Evolution systems of measures and semigroup properties on evolving manifolds

An evolving Riemannian manifold $(M,g_t)_{t\in I}$ consists of a smooth $d$-dimensional manifold $M$, equipped with a geometric flow $g_t$ of complete Riemannian metrics, parametrized by $I=(-\infty,T)$. Given an additional $C^{1,1}$ family of vector fields $(Z_t)_{t\in I}$ on $M$. We study the family of operators $L_t=\Delta_t +Z_t $ where $\Delta_t$ denotes the Laplacian with respect to the metric $g_t$. We first give sufficient conditions, in terms of space-time Lyapunov functions, for non-explosion of the diffusion generated by $L_t$, and for existence 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 measures, 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.


Introduction
Let M be a d-dimensional differentiable manifold equipped with a family of complete Riemannian metrics (g t ) t∈I which is C 1 in t and evolves according to ∂ ∂t g t = 2h t , t ∈ I, where I = (−∞, T ) for some T ∈ (−∞, +∞], and h t a time-dependent 2-tensor on T M . Denote by ∇ t , ∆ t the Levi-Civita connection, resp. Laplacian on M , both with respect to the metric g t . For a given C 1,1 family (Z t ) t∈I of vector fields on M , we study the time-dependent second order differential elliptic operator L t = ∆ t + Z t .
In this paper, we develop the basis for a general theory of the following backward Cauchy problem: ∂ s u(·, x)(s) = −L s u(s, ·)(x) , (s, t) ∈ Λ, x ∈ M, (1.1) where φ ∈ C 2 (M ) ∩ C b (M ) and Λ := {(s, t) : s ≤ t and s, t ∈ I}. We investigate this problem from a probabilistic point of view. Let X t be the diffusion process generated by L t (called L t -diffusion) which is assumed to be non-explosive before time T (see [2,8,14] for details). As in the time-homogeneous case, we construct L t -diffusions X t via horizontal diffusions u t above X t .
Let F(M ) be the frame bundle over M and O t (M ) the orthonormal frame bundle with respect to the metric g t . We denote by π : where (e i ) d i=1 denotes the canonical orthonormal basis of R d . Furthermore, we denote by (V α, β ) d α, β=1 the standard-vertical fields on F(M ). Then given s ≥ 0, the diffusion u t is constructed for t ≥ s as the solution to the following Stratonovich SDE: (∂ t g t )(u t e α , u t e β )V α, β (u t ) dt, u s ∈ O s (M ), πu s = x, (1.2) where B t is a standard Brownian motion on R d . The projection X t := πu t of u t onto M then gives the wanted L t -diffusion process on M , see [2]. In the next section we complement existing results on non-explosion of X t which is a subject already studied in [14]. The backward Cauchy problem (1.1) is the Kolmogorov equation to the following non-autonomous SDE on M : dX t = u t • dB t + Z t (X t ) dt, X s = x. When it comes to the time-inhomogeneous case, the situation turns out to be more involved. For instance, Saloff-Coste and Zúñiga [19,20] studied the ergodic behavior of time-inhomogeneous Markov chains; more sophisticated and strict conditions are required due to the fact that the generator and the semigroup do not commute and due to the lack of uniqueness of the invariant measure. A first goal will be therefore to construct an evolution system of measures as a family of reference measures which plays a role similar to the invariant measure in the time-homogeneous case.
Let us start by reviewing the notion of an evolution system of measures. A family of Borel probability measures (µ t ) t∈I on M is called an evolution system of measures (see [9]) if coefficients on R d (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. Let ρ t (x, y) be the Riemannian distance from x to y with respect to the metric g t . Fixing o ∈ M , we write ρ t (x) := ρ t (o, x) for simplicity. Let Cut t be the set of the cut-locus of (M, g t ). Let At different places in the paper, some of the hypotheses listed below will be put in force.
(H2) There exists an increasing function ϕ ∈ C 2 (R + ) such that ϕ(0) = 0, lim r→+∞ ϕ(r) = +∞ and for some non-negative function a and a function c on I such that (H3) There exists a function k on I such that There exists a positive constant and a positive function on I such that (1.7) (b) Hypothesis (H1) gives a sufficient condition for non-explosion of L t -diffusions. Hypothesis (H2) ensures existence of an evolution system of measures (µ t ) t∈I , whereas (H3) guarantees uniqueness of the evolution system of measures (µ t ) t∈I .
(c) As indicated, the Lyapunov function ϕ • ρ t is 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 measures. 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 of P s,t as s → −∞. We shall prove that if Hypothesis (H3) holds, then for x ∈ M and (s, t) ∈ Λ, lim where µ t (f ) denotes the average of f with respect to the measure µ t .
In Sections 3-5 we use Hypothesis (H3) as standing assumption. Taking the unique evolution system of measures (µ s ) s∈I as reference measures, we study contractivity properties of the time-inhomogeneous semigroup P s,t . For the sake of brevity, we introduce the following notations:  (i) hypercontractive if it maps L p (M, µ t ) into L q (M, µ s ) for some 1 < p < q < +∞ and (s, t) ∈ Λ such that P s,t (p,t)→(q,s) ≤ 1; EJP 23 (2018), paper 20.

Remark 1.3.
It is easy to see that due to contractivity of the semigroup, the function C p,q can be chosen such that the following properties are satisfied: (i) For fixed s ∈ I, the function C p,q (s, s + ·) : (0, ∞) → (0, ∞) is a non-increasing function; (ii) for fixed t ∈ I, the function C p,q (t − ·, t) : (0, ∞) → (0, ∞) is a non-increasing function.
Note that the function C p,q takes into account both the position and the length of the interval [s, t].
In what follows, we use the abbreviation · p, s := · L p (M, µs) .
In Section 4, we extend the arguments of [18] to consider hypercontractivity and supercontractivity via logarithmic Sobolev inequalities (in short log-Sobolev inequalities).
In fact, under the assumption that R Z t ≥ k(t) for t ∈ I, there is a family of log-Sobolev inequalities with respect to P s,t : Hypercontractivity of P s,t in L p space, 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 operators P s,t is equivalent to the validity of the following family of super-log-Sobolev inequalities for every s ∈ I, f ∈ H 1 (M, µ s ) and some positive decreasing function β s . Note that the function β s may depend on the current time s which generalizes the notion of super-log-Sobolev inequalities for non-autonomous systems on R d in [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 (µ t ) t∈I or (P s,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 in L p (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. EJP 23 (2018), paper 20.

Non-explosion
Recall that ρ t (x) denotes the distance function ρ t (o, x) with respect to a fixed reference point o ∈ M . A sufficient condition for non-explosion of L t -diffusions can be given as follows.
Theorem 2.1. Suppose that Hypothesis (H1) holds. Then L t -diffusion process X t is non-explosive before time T .
Proof. Without loss of generality, we suppose that the L t -process X t starts from x at time s. For fixed t * ∈ (s, T ], there exists c := sup t∈[s,t * ] m(t) > 0 such that Then, by the Itô formula for the radial part of X t (see [14, Theorem 2]), we obtain According to Hypothesis (H1), ϕ is an increasing function such that ϕ(r) → +∞ as r → ∞. Thus, there exists m ∈ N + such that ϕ(n) > 0 for all n ≥ m and Therefore we have P{ζ ≥ t * } = 1. Since t * is arbitrary, we obtain 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.
Let ψ ∈ C(R + ) and h ∈ C(I) be non-negative such that for any t ∈ I, then the L t -diffusion process is non-explosive.
Proof. Suppose that the process X t generated by L t starts from x at time s ∈ I. For fixed t * ∈ (s, T ], let c = sup t∈[s,t * ] h(t) and It is easy to see from condition (2.2) that ϕ is an increasing function on R + with ϕ(r) → ∞ as r → ∞, satisfying This completes the proof.

Evolution systems of measures
For t ∈ I consider the linear second order differential operator L t given on a smooth function f by As indicated, Hypothesis (H1) guarantees the existence of a unique Markov semigroup P s,t generated by L t . Indeed, for fixed t ∈ I and f ∈ According to the uniqueness of solutions to Eq. (2.3), we obtain P s,r P r,t = P s,t , s ≤ r ≤ t < T.
and for any f ∈ B b (M ), (s, t) ∈ Λ and x ∈ M , the backward Kolmogorov equation is given Based on Hypothesis (H2) or (H3), one can prove existence and uniqueness of an evolution system.   Proof. (a) We first show existence. Given t ∈ I, a family of measures can be constructed as follows (see e.g. [6] for details).
We claim that under Hypothesis (H2), the family of measures (µ s,t ) s∈(−∞,t] is compact. Suppose that X t starts from o at time s. Under Hypothesis (H2), applying the Itô formula to the radial process ρ t (X t ), we get Using the condition ϕ(0) = 0 and letting n → ∞, we arrive at Therefore, according to the monotonicity of H, we have In addition, since ϕ is a compact and increasing function such that ϕ(r) → +∞ as r → +∞, we know that (µ s,t ) s∈(−∞,t] is a family of compact measures, i.e., for each n ∈ Z, there exists a sequence (t n k ), t n k → +∞ as k → +∞ such that µ tn k ,n * µ n .
Let µ s := P * n,s µ n . It is easy to check that the family µ s satisfies Eq. (1.4), i.e., for φ ∈ B b (M ), µ s (P s,t φ) = P * n,s µ n (P s,t φ) = µ n (P n,t φ) = µ t (φ). By this and the bound (2.6), we get the existence of an evolution system (µ s ) s∈I . Moreover, we have the estimate which completes the proof of Eq. (2.4).
(b) If Hypothesis (H3) holds, we claim that there exists a unique evolution system of where r : [0, ρ t (x)] → M is a g t -geodesic connecting o and x. By this formula and the index lemma, we have where G is the solution to the equation Under Hypothesis (H3), by [14, Lemma 9], we have It is easy to see that F t (s) is non-increasing in s and lim r→0 rF t (r) < ∞. Hence, by means of the positive function in Hypothesis (H3), we obtain By a similar argument as in part (a), we obtain an evolution system of measures such that We now use a coupling method to prove uniqueness of the evolution system. Let (X t , Y t ) be a parallel coupling starting from (x, y) at time s. Then, by [7] or [13], we know that if R Z t ≥ k(t), t ∈ I, then Let (µ t ) t∈I be an evolution system of measures. Then, we have the estimate: If there exists another evolution system of probability measures (ν t ) t∈I , then ν t (f ) is also the limit of P s,t f (o) as s → −∞, and hence ν t = µ t .
Directly from Eq. (2.9) we have the following asymptotic results.
Proof. Let (X t , Y t ) be parallel coupling process associated to L t . For any f ∈ C 1 (M ) being constant outside a compact set, we have Proof. If sup s∈(−∞,t] ρ s (x) < ∞ for any x ∈ M and t ∈ 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: for s ∈ [T, +∞), x ∈ M.

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 manifold M is the Euclidean space R d and the geometric flow g t is given by for some positive function g ∈ C 1 (I). Consider the operator L t = ∆ t + Z t = g(t) −1 (∆ + Z). It is easy to see that k ε = 0 and |Z t | t (o) = g(t) −1/2 |Z|(o) where k ε is defined as in (1.5). Moreover, assume that there exists constant C such that ∇Z ≤ C. Then the curvature Hence, by Theorem 2.3, if we can choose > 0 such that then the non-autonomous system has an unique evolution system of measures.
Example 2.8. The evolving manifold M carries a backward Ricci flow (g t ):  for some positive function on I, the non-autonomous diffusion system has an unique evolution system of measures. Here, for instance, if k(t) = |t| −α with 0 < α < 1, choosing (t) = |t| −α , it is easy to check that (2.11) holds.

Example 2.9.
Suppose that M is a hypersurface parameterized locally by X = {x i } in R d and evolving by its backward mean curvature flow, t ∈ (−∞, T ]. Let {H ij } be the second fundamental form of M and H = g ij H ij its mean curvature. It is well known that ∂ ∂t g ij = 2HH ij ; Consider the process (X t ) generated by L t = ∆ t . Then Assume that R Z t ≥ k(t) and that there exists o ∈ M such that k ε ≤ K for some constant K.

Hence, by Theorem 2.3, if
for some non-negative function C 1 and some function C 2 . Then Combining this with the inequality coth(s) ≤ 1 + s −1 , we obtain that for any positive function on I, Then by Theorem 2.3, if C 1 and C 2 satisfy for some positive function , there exists an evolution system of measures for this system.

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 ∇ s P s,t f (see [7,Corollary 3.2]). To this end, define an R d ⊗ R d -valued process (Q s,t ) (s,t)∈Λ as the solution to the following ordinary differential equation where · is the operator norm on R d . The following derivative formula is taken from [7].

Proposition 3.1.
Assume that R Z t ≥ k(t) for some continuous function k on I. Let (s, t) ∈ Λ. Then for f ∈ C 1 (M ) such that f is constant outside a compact set, and for any h ∈ C 1 b ([s, t]) satisfying h(s) = 0 and h(t) = 1, we have where Q * s,t is the transpose of Q s,t .
The following gradient estimate can be derived from Proposition 3.1.

Theorem 3.2.
Suppose that Hypothesis (H3) holds. Let (µ t ) t∈I be the evolution system of measures for P s,t . Then, (a) for every f ∈ C 1 (M ) such that f is constant outside a compact set and 1 ≤ p < ∞, 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).
For p ∈ (1, ∞) and t − s ≤ 1, by using the integration by parts formula, we have Integrating both sides of the inequality above with respect to µ s , we arrive at Integrating both sides by µ s and minimizing the coefficient in r, 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 R Z t ≥ k(t) for some function k ∈ C(I).

Log-Sobolev inequality and hypercontractivity
In this section, we prove hypercontractivity for P s,t . Let us first introduce the following log-Sobolev inequality, which is essential to the proof of our hypercontractivity theorem. Proposition 4.1. If R Z t ≥ k(t) for some function k ∈ C(I) then for any p ∈ (1, ∞), Proof. Without loss of generality, we suppose f > δ > 0. Otherwise, let f δ = (f 2 + δ) 1/2 . Then by letting δ → 0, we obtain the conclusion. Consider the process (P r,t f 2 ) log(P r,t f 2 )(X r∧τn ) where as above τ n = inf{t ∈ (s, T ] : ρ t (X t ) ≥ n}, n ≥ 1. Applying Itô's formula, we have where M r is a local martingale. By this and the estimate, EJP 23 (2018), paper 20.
we obtain d(P r,t f 2 ) log(P r,t f 2 )(X r ) Integrating both sides from s to t ∧ τ n , we have Then again by dominated convergence theorem, letting n ↑ +∞, we obtain The log-Sobolev inequality leads to hypercontractivity of (P s,t ).

Theorem 4.2.
Suppose that Hypothesis (H3) holds and that (µ t ) is the evolution system of measures for P s,t . Let r ≤ s ≤ t < T and p, q ∈ (1, ∞) such that Proof. For the sake of conciseness, we assume f > δ > 0, otherwise a similar argument as in the proof of Proposition 4.1 can be used. Consider the process Using the Itô formula, we have that for s < τ n ∧ t, d(P s,t f ) q(s) (X s ) = dM s + (L s + ∂ s )(P s,t f ) q(s) (X s ) ds = dM s + (P s,t f ) q(s) q(s)(q(s) − 1)|∇ s log P s,t f | 2 s + q (s) log P s,t f (X s ) ds.
Therefore, for r ≤ s ≤ t < T , By using the dominated convergence theorem and letting n → +∞, we have EJP 23 (2018), paper 20.
which implies d ds P r,s (P s,t f ) q(s) = q (s)P r,s ((P s,t f ) q(s) log P s,t f ) Therefore, for (P r,s (P s,t f ) q(s) ) 1/q(s) , we have d ds (P r,s (P s,t f ) q(s) ) 1/q(s) where the last inequality comes from the fact that q > 0 along with the log-Sobolev inequality (4.1) with f replaced by (P s,t f ) q(s)/2 . According to the definition of q(s), we have d ds P r,s (P s,t f ) q(s) 1/q(s) ≤ 0.
Integrating both sides from r to s, we obtain P r,s (P s,t f ) q(s) 1/q(s) ≤ (P r,t f p ) 1/p .
From this and the fact that q(s)/p ≤ 1, it follows that µ r (P r,s (P s,t f ) q(s) ) ≤ µ r (P r,t f p ) q(s)/p ≤ (µ r (P r,t f p )) q(s)/p , which implies This completes the proof.

Supercontractivity and ultraboundedness
This section is devoted to supercontractivity and ultraboundedness for the semigroup P s,t under Hypothesis (H3).

Super log-Sobolev inequality and boundedness of semigroup
We present a supercontractivity result first.
Theorem 5.1. Suppose that Hypothesis (H3) holds. Let (µ t ) be the evolution system of measures associated with P s,t . Then the following properties are equivalent: EJP 23 (2018), paper 20.
(a) The semigroup P s,t is supercontractive.
First, we give a lemma which makes the proof of this theorem more concise.
Lemma 5.2. Suppose Hypothesis (H2) holds. Let (µ t ) be an evolution system of measures for P s,t . If f ∈ C 1,2 (I × M ) ∩ C(I, L 1 (M, µ r )) and there exists some function for every r ∈ I. Proof.
On the other hand, using Kolmogorov's formula, we have We complete the proof by applying the dominated convergence theorem.
Proof of Theorem 5.1. First we prove "(b) ⇒ (a)". Let (s, t) ∈ Λ and f ∈ C ∞ c (M ) such that f > δ > 0. By Lemma 5.2, we need to check the following to handle the derivative of µ s (P s,t f ) q(s) with respect to s: Under Hypothesis (H3), by Theorem 3.2 (c), there exists a positive constant c(s, t) such Combining all estimates above, we obtain Furthermore, for P s,t f q(s),s , we have Replacing f in the log-Sobolev inequality (5.1) by f p/2 , we get Now again replacing f and p by P s,t f and q(s) in the inequality above, respectively, we Next, we prove "(a) ⇒ (b)". Suppose that there exists C p,q (s, t) and 1 < p < q such that P s,t (p,t)→(q,s) ≤ C p,q (s, t).
Recall the log-Sobolev inequality with respect to P s,t , we are able to integrate both sides of Eq. (5.5) with respect to µ s , Now, we need to deal with the term µ s (P s,t f 2 log P s,t f 2 ). For any h ∈ (0, 1 − 1 p ), by the Riesz-Thorin interpolation theorem, we get where r h = ph p−1 ∈ (0, 1), 1 Set f 2,t = 1. Then from Eq. (5.7), we have by dominated convergence, we obtain or equivalently, Combining this with Eq. (5.6), we arrive at i.e.β t is a positive function on (0, ∞) and 2 ≤ p ≤ q. We complete the proof by letting γ t = r and then β t (r) =β t (γ −1 t (r)).
Proof. Letting p = 2 and q → +∞ in Eq. (5.8), we know from Eq. (5.9) that for f ∈ C ∞ 0 (M ) with f 2,t = 1, Letting r = 8(t − s), we obtain (i) directly. Given (s, t) ∈ Λ. Let q and N be two functions in C 1 ((−∞, t]) such that q < 0, which will be given later. It follows from Eq.   for some positive constants c 1 , c 2 depending on s, t. Hence, if µ s (exp(λρ 2 s )) < ∞ for any λ > 0 and s ∈ I, then P s,t is supercontractive, i.e. which implies h s,n (λ) ≤ 2e Ms for s ∈ I. As M s is independent of n, letting n go to infinity, we obtain µ s (e λρs ) < ∞, for s ∈ I.
(b) If P s,t exp (λρ 2 t ) ∞ < ∞ for any λ > 0 and (s, t) ∈ Λ, then we know from Eq. (5.15) that for any p > 1 and f ∈ C b (M ) satisfying f > 0 and f p,t = 1, On the other hand, if P s,t (p,t)→∞ < ∞ for all p > 1, then

Other criteria on supercontractivity and ultraboundedness
It is straightforward to check that Hypothesis (H3) implies Hypothesis (H2) for ϕ(r) = r 2 , r > 0. As far as supercontractivity and ultraboundedness of P s,t is concerned, we have the following results in terms of other types of space-time Lyapunov conditions. Theorem 5.6. Let γ ∈ C((0, ∞)) be a positive increasing function such that lim r→+∞ γ(r) r = +∞.
holds for t ∈ I, c > 0 and x / ∈ Cut t (o), then P s,t has an evolution system of measures (µ s ) and P s,t is supercontractive with respect to (µ s ).