∞ exp − Z t r (2k(s) − `(s)) ds (1 + `(r)−1) dr < ∞
Example 2.10. Consider an evolving manifold(M, gt)satisfying the following curvature
condition:
Rict≥ −C1(t)(1 + ρ2t); ∂tρt+ Ztρt≤ C2(t)(1 + ρt),
for some non-negative functionC1and some functionC2. Then (Lt+ ∂t)ρt≤ 2ρt∆tρt+ 2C2(t)(ρt+ ρ2t) + 2 ≤ 2ρt q (d − 1)C1(t) (1 + ρ2t) coth q C1(t) (1 + ρ2t)/(d − 1) ρt + 2C2(t) (ρt+ ρ2t) + 2.
Combining this with the inequalitycoth(s) ≤ 1 + s−1, we obtain that for any positive function`onI, (Lt+ ∂t)ρ2t ≤ 2 p (d − 1)C1(t) + C2(t) (ρt+ ρ2t) + 2d ≤2p(d − 1)C1(t) + 2C2(t) + `(t) ρ2t +p(d − 1)C1(t) + C2(t) 2 `−1(t) + 2d
Then by Theorem 2.3, ifC1andC2satisfy Z t −∞ exp Z t r 2p(d − 1)C1(s) + 2C2(s) + `(s) ds × p (d − 1)C1(r) + C2(r) 2 `−1(r) + 2d dr < ∞
for some positive function`, there exists an evolution system of measures for this system.
3
Gradient estimates
We now turn to gradient estimates for the semigroup. It is well known that the so-called Bismut formula is a powerful tool to derive gradient estimates of semigroups in the fixed metric case (see [4, 10]). Let us first recall a Bismut type formula for∇sP
s,tf
(see [7, Corollary 3.2]). To this end, define anRd⊗ Rd-valued process(Q
s,t)(s,t)∈Λ as the
solution to the following ordinary differential equation
dQs,t dt = −R
Z
t(ut)Qs,t, Qs,s=id, (s, t) ∈ Λ, (3.1)
where ut is the horizontal Lt-diffusion process X (s,x)
t with π(us) = x, and RZt(ut) ∈ Rd⊗ Rdsatisfies RZ t(ut)a, b Rd = R Z t(uta, utb), a, b ∈ Rd. IfRZ t ≥ k(t),t ∈ I then we have kQr,tk ≤ exp − Z t r k(s) ds , (r, t) ∈ Λ, (3.2)
Proposition 3.1. Assume thatRZ
t ≥ k(t) for some continuous function k on I. Let (s, t) ∈ Λ. Then forf ∈ C1(M )such thatf is constant outside a compact set, and for any
h ∈ C1
b([s, t])satisfyingh(s) = 0andh(t) = 1, we have
u−1s ∇sP s,tf (x) = E(s,x)Q∗s,tu −1 t ∇ tf (X t) = 1 √ 2E (s,x) f (Xt) Z t s h0(r)Q∗s,rdBr (3.3) whereQ∗s,tis the transpose ofQs,t.
The following gradient estimate can be derived from Proposition 3.1.
Theorem 3.2. Suppose that Hypothesis (H3) holds. Let(µt)t∈I be the evolution system
of measures forPs,t. Then,
(a) for everyf ∈ C1(M )such thatf is constant outside a compact set and1 ≤ p < ∞, |∇sP s,tf |s p,s≤ exp − Z t s k(r) dr |∇tf | t p,t, (s, t) ∈ Λ; (3.4)
(b) for any1 < p < ∞, there exists a positive constantC1= C1(p)such that for every f ∈Bb(M ), |∇sP s,tf |s p,s≤ C1 max r∈[s,(t−1)∨s] Z (r+1)∧t r exp Z r s k(u) du dr !−1 kf kp,t for all(s, t) ∈ Λ;
(c) forf ∈Bb(M ), there exists a positive constantC1= C1(p)such that for all(s, t) ∈ Λ.
|∇sP s,tf |s ∞≤ C1 max r∈[s,(t−1)∨s] Z (r+1)∧t r exp Z r s k(u) du dr !−1 kf k∞.
Proof. By the first equality in (3.3) and inequality (3.2), the first assertion in (a) can be derived directly. It is also easy to see that (c) follows from (b). Hence, it suffices to prove (b).
Forp ∈ (1, ∞)andt − s ≤ 1, by using the integration by parts formula, we have
|∇sP s,tf |ps(x) = 1 √ 2 E (s,x)hf (X t) Z t s h0(r)Q∗s,rdBr i p ≤√1 2Ps,t|f | p(x) E(s,x) Z t s h0(r)Q∗s,rdBr qp/q ≤ c p p √ 2Ps,t|f | p(x) E(s,x) Z t s h02(r)kQs,rk2dr q/2p/q ≤ c p p √ 2Ps,t|f | p(x) E(s,x) Z t s h02(r) exp −2 Z r s k(u) du dr q/2p/q (3.5) where h(r) = Rr s exp Rρ s k(u) du dρ Rt sexp Rρ s k(u) du dρ .
It then follows that
Integrating both sides of the inequality above with respect toµs, we arrive at µs(|∇sPs,tf |ps) ≤ cp p √ 2µt(|f | p) Z t s exp Z r s k(u) du dr −p . (3.6) It leaves us to check the case fort − s > 1. For anyr ∈ [s, t − 1], combining Eq. (3.4) and Eq. (3.6), we have
|∇sP s,rPr,tf (x)|ps≤ exp −p Z r s k(r) dr Ps,r|∇rPr,tf |pr(x) ≤ c p p √ 2exp −p Z r s k(r) dr Z r+1 r exp Z ρ r k(u) du dρ −p Ps,r(Pr,r+1|Pr+1,tf |p)(x) ≤ c p p √ 2 Z r+1 r exp Z ρ s k(u) du dρ −p Ps,t|f |p(x).
Integrating both sides byµsand minimizing the coefficient inr, we obtain the desired
conclusion.
Remark 3.3. In Theorem 3.2 (c), the inequality does not need an evolution system of
measures as the reference measures. So the condition for this result can be weaken by only using
RZ t ≥ k(t)
for some functionk ∈ C(I).
4
Log-Sobolev inequality and hypercontractivity
In this section, we prove hypercontractivity for Ps,t. Let us first introduce the
following log-Sobolev inequality, which is essential to the proof of our hypercontractivity theorem.
Proposition 4.1. IfRZ
t ≥ k(t)for some functionk ∈ C(I)then for anyp ∈ (1, ∞), Ps,t(f2log f2) ≤ 4 Z t s exp −2 Z t r k(u) du dr Ps,t|∇tf |2t + Ps,tf2log Ps,tf2, (s, t) ∈ Λ, (4.1) holds forf ∈ C1 c(M ).
Proof. Without loss of generality, we supposef > δ > 0. Otherwise, letfδ = (f2+ δ)1/2.
Then by lettingδ → 0, we obtain the conclusion.
Consider the process(Pr,tf2) log(Pr,tf2)(Xr∧τn)where as above
τn= inf{t ∈ (s, T ] : ρt(Xt) ≥ n}, n ≥ 1. (4.2)
Applying Itô’s formula, we have
d(Pr,tf2) log(Pr,tf2)(Xr) = dMr+ (Lr+ ∂r)(Pr,tf2log Pr,tf2)(Xr) dr = dMr+ 1 Pr,tf2 |∇rPr,tf2|2r (Xr) dr, s < r < τn∧ t,
we obtain d(Pr,tf2) log(Pr,tf2)(Xr) ≤ dMr+ 4 exp −2 Z t r k(u) du Pr,t|∇tf |2t(Xr) dr, s < r < τn∧ t.
Integrating both sides fromstot ∧ τn, we have E(s,x)f2log f2(X t∧τn) ≤ Ps,tf2log Ps,tf2 (x) + E(s,x) Z t∧τn s 4 exp −2 Z t r k(u) du Pr,t|∇tf |2t(Xr) dr .
Then again by dominated convergence theorem, lettingn ↑ +∞, we obtain
Ps,t(f2log f2) ≤ 4 Z t s exp −2 Z t r k(u) du dr Ps,t|∇tf |2t+ Ps,tf2log Ps,tf2.
The log-Sobolev inequality leads to hypercontractivity of(Ps,t).
Theorem 4.2. Suppose that Hypothesis (H3) holds and that(µt)is the evolution system
of measures forPs,t. Letr ≤ s ≤ t < T andp, q ∈ (1, ∞)such that q ≤ exp − Z t . Z s1 r exp −2 Z s1 u k(z) dz du −1 ds1 ! (p − 1) + 1. ThenPs,t: Lp(M, µt) → Lq(M, µs)satisfies kPs,tk(p,t)→(q,s)≤ 1.
Proof. For the sake of conciseness, we assumef > δ > 0, otherwise a similar argument as in the proof of Proposition 4.1 can be used. Consider the process
(Ps,tf )q(s)(Xs∧τn), s ∈ [r, t], where q(·) = exp − Z t . Z s1 r e−2Rus1k(z) dz du −1 ds1 ! (p − 1) + 1.
Using the Itô formula, we have that fors < τn∧ t,
d(Ps,tf )q(s)(Xs) = dMs+ (Ls+ ∂s)(Ps,tf )q(s)(Xs) ds = dMs+ (Ps,tf )q(s) q(s)(q(s) − 1)|∇slog Ps,tf |2s+ q 0(s) log P s,tf (Xs) ds. Therefore, forr ≤ s ≤ t < T, E(r,x)(Ps,tf )q(s)(Xs∧τn) − (Pr,tf ) q(r)(x) = Z s r
q(u)(q(u) − 1)E(r,x)h(Pu,tf )q(u)−2|∇uPu,tf |2u(Xu∧τn)
i
+ q0(u)E(r,x)h(Pu,tf )q(u)log Pu,tf (Xu∧τn)
i du.
By using the dominated convergence theorem and lettingn → +∞, we have
Pr,s(Ps,tf )q(s)(x) − (Pr,tf )q(r)(x) =
Z s r
q(u)(1 − q(u))Pr,u
(Pu,tf )q(u)−2|∇uPu,tf |2u
(x) + q0(u)Pr,u((Pu,tf )q(u)log Pu,tf )(x)
which implies d dsPr,s(Ps,tf ) q(s)= q0(s)P r,s((Ps,tf )q(s)log Ps,tf ) + q(s)(q(s) − 1)Pr,s((Ps,tf )q(s)−2|∇sPs,tf |2s).
Therefore, for(Pr,s(Ps,tf )q(s))1/q(s), we have d ds(Pr,s(Ps,tf ) q(s))1/q(s) = (Pr,s(Ps,tf )q(s))1/q(s) −q 0(s) q(s)2log Pr,s(Ps,tf ) q(s)+ 1 q(s) ∂s(Pr,s(Ps,tf )q(s)) Pr,s(Ps,tf )q(s) = (Pr,s(Ps,tf )q(s))1/q(s) − q 0(s) q(s)2log Pr,s(Ps,tf ) q(s) + q 0(s) q(s)2 Pr,s(Ps,tf )q(s)log(Ps,tf )q(s) Pr,s(Ps,tf )q(s) +(q(s) − 1)Pr,s (Ps,tf ) q(s)−2|∇sP s,tf |2s Pr,s(Ps,tf )q(s) ≤ (Pr,s(Ps,tf )q(s)) 1−q(s) q(s) Z s r exp −2 Z s u k(t) dt du q0(s) − q(s) + 1 × Pr,s (Ps,tf )q(s)−2|∇sPs,tf |2s
where the last inequality comes from the fact thatq0 > 0along with the log-Sobolev inequality (4.1) withf replaced by(Ps,tf )q(s)/2. According to the definition ofq(s), we
have d ds Pr,s(Ps,tf )q(s) 1/q(s) ≤ 0.
Integrating both sides fromrtos, we obtain
Pr,s(Ps,tf )q(s) 1/q(s)
≤ (Pr,tfp)1/p.
From this and the fact thatq(s)/p ≤ 1, it follows that
µr(Pr,s(Ps,tf )q(s)) ≤ µr(Pr,tfp)q(s)/p≤ (µr(Pr,tfp))q(s)/p,
which implies
kPs,tf kq(s),s≤ kf kp,t.
This completes the proof.
5
Supercontractivity and ultraboundedness
This section is devoted to supercontractivity and ultraboundedness for the semigroup
Ps,tunder Hypothesis (H3).
5.1 Super log-Sobolev inequality and boundedness of semigroup
We present a supercontractivity result first.
Theorem 5.1. Suppose that Hypothesis (H3) holds. Let(µt)be the evolution system of
measures associated withPs,t. Then the following properties are equivalent:
(b) The family of super-log-Sobolev inequalities Z f2log f 2 kf k2 2,s dµs≤ r |∇sf | s 2 2,s+ βs(r)kf k 2 2,s, r > 0, (5.1)
hold for every f ∈ H1(M, µ
s), s ∈ I, and some positive non-increasing function βs: (0, +∞) → (0, +∞).
First, we give a lemma which makes the proof of this theorem more concise.
Lemma 5.2. Suppose Hypothesis (H2) holds. Let(µt)be an evolution system of
mea-sures for Ps,t. If f ∈ C1,2(I × M ) ∩ C(I, L1(M, µr))and there exists some function g ∈ Bb(M )such that|(∂r+ Lr)f | ≤ gfor allr ∈ I, then
d dr Z f (r, x) µr(dx) = Z (∂r+ Lr)f (r, x) µr(dx) (5.2) for everyr ∈ I.
Proof. Forf ∈ C1,2(I × M ) ∩ C(I, L1(M, µ
r)), we have Z
f (r, x) µr(dx) = Z
Ps,rf (r, x) µs(dx), s < r ≤ T.
On the other hand, using Kolmogorov’s formula, we have
d
drPs,rf (r, x) = Ps,r(Lr+ ∂r)f (r, x).
We complete the proof by applying the dominated convergence theorem.
Proof of Theorem 5.1. First we prove “(b)⇒(a)”. Let(s, t) ∈ Λ andf ∈ Cc∞(M )such
thatf > δ > 0. By Lemma 5.2, we need to check the following to handle the derivative ofµs(Ps,tf )q(s)with respect tos:
(Ls+∂s)(Ps,tf )q(s)
= Ls(Ps,tf )q(s)− q(s)(Ps,tf )q(s)−1(LsPs,tf ) + q0(s)(Ps,tf )q(s)log Ps,tf = q(s)(q(s) − 1)|∇sPs,tf |2s(Ps,tf )q(s)−2+ q0(s)(Ps,tf )q(s)log Ps,tf.
Under Hypothesis (H3), by Theorem 3.2 (c), there exists a positive constantc(s, t)such that |∇sP s,tf |2s ∞≤ c(s, t) kf k2∞. Moreover,kPs,tf k∞≤ kf k∞and (Ps,tf )q(s)log+(Ps,tf ) ≤ (Ps,tf )q(s)+1≤ kf kq(s)+1∞ .
Combining all estimates above, we obtain
k(Ls+ ∂s)(Ps,tf )q(s)k∞< ∞.
Now using Lemma 5.2, we have
Furthermore, forkPs,tf kq(s),s, we have d dskPs,tf kq(s),s = kPs,tf k −q(s)+1 q(s),s (q(s) − 1)µs(|∇sPs,tf |2s(Ps,tf )q(s)−2) +q 0(s) q(s)kPs,tf k −q(s)+1 q(s),s µs((Ps,tf )q(s)log Ps,tf ) −q 0(s) q(s)kPs,tf kq(s),slog kPs,tf kq(s),s. (5.3)
Replacingf in the log-Sobolev inequality (5.1) byfp/2, we get Z fplog f p kfp/2k2 2,s ! dµs≤ r p2 4 Z fp−2|∇sf |2sdµs+ βs(r)kfp/2k22,s.
Now again replacingf andpbyPs,tf andq(s)in the inequality above, respectively, we
obtain Z (Ps,tf )q(s)log(Ps,tf ) dµs− kPs,tf k q(s) q(s),slog kPs,tf kq(s),s ≤ rq(s) 4 Z (Ps,tf )q(s)−2|∇sPs,tf |2sdµs+ βs(r) q(s) kPs,tf k q(s) q(s),s.
Combining this with Eq. (5.3) yields
d dskPs,tf kq(s),s≤ βs(r)q0(s) q(s)2 kPs,tf kq(s),s, (s, t) ∈ Λ, where q(s) = e4r−1(t−s)(p − 1) + 1, q(t) = p. It follows that kPs,tf kq(s),s≤ exp Z t s βu(r)q0(u) q(u)2 du kf kp,t. (5.4)
Ifq(s) = q, thenr = 4(t − s) (log(q − 1)/(p − 1))−1. Taking thisrinto Eq. (5.4) yields
kPs,tf kq,s≤ exp Z t s βu(4(t − s) (log(q − 1)/(p − 1)) −1 )q0(u) q(u)2 du ! kf kp,t.
Next, we prove “(a)⇒(b)”. Suppose that there existsCp,q(s, t)and1 < p < q such
that
kPs,tk(p,t)→(q,s)≤ Cp,q(s, t).
Recall the log-Sobolev inequality with respect toPs,t, Ps,t(f2log f2) ≤ 4 Z t s e−2Rrtk(u) du dr Ps,t|∇tf |2t+ Ps,tf2log(Ps,tf2), f ∈ C0∞(M ). (5.5) From this and the fact that
log+(Ps,tf2) ≤ Ps,tf2≤ kf k2∞,
Now, we need to deal with the termµs(Ps,tf2log Ps,tf2). For anyh ∈ (0, 1 − 1p), by the
Riesz-Thorin interpolation theorem, we get
kPs,tf kqh,s≤ Cp,q(s, t) rhkf k ph,t, f ∈ L p(M, µ s), (5.7) whererh=p−1ph ∈ (0, 1), p1 h = 1 − rh+ rh p and 1 qh = 1 − rh+ rh q , i. e., rh= ph p − 1, ph= 1 1 − h, qh= 1 −p(q − 1) q(p − 1)h −1 .
Setkf k2,t= 1. Then from Eq. (5.7), we have Z
Ps,t|f |2(1−h) qh
dµs≤ Cp,q(s, t)rhqh,
which further implies
1 h " Z Ps,t|f |2(1−h) qh dµs− Z Ps,t|f |2dµs qh/ph# = 1 h Z Ps,t|f |2(1−h) qh dµs− 1 ≤ 1 h Cp,q(s, t) rhqh− 1. As lim h→0 1 h Cp,q(s, t) rhqh− 1 = p p − 1log Cp,q(s, t),
by dominated convergence, we obtain
p(q − 1) q(p − 1) Z Ps,tf2log Ps,tf2dµs− Z Ps,t(f2log f2) dµs≤ p p − 1log Cp,q(s, t), or equivalently, µs(Ps,tf2log Ps,tf2) ≤ q(p − 1) p(q − 1)µt(f 2log f2) + q q − 1log Cp,q(s, t).
Combining this with Eq. (5.6), we arrive at
µt(f2log f2) ≤ γt(t − s) µt(|∇tf |2t) + ˜βt(t − s) (5.8) wheref ∈ C0∞(M ),kf k2,t= 1and γt(t − s) = 4p(q − 1) q − p Z t t−(t−s) exp −2 Z t r k(u) du dr, ˜ βt(t − s) = pq q − plog Cp,q(s, t), (5.9)
i.e. β˜tis a positive function on(0, ∞)and2 ≤ p ≤ q. We complete the proof by letting γt= rand then
βt(r) = ˜βt(γt−1(r)).
Next, we study the ultraboundedness by using the super-log-Sobolev inequality (5.1).
Theorem 5.3. Suppose that Hypothesis (H3) holds. Let(µt)be an evolution system of
measures associated withPs,t.
(i) If the functionkin Hypothesis (H3) is almost surely non-negative andPs,tsatisfies kPs,tk(2,t)→∞≤ C2,∞(s, t),
(ii) Conversely, assume Eq. (5.1) holds for some positive non-increasing function β : (0, +∞) → (0, +∞), which is independent ofs. If there exists a functionr ∈ C([2, ∞))
such that t0:= Z ∞ 2 r(p) p − 1dp < ∞,
then fort − s ≥ t0, we have
kPs,tk(2,t)→∞≤ exp Z ∞ 2 β(r(p)) p2 dp .
Proof. Lettingp = 2andq → +∞in Eq. (5.8), we know from Eq. (5.9) that forf ∈ C0∞(M )
withkf k2,t= 1, µt(f2log f2) ≤ 8 Z t s e−2Rrtk(u) du dr µt(|∇tf |2t) + 2 log C2,∞(s, t) ≤ 8(t − s)µt(|∇tf |2t) + 2 log C2,∞(s, s + (t − s)).
Lettingr = 8(t − s), we obtain (i) directly.
Given(s, t) ∈ Λ. LetqandN be two functions inC1((−∞, t])such thatq0< 0, which will be given later. It follows from Eq. (5.3) that forf ∈ Cc∞(M )such thatf > 0,
d dse −N (s)kP s,tf kq(s),s =q 0(s) q(s)e −N (s)kP s,tf k −q(s)+1 q(s),s µs (Ps,tf )q(s)log Ps,tf − kPs,tf k q(s) q(s),slog kPs,tf kq(s),s +q(s)(q(s) − 1) q0(s) µs |∇sPs,tf |2s(Ps,tf )q(s)−2 − N0(s)q(s) q0(s)kPs,tf k q(s) q(s),s . (5.10) From this, and applying the super-log-Sobolev inequality (5.1) to(Ps,tf )q(s)/2, we obtain
d dse −N (s)kP s,tf kq(s),s ≥ e−N (s)kPs,tf k −q(s)+1 q(s),s q0(s) q(s) q(s)(q(s) − 1) q0(s) + r q(s) 4 µs |∇sP s,tf |2s(Ps,tf )q(s)−2 + 1 q(s)β(r) − N 0(s)q(s) q0(s) kPs,tf k q(s) q(s),s . (5.11)
Letq(s)andN (s)be the solutions to the following equations respectively:
q0(s) = −4(q(s) − 1)
r ◦ q(s) , q(t) = 2; N0(s) = q
0(s)β(r ◦ q(s))
q(s)2 , N (t) = 0.
It is easy to see thatq0< 0. Then from this and (5.11), we conclude that:
e−N (s)kPs,tf kq(s),s≤ kf k2,t. (5.12) Defining t0:= Z ∞ 2 r(p) 4(p − 1)dp < ∞,
we claim that q(s) → +∞ as s → t − t0, and then N (s) → R∞
2
β(r(p))
p2 dp. Indeed,
i.e.q(t − t0) = ∞.By this and Eq. (5.12), we have kPt−t0,tk(2,t)→∞≤ exp Z ∞ 2 β(r(p)) p2 dp .
5.2 Dimension-free Harnack inequality and boundedness of semigroup
Next, we will use integrability of the Gaussian functioneλρ2
t (for λ > 0andt ∈ I) with respect to the families of measures(µs)s∈I or(Ps,t)(s,t)∈Λto give another criterion
which is equivalent to supercontractivity or ultraboundedness. To this end, we need the following preliminary result which is a dimension-free Harnack-type estimate forPs,t
(see [7]).
Lemma 5.4. Assume that RZ
t ≥ k(t) holds for t ∈ I. For every f ∈ Cb(M ), p > 1, (s, t) ∈ Λandx, y ∈ M, we have the inequality:
|Ps,tf |p(x) ≤ (Ps,t|f |p) (y) exp p 4(p − 1) Z t s exp 2 Z r s k(u) du dr −1 ρ2s(x, y) ! . (5.13) The main result of this section is the following:
Theorem 5.5.Suppose that Hypothesis (H3) holds. Let(µs)be the evolution system of
measures. Then
(i) Ps,tis supercontractive with respect to(µs)if and only ifµt exp(λρ2t) < ∞for any λ > 0andt ∈ I;
(ii) Ps,tis ultrabounded with respect to(µs)if and only if
Ps,texp(λρ2t)
∞< ∞for any λ > 0and(s, t) ∈ Λ.
Proof. (a)From Hypothesis (H3) we know thatRZ
t ≥ k(t)fort ∈ I. It follows by the
Harnack inequality (5.13) that for(s, t) ∈ Λ,p > 1andf ∈ Cb(M ),
|Ps,tf |p(x) ≤ Ps,t|f |p(y) exp p 4(p − 1) Z t s exp 2 Z r s k(u) du dr −1 ρs(x, y)2 ! . Ifµt(|f |p) = 1, then 1 ≥ |Ps,tf (x)|p Z exp − p 4(p − 1) Z t s exp 2 Z r s k(u) du dr −1 ρs(x, y)2 ! µs(dy) ≥ |Ps,tf (x)|pµs(Bs(o, R)) exp − p(ρs(x) + R)2 4(p − 1) Z t s exp 2 Z r s k(u) du dr −1! , (5.14) whereBs(o, R) := {y ∈ M : ρs(y) ≤ R}. Since(µs)is compact, there existsR > 0, which
may depend ons, such that
µs(Bs(o, R)) = µs({x : ρs(x) ≤ R}) ≥ 1 − µs(ρ2s) R2 ≥ 1 − H2(s) R2 ≥ 2 −p.
By this and Eq. (5.14), we arrive at
which further implies |Ps,tf (x)| ≤ 2 exp Z t s exp 2 Z r s k(u) du dr −1ρ2 s(x) + R2 4(p − 1) ! , s < t. (5.15) Therefore, we have kPs,tf kq,s≤µs exp q(c1+ c2ρ2s) 1/q
for some positive constantsc1, c2depending ons, t. Hence, ifµs(exp(λρ2s)) < ∞for any λ > 0ands ∈ I, thenPs,tis supercontractive, i.e.
kPs,tk(p,t)→(q,s)< ∞
for any1 < p < q < ∞.
Conversely, if the semigroupPs,tis supercontractive, then by Theorem 5.1, we know
that the family of super-log-Sobolev inequalities (5.1) holds. Now our first step is to proveµs(eλρs) < ∞for anys ∈ Iandλ > 0. Letρsn= ρs∧ nandhs,n(λ) = µs(exp (λρns)).
Takingexp(λ 2ρ
n
s)into the super-log-Sobolev inequality (5.1) above, we have
λh0s,n(λ) − hs,n(λ) log hs,n(λ) ≤ hs,n(λ)λ2 r 4+ βs(r) λ2 . This implies 1 λlog hs,n(λ) 0 = λh 0 s,n(λ) − hs,n(λ) log hs,n(λ) λ2h s,n(λ) ≤r 4 + βs(r) λ2 . (5.16)
Integrating both sides of Eq. (5.16) fromλto2λ, we obtain
hs,n(2λ) ≤ h2s,n(λ) exp r 2λ 2+ β s(r) . (5.17)
By this and the fact that there exists a constantMssuch that µs({λρs≥ Ms}) ≤ 1 4exp −r 2λ 2+ β s(r) , it follows that: hs,n(λ) = Z {λρs≥Ms} eλρns dµ s+ Z {λρs<Ms} eλρns dµ s ≤ µs({λρs≥ Ms})1/2µs(e2λρ n s)1/2+ eMs ≤ 1 4exp −r 2λ 2+ β s(r) 1/2 exp r 4λ 2+1 2βs(r) hs,n(λ) + eMs ≤ 1 2hs,n(λ) + e Ms,
which implieshs,n(λ) ≤ 2eMsfors ∈ I. AsMsis independent ofn, lettingngo to infinity,
we obtain
µs(eλρs) < ∞, for s ∈ I.
Our second step is to prove µs(eλρ
2
s) < ∞ for all s ∈ I and λ > 0. Let hs(λ) :=
limn→∞hs,n(λ). Integrating both sides of Eq. (5.16) from1toλand lettingn → ∞, we
wherec0(s) := log µs(exp(ρs)). Now, we observe that for any positive constant, Z ∞ 1 hs(λ) e−( r 4+)λ 2 dλ = Z M dµs Z ∞ 1 eλρse−(r4+)λ 2 dλ < ∞.
On the other hand, it is easy to see that for > 0,
Z M dµs Z ∞ 1 eλρse−(r4+)λ 2 dλ = Z M exp ρ2s/(r + 4) dµs Z ∞ 1 exp − 1 2 √ r + 4λ − ρs/ √ r + 4 2! dλ ≥ √ 2 r + 4 Z M exp ρ2s/(r + 4) dµs Z ∞ √ r+4/2 exp(−t2) dt.
By the arbitrariness ofr, we obtain that there exists a number Nssuch that for any λ > 0,
Z
eλρ2sdµ
s< Ns, s ∈ I,
which completes the proof of (i).
(b) IfkPs,texp (λρ2t)k∞< ∞for anyλ > 0and(s, t) ∈ Λ, then we know from Eq. (5.15)
that for anyp > 1andf ∈ Cb(M )satisfyingf > 0andkf kp,t= 1,
P(s+t)/2,tf (x) ≤ 2 exp Z t (s+t)/2 exp 2 Z r (s+t)/2 k(u) du ! dr !−1 ρ2(s+t)/2(x) + R2 4(p − 1)
which implies that there exist constantsc1andc2such that kPs,tf k∞ ≤ 2 Ps,(t+s)/2exp 2c1+ c2ρ2(s+t)/2(x) Z t s+t 2 exp 2 Z r s+t 2 k(u) du ! dr !−1 ∞ < ∞.
On the other hand, ifkPs,tk(p,t)→∞< ∞for allp > 1, then Ps,texp(λρ2t) ∞≤ kPs,tk(p,t)→∞ exp(λρ2t) p,t< ∞
providedµt(exp(λρ2t))is bounded for allt ∈ I. Hence, it suffices to prove that µt(exp(λρ2t)) < ∞.
Since Ps,t is ultrabounded, Ps,t is supercontractive. Using Theorem 5.5 (i), we get µt(exp(λρ2t)) < ∞. This completes the proof.
5.3 Other criteria on supercontractivity and ultraboundedness
It is straightforward to check that Hypothesis (H3) implies Hypothesis (H2) for
ϕ(r) = r2,r > 0. As far as supercontractivity and ultraboundedness ofPs,tis concerned,
we have the following results in terms of other types of space-time Lyapunov conditions.
Theorem 5.6. Letγ ∈ C((0, ∞))be a positive increasing function such that
lim r→+∞
γ(r)
(i) If
(Lt+ ∂t)ρ2t(x) ≤ c − γ(ρ 2
t(x)) (5.19)
holds fort ∈ I,c > 0andx /∈ Cutt(o), thenPs,thas an evolution system of measures (µs)andPs,tis supercontractive with respect to(µs).
(ii) If (5.19) holds forγsuch thatgλ(r) = rγ(λ log r)is convex on(0, ∞)and such that
for anyλ > 0,
Z ∞ 0
dr
rγ(λ log r) < ∞,
thenPs,thas an evolution system of measures(µs)andPs,tis ultrabounded with
respect to(µs).
(iii) If (5.19) holds forγ(r) = αrδ, whereα > 0andδ > 1, thenP
s,tis ultrabounded with
respect to(µs)and
kPs,tk(2,t)→∞≤ exp
c(t − s)−δ/(δ−1)
holds for some constantc > 0and all(s, t) ∈ Λ.
Proof. It is an immediate consequence of Theorem 2.1 that under condition (5.19) the processX.is non-explosive up to timeT. The idea of following proof is similar to [22, Corollary 5.7.6]. We include a proof for convenience.
(a) LetXtbe a diffusion processes generated byLt. Then by Itô’s formula, d exp(λρ2t(Xt))
= 2λρt(Xt) exp(λρ2t(Xt)) dbt+ 1{Xt∈Cut/ t(o)}(Lt+ ∂t) exp(λρ
2
t(Xt)) dt − d`t, (5.20)
wherebtis a one-dimensional Brownian motion and`tan increasing process supported
on{t ≥ s : Xt∈ Cutt(o)}. By Eq. (5.19), it follows that (Lt+ ∂t) exp(λρ2t) = λ exp(λρ 2 t)(Lt+ ∂t)ρ2t+ 4λ 2ρ2 texp(λρ 2 t) ≤ exp(λρ2t)(c − γ(ρ 2 t)) + 4λ 2ρ2 texp(λρ 2 t) (5.21)
holds outsideCutt(o). Iflim sup r→∞
γ(r)
r = +∞,then there existc1, c2> 0such that for each t ∈ I,
(Lt+ ∂t) exp(λρ2t) − 1 ≤ c1− c2 exp(λρ2t) − 1
holds outsideCutt(o). According to Theorem 2.3, there exists an evolution system of
measures(µs)such thatsups∈Iµs(eλρ
2
s) < ∞. We then obtain the first conclusion by Theorem 5.5 (i).
(b) We use Theorem 5.5 (ii) to give the proof. So it suffices for us to check
kPs,texp(λρ2t)k∞< ∞. By Eq. (5.21), we have (Lt+ ∂t) exp(λρ2t) ≤ λ exp(λρ 2 t)(c − γ(ρ 2 t)) + 4λ 2ρ2 texp (λρ 2 t) ≤ λ exp(λρ2t)(c1− γ(ρ2t)/2), (5.22)
wherec1= 0 ∨ sup c −12γ(r) + 4λr < ∞. According to Eq. (5.22), there exists a positive
constantC(λ)such that
whereλ > 0. For fixedx ∈ M, let
θs(t) := E(s,x)exp(λρ2t(Xt)) , t ≥ s.
We need to show thatθs(t)is uniformly bounded. Since the set{t ∈ [s, T ) : Xt∈ Cutt(o)}
is of measure zero, it follows from Eq. (5.20) and Eq. (5.23) that
E(s,x)λρ2t(Xt∧τn) ≤ exp λρ
2
s(x) + C(λ) E
(s,x)[t ∧ τ n− s],
whereτn:= inf{t ∈ [s, T ) : ρt(Xt) ≥ n}. Sinceτn↑ T asn → ∞, we further conclude that θs(t) ≤ exp(λρs(x)2) + C(λ)(t − s)
for anyλ > 0and(s, t) ∈ Λ. In particular, hereθsis continuous and Mt:= 2 √ 2λ Z t s ρr(Xr) exp λρ2r(Xr) dbr
is a square integrable martingale. By Fubini’s theorem, along with Eq. (5.20) and Eq. (5.22), we have θs(t + r) − θs(t) r ≤ λc1 r Z t+r t θs(u) du − λ 2r Z t+r t E(s,x)exp λρ2 u(Xu) γ(ρ2u(Xu)) du ≤λc1 r Z t+r t θs(u) du − λ 2r Z t+r t
θs(u)γ(λ−1log θs(u)) du
where the second inequality comes from the fact that for λ > 0, the function r 7→ rγ(λ log r)is convex forr ≥ 1. Therefore,
θ0s(t) ≤ λc1θs(t) − λ
2θs(t)γ(λ −1log θ
s(t)), t ∈ [s, T ).
Then, by a similar discussion as in the fixed metric case (see the proof of [22, Corol-lary 5.7.6]), we obtain
θs(t) ≤ G−1(λ(t − s)/4) ∨ c2< ∞, (5.24)
for some positive constantc2where G(r) :=
Z ∞ r
ds
sγ(λ−1log s), r > 1.
In particular, forγ(r) = αrδ forδ > 1andα > 0, we have
G(r) = λ δ
α(δ − 1)(log r) 1−δ.
Combining this with Eq. (5.24), we complete the proof of (ii) and (iii).
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Acknowledgments. This work has been supported by the Fonds National de la
