Nicolas Fournier
email: nicolas.fournier@sorbonne-universite.fr

Camille Tardif
email: camille.tardif@sorbonne-universite.fr

On The Simulated Annealing In R d

Keywords: 2010 Mathematics Subject Classification. 60J60 Simulated annealing, time-inhomogeneous diffusion processes, large time behavior, nonexplosion. We

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction and results

1.1. Main results. We work with the following setting. Actually, c * = sup{E(x, y) : x local minimum of U , y global minimum of U } represents the maximum potential energy required to reach a global minimum y of U when starting from anywhere else.

Assumption (A)

We fix x 0 ∈ R d , c > 0 and β 0 > 0 and consider the time-inhomogeneous S.D.E.

(1)

X t = x 0 + B t -1 2 t 0 β s ∇U (X s )ds where β t = log(e cβ0 + t) c

and where (B t ) t≥0 is a d-dimensional Brownian motion. By Theorem 15 proved in the appendix, since U ≥ 0 under (A), (1) has a pathwise unique non-exploding solution (X t ) t≥0 . Here is our main result.

Theorem 1. Assume (A) and that R d e -α0U (x) dx < ∞ for some α 0 > 0. Fix c > c * , x 0 ∈ R d and β 0 > 0 and consider the unique solution (X t ) t≥0 to [START_REF] Azencott | Simulated annealing. Séminaire Bourbaki[END_REF]. Then lim t→∞ U (X t ) = 0 in probability.

One of the ingredients of the proof is the following proposition, which asserts that, in full generality, the simulated annealing is successful on the event where the process (X t ) t≥0 does not escape to infinity in large time.

Proposition 2. Assume (A), fix c > c * , x 0 ∈ R d and β 0 > 0. For (X t ) t≥0 the solution to [START_REF] Azencott | Simulated annealing. Séminaire Bourbaki[END_REF],

∀ ε > 0, lim t→∞ P lim inf s→∞ |X s | < ∞ and U (X t ) > ε = 0.
1.2. Comments and references. The simulated annealing has been introduced by Kirkpatrick-Gelatt-Vecchi [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF] as a numerical procedure to find a (possibly non unique) global minimum of a function U on a given state space. We refer to Azencott [START_REF] Azencott | Simulated annealing. Séminaire Bourbaki[END_REF] for an early review of the method and its links with the theory of Freidlin-Wentzell [START_REF] Freidlin | Random perturbations of dynamical systems[END_REF].

With our notation and in our context where the state space is R d , the main idea of the simulated annealing is the following. The solution to [START_REF] Azencott | Simulated annealing. Séminaire Bourbaki[END_REF], with β constant, has µ β (dx) = Z -1 β e -βU (x) dx as invariant probability distribution, if Z β = R d e -βU (x) dx < ∞. Using that min U = 0 and that ∇U is locally bounded, we deduce that there is κ > 0 such that Z β ≥ e -1 {U ≤1/β} dx ≥ κβ -d for all β ≥ 1. Hence under the condition that Z α0 < ∞ for some α 0 > 0, it holds that for all ε > 0, [START_REF] Fang | An improved annealing method and its large-time behavior[END_REF] µ β (U > ε) ≤ Z -1

β {U >ε} e -βU (x) dx ≤ β d κ Z α0 e -(β-α0)ε -→ 0 as β → ∞.
Hence one hopes that the solution to [START_REF] Azencott | Simulated annealing. Séminaire Bourbaki[END_REF], with lim t→∞ β t = ∞, satisfies lim t→∞ U (X t ) = 0 in probability. However, it is necessary that β t increases sufficiently slowly to infinity, so that Law(X t ) remains close, for all times, to µ βt . If β t increases too fast to infinity, one may remain stuck near a local minimum of U , as in the classical deterministic gradient method.

A major contribution is due to Holley-Kusuoka-Stroock [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF], see also Holley-Stroock [START_REF] Holley | Simulated annealing via Sobolev inequalities[END_REF]. Replacing R d by a compact manifold M , they showed that when β t c -1 log(1 + t), the simulated annealing procedure is successful, i.e. lim t→∞ U (X t ) = 0 in probability, if and only if c > c * . Their proof is almost purely analytic and very elegant. It relies on precise spectral gap estimates providing an asymptotically optimal Poincaré inequality. They use at many places the compactness of the state space. This kind of proof involving functional inequalities has been extended to the non-compact case of R d by Royer [START_REF] Royer | A remark on simulated annealing of diffusion processes[END_REF] and Miclo [START_REF] Miclo | Recuit simulé sur R n . Étude de l'évolution de l'énergie libre[END_REF], at the price of many growth conditions on U , like Zitt [START_REF] Zitt | Annealing diffusions in a potential function with a slow growth[END_REF], taking advantage of some weak Poincaré inequalities, worked under another set of rather stringent conditions, still implying that all the local minima of U are lying in a compact set. He in particular assumes that |∇U | is bounded and that there is ε > 0 such that, for all x outside a compact, U (x) ≥ log 1+ε |x| and ∆U (x) ≤ 0.

Here we only assume that R d e -α0U (x) dx < ∞ for some α 0 > 0, which seems very natural in view of [START_REF] Fang | An improved annealing method and its large-time behavior[END_REF]. This covers and consequently extends the previously cited works in R d . In particular, nothing forbids U to oscillate, as strongly as it wants, and as far as it wants from compact sets, and thus in particular to have an unbounded set of local minima.

We believe that all the results of the paper hold true when

U ∈ C 2 (R d ) (instead of U ∈ C ∞ (R d )).
We assume that U ∈ C ∞ (R d ) for simplicity, in particular because we want to apply serenely the results of Holley-Kusuoka-Stroock [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF], where the potential is assumed to be smooth, although it is likely they do not really use it.

1.3. Short heuristics. Let us emphasis that our proof relies on the following two main points. An entropy computation, see Heuristic Computation 5, shows that the condition R d e -α0U (x) dx < ∞ implies that lim inf t→∞ |X t | < ∞ a.s. Now, recall that in the compact case, see Holley-Kusuoka-Stroock [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF] or Miclo [START_REF] Miclo | Une étude des algorithmes de recuit simulé sous-admissibles[END_REF], it a.s. holds that lim sup t→∞ U (X t ) = c, where c > 0 is the constant appearing in the definition of (β t ) t≥0 , see [START_REF] Azencott | Simulated annealing. Séminaire Bourbaki[END_REF]. Combining these two points, it seems rather clear from the Borel-Cantelli Lemma that, in the non compact setting, the process should also satisfy lim sup t→∞ U (X t ) = c a.s., see points (d)-(e) in the next subsection for more details concerning a weaker, but sufficient, statement. Hence it will eventually remain in a compact set and Theorem 1 will follow from the compact case.

Apart from Heuristic Computation 5, which seems new and efficient, there are a number of technical issues, that are detailed in the next subsection.

1.4. Plan of the proof. We denote by (X t ) t≥0 the solution to [START_REF] Azencott | Simulated annealing. Séminaire Bourbaki[END_REF]. We assume (A) and the conditions that R d e -α0U (x) dx < ∞ for some α 0 > 0 and c > c * . (a) In Section 2, we prove some auxiliary weak regularization property for the law of the solution to [START_REF] Azencott | Simulated annealing. Séminaire Bourbaki[END_REF]. This allows us, when applying P.D.E. techniques, to do as if the law of X 0 had a bounded density concentrated around x 0 , with a precise bound as a function of β 0 .

(b) In Section 3, we show that lim inf t→∞ |X t | < ∞: the process cannot escape to infinity in large time. This does not use the condition c > c * . The key argument is the following: under the additional assumptions that Law(X 0 ) is smooth and β 0 > α 0 , we prove the important a priori estimate sup t≥0 E[U (X t )] < ∞, see Heuristic Computation 5, which a priori implies that lim inf t→∞ |X t | < ∞ by the Fatou lemma and since lim |x|→∞ U (x) = ∞. We then make all this rigorous and get rid of the additional assumptions using point (a) and that our process does not explode in finite time.

This central a priori estimate is derived from a rather original entropy computation. Let us mention that deducing that sup t≥0 E[U (X t )] < ∞ directly from the Itô formula would necessarily require some stringent conditions on ∇U and ∆U .

(c) In Section 4, we verify in Lemma 7 that, with an abuse of language, U (X t ) → 0 in probability as t → ∞ on the event where sup t≥0 |X t | < ∞. This is easy, by localization, in view of the results of Holley-Kusuoka-Stroock [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF]

|X t | ≤ C B ≥ 1 2 .
This is rather natural: in the compact setting, it is well-known, see [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF] or Miclo [START_REF] Miclo | Une étude des algorithmes de recuit simulé sous-admissibles[END_REF], that lim sup t→∞ U (X t ) = c a.s. It would not be too difficult to deduce that in the non-compact case, there exists

C t0,x0 > 0 such that P t0,x0 (sup t≥0 |X t | ≤ C t0,x0 ) ≥ 1/2.
The main issue is to show that C t0,x0 does not depend too much on t 0 and x 0 . This is tedious, and we have to revisit the proof of [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF].

(e) In Section 5, we prove Proposition 2: by (d), on the event lim inf s→∞ |X s | < ∞, our process will eventually be absorbed in a compact set, so that sup s≥0 |X s | < ∞, whence the success of the simulated annealing by point (c). Our proof completely breaks down for slower freezing schemes, i.e. if β t log t as t → ∞: in such a case, point (d) above cannot hold true, even non uniformly in t 0 and x 0 .

Observe that we do not assume any Lyapunov condition, which would involve ∆U and ∇U and would forbid U to oscillate too strongly.

As already mentioned and in view of (2), our only assumption, i.e. the existence of an invariant probability measure for some (low) temperature, is very natural and allows for potentials with a very general shape.

1.6. On the main assumption. However, we have shown in a previous paper with Monmarché [START_REF] Fournier | Simulated annealing in R d with slowly growing potentials[END_REF] that things may work even when R d e -αU (x) dx = ∞ for all α > 0. In [3, Theorem 1], we show that under (A) and when x • ∇U (x) ≥ a/ log |x| outside a compact, the simulated annealing works if c > c * and a > c(d -2)/2. This applies e.g. when U (x) = a log log(1 + |x| 2 ) outside a compact.

We also show in [3, Proposition 2], if U (x) = a log log(1 + |x| 2 ) with a < c(d -2)/2, then X t escapes to infinity as t → ∞ with positive probability, so that the simulated annealing fails, even when c > c * (since here c * = 0).

Observe that the condition x • ∇U (x) ≥ a/ log |x| outside a compact is much stronger than its integrated version U (x) ≥ a log log(1+|x| 2 ) (outside a compact), in particular, all the local minima of U have to belong to a compact.

One might think that the above results could be reinforced as follows: the simulated annealing is successful if U (x) ≥ a log log(1 + |x| 2 ) with a > c(d -2)/2 (and c > c * ), and fails if U (x) ≤ a log log(1 + |x| 2 ) with a < c(d -2)/2. This is not case. Indeed, we found some intricate conditions [3, Assumption (H 2 (α)) or Assumption (H 3 (α, β))], allowing for some very oscillating potentials, under which the simulated annealing is successful. In particular, it is stated in [3, Proposition 3] that if d ≥ 2, for all p ≥ 1, we can find a potential U such that log •p (1 + |x| 2 ) ≤ U (x) ≤ 3 log •p (1 + |x| 2 ) outside a compact, for which c * = 1 and for which the simulated annealing works for all c ∈ (1, 2). Note here that if d ≥ 3, the condition U (x) ≤ a log log(1 + |x| 2 ) is fulfilled (outside a compact) with any a > 0, e.g. a = (d -2)/2, but if c = 3/2, the simulated annealing is successful although a < c(d -2)/2. Thus, without the condition that R d exp(-α 0 U (x))dx < ∞ for some α 0 > 0, the situation may be very intricate and really depend on the shape of U . In particular, the above results show that for two potentials U and V satisfying (A), such that U (x) ≤ V (x) for all x ∈ R d (which suggests that U is less confining than V ), it is possible that the simulated annealing process related to V escapes to infinity with positive probability, while the simulated annealing procedure works for U . 1.7. Non-explosion. The non-explosion of the solution to (1), using only that U ≥ 0, is checked in the appendix. Actually, we treat, without major complication, the more general case where β : R + → (0, ∞) is any smooth function and where U : R d → R is smooth and satisfies U (x) ≥ -L(1 + |x| 2 ) for some constant L > 0. This is not so easy, since we do not want to assume any local condition on ∇U . We use purely deterministic techniques inspired by the seminal work of Grigor'yan [START_REF] Grigor'yan | Stochastically complete manifolds[END_REF], also exposed in [START_REF] Grigor'yan | Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds[END_REF]Section 9] and by the paper of Ichihara [START_REF] Ichihara | Explosion problems for symmetric diffusion processes[END_REF], both dealing with more general but time-homogeneous processes.

Let us mention that in the homogeneous case, Ichihara uses the P.D.E. satisfied by v(x) = E x [e -σ1 ], where σ 1 = inf{t ≥ 0 : |X t | ≤ 1}, while Grigor'yan rather studies the P.D.E. satisfied by w(t, x) = P x [ζ < t], where ζ is the life-time of the solution. In the inhomogeneous setting, we study, roughly, the P.D.E. satisfied by u(t, x) = E t,x [e -ζ ], where ζ is the life-time of the solution. The situation is slightly more complicated, but we manage to take advantage of some computations found in [START_REF] Grigor'yan | Stochastically complete manifolds[END_REF] and [START_REF] Ichihara | Explosion problems for symmetric diffusion processes[END_REF] to show that u ≡ 0.

Weak regularization

We prove some weak regularization that will allow us, when using P.D.E. techniques, to replace the Dirac initial condition δ x0 by some bounded functions concentrated around x 0 . One might invoke the Hörmander theorem, but since we need a precise bound as a function of β 0 (see Lemma 14 below), we will rather use the following weaker lemma based on stopping times. Lemma 3. Assume (A) and fix c > 0. For any A > 1, there is a constant

C (1)
A such that for any x 0 ∈ {U ≤ A}, any β 0 > 0, denoting by (X t ) t≥0 the corresponding solution to (1), there exists a stopping time τ ∈ [0, 1] such that sup t∈[0,τ ] |X t -x 0 | ≤ 1 and such that the law of (τ, X τ ) has a density bounded by exp(C (1)

A (β 0 + 1))1 {[0,1]×B(x0,1)} .
Proof. We fix x 0 ∈ {U ≤ A} and β 0 > 0. We introduce some random variable R, uniformly distributed in [1/2, 1] and independent of (X t ) t≥0 . We claim that

τ = inf{t ≥ 0 : |X t -x 0 | = R} ∧ R satisfies the requirements of the statement. First, τ ≤ R ≤ 1 and sup t∈[0,τ ] |X t -x 0 | ≤ R ≤ 1.
Next, we consider a d-dimensional Brownian motion (W t ) t≥0 independent of R and we set τ = inf{t ≥ 0 : |W t | = R} ∧ R. We introduce the martingale

L t = - 1 2 t∧τ 0 β s ∇U (x 0 + W s ) • dW s ,
as well as its exponential E t = exp(L t -1 2 L t ), which is uniformly integrable by the Novikov criterion, see Revuz-Yor [14, Proposition 1.15 p 332], because

L ∞ ≤ 1 4 (sup B(x0,1) |∇U | 2 ) 1 0 β 2
s ds is bounded. The Girsanov theorem tells us that under E ∞ • P, the process

B t∧τ = W t∧τ + 1 2 t∧τ 0 β s ∇U (x 0 + W s )ds, t ≥ 0
is a (stopped) Brownian motion, so that x 0 + W t∧τ is a (stopped) solution to [START_REF] Azencott | Simulated annealing. Séminaire Bourbaki[END_REF]. Hence for all measurable φ : R

+ × R d → R + , E[φ(τ, X τ )] = E[φ(τ , x 0 + W τ )E ∞ ].
By the Itô formula,

β t∧τ U (x 0 + W t∧τ ) = β 0 U (x 0 ) + t∧τ 0 β s U (x 0 + W s )ds -2L t + 1 2 t∧τ 0 β s ∆U (x 0 + W s )ds, whence, since U ≥ 0 and β ≥ 0, L ∞ ≤ 1 2 β 0 U (x 0 ) + τ 0 β s U (x 0 + W s )ds + 1 2 τ 0 β s ∆U (x 0 + W s )ds . Recalling that τ ≤ 1, that sup [0,τ ] |W s | ≤ R ≤ 1 that x 0 ∈ {U ≤ A}, that β s ≤ 1/c and that sup [0,1] β s ≤ β 0 + 1/c, we deduce that L ∞ ≤ 1 2 β 0 + 1 c A + sup x∈{U ≤A} sup y∈B(x,1) U (y) + 1 2 |∆U (y)| ≤ C A (1 + β 0 ),
for some finite constant C A > 0 depending on A and c. We used that 1+β0) and for all measurable φ : R

∪ x∈{U ≤A} B(x, 1) is bounded because lim |x|→∞ U (x) = ∞. Hence E ∞ = exp(L ∞ -1 2 L ∞ ) ≤ exp(L ∞ ) ≤ e C A (
+ ×R d → R + , (4) E[φ(τ, X τ )] ≤ e C A (1+β0) E[φ(τ , x 0 + W τ )].
We now verify that (τ , W τ ) has a bounded density, necessarily supported in [0, 1] × B(0, 1). For r > 0, we introduce τ r = inf{t > 0 : |W t | = r}. We have τ = τ R ∧ R, so that the density of (τ , W τ ) is bounded by the sum of the densities of (τ

R , W τ R ) and (R, W R ). Recall that R ∼ U([1/2, 1]). The density of (R, W R ) is 2e -|x| 2 /(2r) /(2πr) d/2 1 {r∈[1/2,1],x∈R d } , which is bounded.
Next, denoting by µ r (s) the density of τ r , we have by scaling that µ r (s) = r -2 µ 1 (r -2 s), because τ r has the same law as r 2 τ 1 . One may then check that the density of (τ

R , W τ R ) is |x| -d-1 µ 1 (|x| -2 r)1 {r>0,|x|∈[1/2,1]} , up to some normalization constant. Since µ 1 is bounded, so is the density of (τ R , W τ R ).
Denoting by C the bound of the density of (τ , W τ ), we conclude from ( 4) that (τ, X τ ) has a density bounded by Ce C A (1+β0) 1 {s∈[0,1],x∈B(x0,1)} . The conclusion follows.

No escape in large time

In this section, we prove that lim inf

t→∞ |X t | < ∞. Proposition 4. Assume (A) and fix c > 0, x 0 ∈ R d and β 0 > 0. Suppose that there is α 0 > 0 such that R d e -α0U (x) dx < ∞. For (X t ) t≥0 the solution to (1), lim inf t→∞ |X t | < ∞ a.s.
The crucial point is the following uniform in time a priori estimate.

Heuristic Computation 5. Assume (A), fix c > 0, β 0 > 0 and assume that there is α 0 ∈ (0, β 0 ) such that R d e -α0U (x) dx < ∞. Let f 0 be a probability density on R d . Let (X t ) t≥0 be the solution to (1) starting from X 0 with law f 0 . If

κ(f 0 ) = R d f 0 (x) log(1 + f 0 (x)e β0U (x) )dx < ∞, setting a 0 = [ R d e -α0U (x) dx] -1 , we have, if f t = Law(X t ) has a smooth enough density, sup t≥0 E[U (X t )] ≤ κ(f 0 ) -log(a 0 ) β 0 -α 0 .
This relies on a rather indirect entropy computation. As already mentioned, obtaining a uniform in time moment bound, using the Itô formula, would require much more stringent conditions involving ∇U and ∆U . Observe that the computation below is rather original, in that we do not differentiate the true relative entropy x) dx, but rather the relative entropy without normalization constant R d f t (x) log(f t (x)e βtU (x) )dx. Strangely, using the true relative entropy functional does not seem to provide interesting results.

R d f t (x) log(f t (x)Z βt e βtU (x) )dx, where Z β = R d e -βU (
Informal proof. The law f t of X t weakly solves

∂ t f t (x) = 1 2 div[∇f t (x) + β t f t (x)∇U (x)] = 1 2 div[e -βtU (x) ∇(f t (x)e βtU (x) )]. (5)
We do as if f t was a smooth strong solution to [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] and proceed to integrations by parts without justification. For any smooth φ : R + → R, we have, setting ψ(u) = uφ (u) -φ(u) for all u ≥ 0,

d dt R d φ(f t (x)e βtU (x) )e -βtU (x) dx (6) = R d ∂ t f t (x)φ (f t (x)e βtU (x) ) + β t U (x)f t (x)φ (f t (x)e βtU (x) ) -β t U (x)φ(f t (x)e βtU (x) )e -βtU (x) dx = - 1 2 R d |∇(f t (x)e βtU (x) )| 2 φ (f t (x)e βtU (x) )e -βtU (x) dx + β t R d U (x)ψ(f t (x)e βtU (x) )e -βtU (x) dx.
For the last equality (first term), we used ( 5) and an integration by parts.

We now apply [START_REF] Grigor'yan | Stochastically complete manifolds[END_REF] with the convex function φ(u) = u log(1 + u), for which ψ(u) = u 2 1+u ≤ u, to find, throwing away the nonpositive term, h t ≤ β t u t , where we have set

h t = R d f t (x) log(1 + f t (x)e βtU (x) )dx and u t = R d U (x)f t (x)dx = E[U (X t )]. But h t ≥ R d f t (x) log(f t (x)e βtU (x) )dx = R d f t (x) log(f t (x))dx + β t u t ≥ (β t -α 0 )u t + log(a 0 ). We used that R d f (x) log(f (x)/g(x)
)dx ≥ 0 for any pair of probability densities f and

g on R d , whence R d f t (x) log(f t (x))dx ≥ R d f t (x) log(a 0 exp(-α 0 U (x))dx = log(a 0 ) -α 0 u t . We conclude, since h 0 = κ(f 0 ), that (β t -α 0 )u t ≤ h t -log(a 0 ) ≤ κ(f 0 )-log(a 0 )+ t 0 β s u s ds = κ(f 0 )-log(a 0 )+ t 0 β s β s -α 0 (β s -α 0 )u s ds,
whence, by the Gronwall lemma,

(β t -α 0 )u t ≤ [κ(f 0 ) -log(a 0 )] exp t 0 β s ds β s -α 0 = [κ(f 0 ) -log(a 0 )] β t -α 0 β 0 -α 0 . Consequently, E[U (X t )] = u t ≤ [κ(f 0 ) -log(a 0 )]/[β 0 -α 0 ] for all t ≥ 0.
We now try to deduce from this informal computation the rigorous results we need.

Lemma 6. If ∇U is bounded together with all its derivatives and if the initial density f 0 belongs to C c (R d ), the a priori estimate of Heuristic Computation 5 rigorously holds true for the solution to (1) starting from X 0 ∼ f 0 .

Proof. We first justify rigorously [START_REF] Grigor'yan | Stochastically complete manifolds[END_REF], for all t ∈ (0, ∞), with φ(u) = u log(1+u). Recall that f t is the law of X t . Since U has at most linear growth and ∇U is bounded, it is (widely) enough to check that

(f t (x)) t>0,x∈R d is a strong solution to (5), i.e. 2∂ t f t (x) = ∆f t (x)+β t ∇U (x)•∇f t (x)+β t f t (x)∆U (x)
on (0, ∞) × R d and satisfies, for all 0 < t 0 < t 1 , for some constants C t0,t1 > 0 and λ t0,t1 > 0,

∀t ∈ [t 0 , t 1 ], x ∈ R d , f t (x) + |∂ t f t (x)| + |∇f t (x)| + |D 2 f t (x)| ≤ C t0,t1 exp(-λ t0,t1 |x| 2 ).
To prove those bounds, we use classical results found in Friedman [START_REF] Friedman | Partial differential equations of parabolic type[END_REF], that apply to uniformly parabolic equations with bounded and Lipschitz coefficients (actually, Hölder is enough): by [5, Chapter 1, Theorem 12], we have f t (x) = R d Γ(x, t; ξ, 0)f 0 (ξ)dξ, with, for some C T > 0 and λ > 0, for all t ∈ [0, T ], all x ∈ R d , [START_REF] Grigor'yan | Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds[END_REF] |Γ(x, t; ξ, 0

)| + t 1/2 |∇ x Γ(x, t; ξ, 0)| + t|D 2 Γ(x, t; ξ, 0)| ≤ C T t -d/2 e -λ|x-ξ| 2 /t .
The above estimates for Γ and ∇Γ are nothing but [5, Chapter 1, Equations (6.12) and (6.13)], and the estimate on D 2 Γ is proved similarly, using [5, Chapter 1, Equation (4.11)]. The Gaussian upper-bounds of

f t (x), |∇f t (x)| and |D 2 f t (x)| follow, because f 0 ∈ C c (R d ).
Finally, the bound on

∂ t f t (x) follows from the fact that 2|∂ t f t (x)| ≤ |∆f t (x)| + β t ||∇U || ∞ |∇f t (x)| + β t ||∆U || ∞ f t (x).
Hence, all the arguments in Heuristic Computation 5 are correct for t ∈ (0, ∞), and we conclude that for all t 0 > 0, sup t≥t0 E[U (X t )] ≤ (β t0 -α 0 ) -1 (κ(f t0 ) -log(a 0 )). To complete the proof, the only issue is to show that lim t0→0+ κ(f t0 ) = κ(f 0 ). This can be deduced from the continuity of Section 7], and the fact that there are C T > 0 and λ T > 0 such that f t (x) ≤ C T e -λ T |x| 2 for all t ∈ [0, T ] and x ∈ R d . This follows from [START_REF] Grigor'yan | Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds[END_REF], the fact that

f t (x) on [0, ∞) × R d , see [5, Chapter 1,
f t (x) = R d Γ(x, t; ξ, 0)f 0 (ξ)dξ and that f 0 ∈ C c (R d ).
We can now prove the main result of this section.

Proof of Proposition 4. We assume (A) and that R d e -α0U (x) dx < ∞ for some α 0 > 0. We fix c > 0, x 0 ∈ R d and β 0 > 0 and aim to check that for (X t ) t≥0 the solution to (1), lim inf t→∞ |X t | < ∞ a.s. We divide the proof in four steps.

Step 1. We of course may assume additionally that β 0 > α 0 : fix t 0 ≥ 0 large enough so that β t0 > α 0 and observe that (X t0+t ) t≥0 solves (1), with x 0 replaced by X t0 and β 0 replaced by β t0 (and with the Brownian motion (B t0+t -B t0 ) t≥0 ). Since lim inf t→∞ |X t | = lim inf t→∞ |X t0+t |, the conclusion follows.

Step 2. From now on, we assume that β 0 > α 0 . We introduce the stopping time τ ∈ [0, 1] as in Lemma 3. We recall that sup [0,τ ] |X t -x 0 | ≤ 1 and that for h ∈ L 1 ([0, 1] × B(x 0 , 1)) the density of (τ, X τ ), there is C > 0 (depending on x 0 and β 0 ) such that h(u, x) ≤ C1 {u∈[0,1],x∈B(x0,1)} .

Step 3. For n ≥ |x 0 | + 1, we introduce U n ∈ C ∞ (R d ) such that U n (x) = U (x) for all x ∈ B(0, n) and U n (x) = |x| as soon as |x| ≥ n + 1, with furthermore U n (x) ≥ min(U (x), |x|) -1 for all x ∈ R d . Then ∇U n is bounded together all its derivatives. We denote by (X n t ) t≥0 the solution to (1), with U n instead of U . By a classical uniqueness argument (using that ∇U is locally Lipschitz continuous), X and X n coincide until they reach B(0, n) c . In particular, X t = X n t for all t ∈ [0, τ ] and, setting

ζ n = inf{t ≥ 0 : |X τ +t | ≥ n} = inf{t ≥ 0 : |X n τ +t | ≥ n}, it a.s. holds that X n τ +t = X τ +t for all t ∈ [0, ζ n ]. Since lim n ζ n = ∞ a.
s., we conclude that for all t ≥ 0, lim n U n (X n τ +t ) = U (X τ +t ) a.s. As we will check in Step 4, [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF] sup

n≥|x0|+1 sup t≥0 E[U n (X n τ +t )] < ∞.
By the Fatou lemma, we will conclude that sup t≥0 E[U (X τ +t )] < ∞. By the Fatou Lemma again, this will imply that E[lim inf t→∞ U (X t )] < ∞. Since lim |x|→∞ U (x) = ∞ by (A), this will show that lim inf t→∞ |X t | < ∞ a.s. and thus complete the proof.

Step 4. Here we verify [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF]. Denote, for x ∈ R d and β > 0, by f n,x,β t the law at time t of the solution to (1) with x 0 = x, with β 0 replaced by β and with U n instead of U . We then have, since

h is the density of (τ, X τ ) = (τ, X n τ ), E[U n (X n τ +t )] = E[E[U n (X n τ +t )|F τ ]] = [0,1]×B(x0,1) h(u, x) R d U n (y)f n,x,βu t (dy) dudx.
Consider any probability density

f 0 ∈ C c (R d ) such that f 0 > c1 B(x0,1)
, for some constant c > 0.

We thus have h(u, x) ≤ C1 {u∈[0,1],x∈B(x0,1)} ≤ (C/c)f 0 (x), and write

E[U n (X n τ +t )] ≤ C c sup u∈[0,1] R d f 0 (x) R d U n (y)f n,x,βu t (dy) dx = C c sup u∈[0,1] E[U n (Y n,u t )],
where (Y n,u t ) t≥0 is the solution to (1) starting from X 0 ∼ f 0 , with β 0 replaced by β u and U by U n . To conclude the step, it only remains to verify that sup

n≥|x0|+1 sup u∈[0,1] sup t≥0 E[U n (Y n,u t )] < ∞. But Lemma 6 tell us that, setting κ n,u (f 0 ) = R d f 0 (x) log(1 + f 0 (x)e βuUn(x) )dx and a n = [ R d e -α0Un(x) dx] -1 , it holds that E[U n (Y n,u t )] ≤ κ n,u (f 0 ) -log a n β u -α 0 .
This last quantity is uniformly bounded, because

• β u ≥ β 0 > α 0 for all u ∈ [0, 1] (by Step 1); • sup n≥|x0|+1,u∈[0,1] κ n,u (f 0 ) < ∞, since f 0 ∈ C c (R d ), β u ≤ β 1 for all u ∈ [0, 1] and U n (x) = U (x) for all x ∈ Supp f 0 if n is large enough; • sup n≥|x0|+1 (-log a n ) < ∞, since R d e -α0Un(x) dx ≤ e α0 [ R d e -α0U (x) dx + R d e -α0|x| dx] < ∞, recall that U n (x) ≥ min{U (x), |x|} -1.

Localization and absorption

Here we prove that on the event where sup t≥0 |X t | < ∞, the simulated annealing procedure is successful. We also check that each time the process (X t ) t≥0 comes back in a given compact, it has a large probability to be absorbed forever in a (larger) compact.

Lemma 7. Assume (A), fix c > c * , x 0 ∈ R d and β 0 > 0 and consider the solution (X t ) t≥0 to (1). For any ε > 0,

lim t→∞ P sup s≥0 |X s | < ∞ and U (X t ) > ε = 0.
Proposition 8. Assume (A) and fix c > c * . For any A ≥ 1, there is b A > 1 and K A > A such that if x 0 ∈ {U ≤ A} and β 0 ≥ b A , for (X t ) t≥0 the solution to (1), we have

P sup t≥0 U (X t ) ≤ K A ≥ 1 2 .
The rest of the section is dedicated to the proof of these two results. Lemma 7 will easily follow from a result of Holley-Kusuoka-Stroock [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF] concerning the compact case.

Concerning Proposition 8, let us recall from Holley-Kusuoka-Stroock [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF], see also Miclo [START_REF] Miclo | Une étude des algorithmes de recuit simulé sous-admissibles[END_REF], that in the compact setting, lim sup t→∞ U (X t ) = c a.s. and moreover for any ε > 0, if x 0 belongs to a connected component of {U ≤ c + ε} containing a global minimum of U , it holds that P(sup t≥0 U (X t ) ≤ c + ε) > 0. This immediately extends to the non-compact setting, since the set {U ≤ c + ε} is compact. Unfortunately, such a result is not uniform in β 0 > 0, and we really need a uniform bound, see Step 2 of the proof of Proposition 2 in Section 5. We believe it is not possible to deduce Proposition 8 from [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF][START_REF] Miclo | Une étude des algorithmes de recuit simulé sous-admissibles[END_REF]. At this end, we have to work hard, following the ideas of [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF], taking much less care about many constants and obtaining much less precise results (e.g. it might be possible to control K A in Proposition 8) but carefuly tracking the dependence in β 0 and x 0 .

In the whole section, we assume (A) and work with some fixed c > c * . We introduce some notation.

Notation 9. Let K ≥ 1. (a) We consider L K > 0 such that {U ≤ K} ⊂ [-(L K -1), (L K -1)] d . We denote by M K the torus [-L k , L K ) d , that is R d quotiented by the equivalence relation x ∼ y if and only if for all i = 1, . . . , d, (x i -y i )/(2L K ) ∈ Z. (b) We also consider U K ∈ C ∞ (M K ) such that min M K U K = 0, such that U K (x) = U (x) for all
x ∈ {U ≤ K}, and such that

c K * = sup{E K (x, y) : x, y ∈ M K } ≤ c * , where E K (x, y) = inf{max t∈[0,1] U K (γ t ) -U K (x) -U K (y) : γ ∈ C([0, 1], M K ), γ 0 = x, γ 1 = y}. (c) For x 0 ∈ {U ≤ K} ⊂ M K and β 0 > 0, we introduce the inhomogeneous M K -valued diffusion (9) X K t = x 0 + B t - 1 2 t 0 β s ∇U K (X K s )ds modulo 2L K ,
where (B t ) t≥0 is a d-dimensional Brownian motion, where β t = c -1 log(e cβ0 + t) as in (1) and

for x = (x 1 , . . . , x d ) ∈ R d , x modulo 2L K = x i -2L K x i + L K 2L K i=1,...,d ∈ [-L K , L K ) d .
For point (b), it suffices to choose a smooth version of U K = min{U, K}, see [3, Step 1 of the proof of Lemma 6].

Observe that the same constant c K * appears in [START_REF] Fang | An improved annealing method and its large-time behavior[END_REF], where Fang-Qian-Gong study a modified simulated annealing process in which, roughly, β t does not go to infinity when the process lies in the region {U ≥ K}. This is a natural modification, since there is no reason to spend a lot of time in this region.

Since U K = U on {U ≤ K} and since U is locally Lipschitz continuous, a simple uniqueness argument shows the following.

Remark 10. For any K ≥ 1, any x 0 ∈ {U ≤ K}, any β 0 > 0, for (X t ) t≥0 the solution to (1) and (X K t ) t≥0 the solution to (9), both driven by the same Brownian motion, it holds that sup

t≥0 U K (X K t ) ≤ K = sup t≥0 U K (X K t ) ≤ K, sup t≥0 |X K t -X t | = 0 = sup t≥0 U (X t ) ≤ K .
We can now give the Proof of Lemma 7. By [8, Theorem 2.7] and since c > c

* ≥ c K * , U K (X K t ) → 0 in probability, as t → ∞, for each K ≥ 1. We fix η > 0. Since lim |x|→∞ U (x) = ∞, there is K η > 0 such that P(sup s≥0 |X s | < ∞, sup s≥0 U (X s ) > K η ) ≤ η.
We then write, using Remark 10,

P sup s≥0 |X s | < ∞ and U (X t ) > ε ≤η + P sup s≥0 U (X s ) ≤ K η and U (X t ) > ε =η + P sup s≥0 U Kη (X Kη s ) ≤ K η and U Kη (X Kη t ) > ε ≤η + P U Kη (X Kη t ) > ε .
We conclude that lim sup t→∞ P(sup s≥0 |X s | < ∞ and U (X t ) > ε) ≤ η, whence the result since η > 0 is arbitrarily small.

We next introduce the invariant probability measure of the time-homogeneous version of (9).

Remark 11. There is a constant κ 0 > 0 such that, for all K ≥ 1, all β > 0, it holds that

Z K β := M K exp(-βU K (x))dx ≥ κ 0 (β + 1) -d .
We also have Z K β ≤ (2L K ) d . We introduce the probability density

µ K β (x) = (Z K β ) -1 exp(-βU K (x)), x ∈ M K . Proof. Since min R d U = 0, there is x * ∈ R d such that U (x * ) = 0. Fix r * > 0 such that B(x * , r * ) ⊂ {U ≤ 1} ⊂ M K . Denote by C = sup B(x * ,r * ) |∇U |. For all K ≥ 1, all x ∈ B(x * , r * ), we have that x ∈ {U ≤ 1} ⊂ M K and U K (x) = U (x) ≤ C|x -x * |.
Hence for all β > 0,

Z K β ≥ B(x * ,r * ) exp(-βC|x -x * |)dx ≥ e -1 Vol(B(x * , r * ∧ (1/(Cβ))),
from which the lower-bound follows. The upper-bound is trivial.

As a final preliminary, we recall the crucial spectral gap estimate of Holley-Kusuoka-Stroock [8, Theorem 1.14 and Remark 1.16], in the special case of the torus. We use that c K * ≤ c * , see Notation 9-(b) (in the notation of [START_REF] Holley | Asymptotics of the spectral gap with applications to the theory of simulated annealing[END_REF], m = c K * ). Lemma 12 (Holley-Kusuoka-Stroock). Fix K ≥ 1. There is a constant γ K > 0 such that for all φ ∈ C 1 (M K ), for all β > 0,

M K |∇φ(x)| 2 µ K β (x)dx ≥ λ K (β) M K φ(x) - M K φ(y)µ K β (y)dy 2 µ K β (x)dx, with λ K (β) = γ K (β + 1) 2-5d exp(-βc * ).
The constant γ K drastically depends on K but, as we will see, this is not an issue.

Lemma 13. Fix K ≥ 1. There is a constant b

K > 0 such that if β 0 ≥ b (1)
K , then for any density f K 0 ∈ C(M K ), for f K t the density of X K t , the solution to (9) starting from

X K 0 ∼ f K 0 , ∀ t ≥ 0, M K (f K t (x)) 2 µ K βt (x) dx ≤ max 2, M K (f K 0 (x)) 2 µ K β0 (x) dx .
Proof. The function (f K t (x)) t≥0,x∈M K is a weak solution to the uniformly parabolic equation

∂ t f K t (x) = 1 2 div(∇f K t (x)+β t f K t (x)∇U K (x)
). It can be seen as a periodic solution of the same equation in R d , with U K and f K 0 replaced by their periodic continuation. We thus can apply some classical results, see Friedman [5, Chapter 1, Theorems 10 and 12] and conclude that (

f K t (x)) t≥0,x∈M K belongs to C([0, ∞) × M K ) ∩ C 1,2 ((0, ∞) × M K ).
The periodic continuation of f K 0 has an infinite mass, but this is allowed by [START_REF] Friedman | Partial differential equations of parabolic type[END_REF]. Since furthermore M K is bounded, all the computations below are easily justified.

We introduce

ϕ(t) = M K (f K t (x)) 2 µ K βt (x) dx = Z K βt M K (f K t (x)) 2 e βtU K (x) dx. Since (Z K βt ) = -β t M K U K (
x)e -βtU K (x) dx ≤ 0, we have, for all t > 0,

ϕ (t) ≤ Z K βt M K 2[∂ t f K t (x)]f K t (x)e βtU K (x) dx + β t Z K βt M K U K (x)(f K t (x)) 2 e βtU K (x) dx. Recalling that ∂ t f K t (x) = 1 2 div(∇f K t (x) + β t f K t (x)∇U K (x)
), proceeding to an integration by parts in the first term and to a rough upper-bound in the second one, we find

ϕ (t) ≤ -Z K βt M K |∇f K t (x) + β t f K t (x)∇U K (x)| 2 e βtU K (x) dx + β t ||U K || ∞ ϕ(t) = - M K ∇ f K t (x) µ K βt (x) 2 µ K βt (x)dx + β t ||U K || ∞ ϕ(t).
By Lemma 12 with φ(x) = f K t (x)/µ K βt (x), for which M K φ(y)µ K βt (y)dy = 1 , we conclude that

ϕ (t) ≤ -λ K (β t ) M K f K t (x) µ K βt (x) -1 2 µ K βt (x)dx+β t ||U K || ∞ ϕ(t) = -λ K (β t )[ϕ(t)-1]+β t ||U K || ∞ ϕ(t).
But we know from Lemma 12 that for all β > 0,

λ K (β) ≥ γ K (β + 1) 2-5d e -βc * ≥ γ K e -β(c+c * )/2
for some other constant γ K > 0, since c > c * . Setting α = (c -c * )/(2c) ∈ (0, 1), so that (c + c * )/2 = c(1 -α), recalling that β t = log(e cβ0 + t)/c, we conclude that

ϕ (t) ≤ - γ K (e cβ0 + t) 1-α [ϕ(t) -1] + ||U K || ∞ c(e cβ0 + t) ϕ(t).
Let b

(1)

K = 1 αc log(2||U K || ∞ /(γ K c)), so that if β 0 ≥ b (1)
K , for all t ≥ 0,

||U K || ∞ c(e cβ0 + t) ≤ γ K 2(e cβ0 + t) 1-α , whence ϕ (t) ≤ - γ K 2(e cβ0 + t) 1-α [ϕ(t) -2].
We classically conclude that indeed, if

β 0 ≥ b (1)
K , then for all t ≥ 0, ϕ(t) ≤ max(2, ϕ(0)). From the previous lemma and the Cauchy-Schwarz inequality, we deduce the following.

Lemma 14. For

A ≥ 1, let D A = 2C (1) A +C (2) A +1+4c and K A = D A +1, where C (1)
A was introduced in Lemma 3 and where C

(2)

A = sup x∈{U ≤A} sup y∈B(x,2) U (y). There is a constant C (3) A > 0 such that, if β 0 ≥ b (1) K A (see Lemma 13) and x 0 ∈ {U ≤ A}, it holds that sup [0,τ ] U K A (X K A t ) ≤ D A a.s. and ∀ t ≥ 0, P(U K A (X K A τ +t ) ≥ D A ) ≤ C (3) A (e cβ0 + t) 2 , where (X K A t
) t≥0 is the solution to (9) starting from x 0 and where τ is the stopping time introduced in Lemma 3 (for the solution (X t ) t≥0 to (1) driven by the same Brownian motion as (X

K A t ) t≥0 ). Proof. We fix A ≥ 1, β 0 ≥ b (1) K A and x 0 ∈ {U ≤ A}. First, since B(x 0 , 1) ⊂ {U ≤ D A } (because D A ≥ C (2) A ), since sup [0,τ ] |X t -x 0 | ≤ 1 and since D A ≤ K A , Remark 10 tells us that X t = X K A t for all t ∈ [0, τ ]. In particular, sup [0,τ ] U K A (X K A t
) ≤ D A and the law of (τ, X K A τ ) = (τ, X τ ) has a density h(u, x) bounded by e C (1) A (β0+1) 1 {u∈[0,1],x∈B(x0,1)} , see Lemma 3.

Denote, for x ∈ R d and β > 0, by f K A ,x,β t the law of the solution of ( 9) with K = K A , with x 0 = x and with β 0 replaced by β. We then have

P(U K A (X K A τ +t ) ≥ D A ) =E[P(U K A (X K A τ +t ) ≥ D A |F τ )] = [0,1]×B(x0,1)
h(u, x)

{U K A ≥D A } f K A ,x,βu t (dy) dudx.
Consider a probability density

f 0 ∈ C c (R d ) such that (2v d ) -1 1 B(x0,1) ≤ f 0 ≤ 1 B(x0,2)
, v d being the volume of the unit ball. We write

P(U K A (X K A τ +t ) ≥ D A ) ≤2v d e C (1) A (β0+1) sup u∈[0,1] R d {U K A ≥D A } f K A ,x,βu t (dy) f 0 (x)dx (10) =2v d e C (1) A (β0+1) sup u∈[0,1] P(U K A (Y A,u t ) ≥ D A ),
where (Y A,u t ) t≥0 is the solution to ( 9) with K = K A , starting from Y 0 ∼ f 0 , with β 0 replaced by β u . We now denote by f A,u t the density of Y A,u t and use the Cauchy-Schwarz inequality to write

P(U K A (Y A,u t ) ≥ D A ) = {U K A ≥D A } f A,u t (x)dx (11) ≤ {U K A ≥D A } µ K A βt+u (x)dx 1/2 M K A [f A,u t (x)] 2 dx µ K A βt+u (x) 1/2
.

By Lemma 13, we know that, since

β u ≥ β 0 ≥ b (1) K A , M K A [f A,u t (x)] 2 dx µ K A βt+u (x) ≤ 2 ∨ M K A [f 0 (x)] 2 dx µ K A βu (x) ≤ 2 ∨ B(x0,2) dx µ K A βu (x) = 2 ∨ B(x0,2) Z K A βu e βuU (x) dx.
Recalling the definition of C

(2)

A and that Z K A βu ≤ (2L K A ) d , see Remark 11, we find ( 12)

M K A [f A,u t (x)] 2 dx µ K A βt+u (x) ≤ 2 ∨ [2 d v d (2L K A ) d e C (2) A βu ] = 2 d v d (2L K A ) d e C (2) A βu .
Next, since the volume of M K A is smaller than (2L K A ) d , we have

{U K A ≥D A } µ K A βt+u (x)dx ≤ (2L K A ) d e -βt+uD A Z K A βt+u .
By Remark 11 again, ( 13)

{U K A ≥D A } µ K A βt+u (x)dx ≤ (2L K A ) d (β t+u + 1) d e -βt+uD A κ 0 ≤ C A e -βt+u(D A -1) ,
for some constant C A > 0 of which we now allow the value to change from line to line. Gathering ( 10)-( 11)-( 12)-( 13), we find

P(U K A (X K A τ +t ) ≥ D A ) ≤ C A e C (1) A (β0+1) sup u∈[0,1] e -βt+u(D A -1)/2+C (2) A βu/2 .
Since s → β s is non-decreasing and

D A -1 > C (2)
A ,

P(U K A (X K A τ +t ) ≥ D A ) ≤ C A e C (1) A βt sup u∈[0,1] e -βt+u(D A -1-C (2) A )/2 ≤ C A e -βt(D A -1-C (2) A -2C (1)
A )/2 .

Recalling finally that D A = 2C

(1)

A + C (2)
A + 1 + 4c and that β t = c -1 log(e cβ0 + t), we conclude that

P(U K A (X K A τ +t ) ≥ D A ) ≤ C A e -2cβt = C A (e cβ0 + t) 2
as desired.

We finally give the Proof of Proposition 8. We fix A ≥ 1 and introduce

D A , K A = D A + 1, b (1)
K A and C (3)
A as in Lemma 14. We will show that one can find b A > b

(1)

K A such that if β 0 > b A and x 0 ∈ {U ≤ A}, the solution (X K A t ) t≥0 to (9) satisfies P sup t≥0 U K A (X K A t ) ≤ K A ≥ 1 2 .
By Remark 10, this will show the result. By Lemma 14, we have sup

[0,τ ] U K A (X K A t ) ≤ D A ≤ K A a.
s., so that we only have to check that

P sup t≥0 U K A (X K A τ +t ) ≤ K A ≥ 1 2 . We consider φ A ∈ C ∞ (R + ), with values in [0, 1], such that φ A = 0 outside [D A , K A ] and such that φ A ((D A + K A )/2) = 1,

and we introduce ψ

A = φ A • U K A : M K A → [0, 1]. Setting L A βt ψ A (x) = 1 2 (∆ψ A (x) -β t ∇ψ A (x) • ∇U K A (x)), we have |L A βt ψ A (x)| ≤ C (4)
A (1

+ β t )1 {U K A (x)≥D A } , where C (4)
A is a constant involving the supremum on M K A of U K A and its two first derivatives.

We now fix b

A > b (1) K A such that for all β 0 ≥ b A , ∞ 0 C (3) A C (4) A [1 + log(e cβ0 + t + 1)] c(e cβ0 + t) 2 dt = ∞ e cβ 0 C (3)
A C

A [1 + log(s + 1)] cs 2 ds ≤ 1 40 .

By Itô's formula and since ψ

A (X K A τ ) = 0 (because U K A (X K A τ ) ≤ D A ), ψ A (X K A τ +t ) = M t + R t
, where (M t ) t≥0 is a martingale issued from 0 and where

R t = t 0 L A βτ+s ψ A (X K A τ +s )ds.
By Lemma 14, since

|L A βτ+t ψ A (x)| ≤ C (4)
A (1 + β t+1 )1 {U K A (x)≥D A } and since β 0 ≥ b A ≥ b (1)
K A , E sup t≥0 |R t | ≤ C (4) A ∞ 0 (1 + β t+1 )P(U K A (X K A τ +t ) ≥ D A )dt ≤ ∞ 0 C (3)
A C

A (1 + β t+1 ) (e cβ0 + t) 2 dt ≤ 1 40 . Consequently, for E = {sup t≥0 |R t | < 1/10}, we have P(E c ) ≤ 1/4. (4)
On E, we have M t = M t∧σ , where σ = inf{t ≥ 0 :

M t / ∈ [-1/10, 11/10]}, because M t + R t = ψ A (X K A τ +t ) takes values in [0, 1]. On {sup t≥0 U K A (X K A τ +t ) ≥ K A }, the process ψ A (X K A τ +t ) must up- cross [0, 1] at least once, so that on E ∩ {sup t≥0 U K A (X K A τ +t ) ≥ K A }, the martingale M t = M t∧σ
must up-cross [1/10, 9/10] at least once. Hence

P sup t≥0 U K A (X K A τ +t ) ≥ K A ≤ P(E c ) + P(E, (M t∧σ ) t≥0 up-crosses [1/10, 9/10]) ≤ 1/4 + p,
where p = P((M t∧σ ) t≥0 up-crosses [1/10, 9/10]).

By Doob's up-crossing inequality, see e.g. Revuz-Yor [14, Proposition 2.1 page 61] we know that, for any continuous martingale (Z t ) t≥0 , any a < b, denoting by U T,a,b the number of up-crossings of

[a, b] by (Z t ) t≥0 during [0, T ], it holds that (b -a)E[U T,a,b ] ≤ E[(Z T -a) -].
Thus for N T the number of up-crossings of [1/10, 9/10] by the martingale (M t∧σ ) t≥0 during [0, T ], it holds that p = lim

T →∞ P(N T ≥ 1) ≤ lim T →∞ E[N T ] ≤ lim T →∞ E[(M T ∧σ -1/10) -] 8/10 ≤ 2/10 8/10 = 1 4 .
We used that M T ∧σ ≥ -1/10 by definition of σ. We conclude that, for all β > b A , we have

P(sup t≥0 U K A (X K A τ +t ) ≥ K A ) ≤ 1/2 as desired.

Success of the simulated annealing

We now show that no escape in large time implies the success of the simulated annealing.

Proof of Proposition 2. We assume (A), fix c > c * , x 0 ∈ R d , β 0 > 0 and consider the solution (X t ) t≥0 to (1). Since lim |x|→∞ U (x) = ∞, our goal is to show that for any fixed ε > 0,

lim t→∞ P lim inf s→∞ U (X s ) < ∞ and U (X t ) > ε = 0.
Step 1. It suffices to show that for each A ≥ 1, setting

Ω A = {lim inf s→∞ U (X s ) < A},
it holds that Ω A ⊂ {sup s≥0 |X s | < ∞}. Indeed, if this hold true, we fix η > 0, consider A η > 0 large enough so that P(A η ≤ lim inf s→∞ U (X s ) < ∞) < η and write

P lim inf s→∞ U (X s ) < ∞ and U (X t ) > ε ≤η + P(Ω Aη and U (X t ) > ε) ≤η + P sup s≥0 |X s | < ∞ and U (X t ) > ε .
Thus lim sup t→∞ P(lim inf s→∞ U (X s ) < ∞ and U (X t ) > ε) ≤ η by Lemma 7. Since η > 0 is arbitrarily small, the conclusion follows.

Step 2. We fix A ≥ 1 and show that for

Ω A = {lim inf s→∞ U (X s ) < A}, we have Ω A ⊂ {sup s≥0 |X s | < ∞}.
We introduce b A > 1 and K A > A as in Proposition 8 and consider t A ≥ 0 large enough so that β t A ≥ b A . We set S 0 = t A and, for all k ≥ 1,

T k = inf{t > S k-1 : U (X t ) ≤ A} and S k = inf{t > T k : U (X t ) ≥ K A },
with the convention that inf ∅ = ∞.

We start from

P(S k+1 < ∞|S k < ∞) =P(T k+1 < ∞, S k+1 < ∞|S k < ∞) =E 1 {T k+1 <∞} P(S k+1 < ∞|F T k+1 ) S k < ∞ .
But on {T k+1 < ∞} and conditionally on F T k+1 , (X T k+1 +t ) t≥0 is a solution to (1), starting from X T k+1 ∈ {U ≤ A}, with β 0 replaced by β T k+1 ≥ β t A ≥ b A . Hence, using Proposition 8, a.s.,

1 {T k+1 <∞} P(S k+1 < ∞|F T k+1 ) = 1 {T k+1 <∞} P sup t≥0 U (X T k+1 +t ) ≥ K A ≤ 1/2.
All this shows that for all k ≥ 1, P(S k+1 < ∞|S k < ∞) ≤ 1/2.

Consequently, there a.s. exists k ≥ 1 such that S k = ∞, and we introduce

k 0 = inf{k ≥ 1 : S k = ∞}. We then have S k0-1 < ∞ = S k0 . By definition of Ω A , it holds that T k0 < ∞ on Ω A . Since U (X t ) < K A for all t ∈ [T k0 , S k0 ) = [T k0 , ∞) (on Ω A ), this implies that Ω A ⊂ lim sup s→∞ U (X s ) ≤ K A ⊂ sup s≥0 |X s | < ∞ as desired.
We conclude the section with the Proof of Theorem 1. We assume (A) and that there is α 0 > 0 such that R d e -α0U (x) dx < ∞. We fix c > 0, x 0 ∈ R d , β 0 > 0 and consider the unique solution (X t ) t≥0 to [START_REF] Azencott | Simulated annealing. Séminaire Bourbaki[END_REF]. By Proposition 4, lim inf t→∞ |X t | < ∞ a.s. If moreover c > c * , lim t→∞ U (X t ) = 0 in probability by Proposition 2.

Appendix: non-explosion

It remains to study the non-explosion of our process. Surprisingly, this is rather tedious, except if assuming some Lyapunov condition, for example that -x•∇U (x) ≤ C(1+|x| 2 ), which forbids too nasty oscillations. We will prove the following result, which is much stronger (but more natural) than what we really need, since U ≥ 0 under (A). Then it holds that ζ = ∞ a.s.

Since ∇U is locally Lipschitz continuous, the existence of a pathwise unique possibly exploding solution is classical. This result is rather natural: as is well-known, the solution to [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF], with U (x) = -(1 + |x| 2 ) α explodes if and only if α > 1. The difficulty relies in the fact that we do not want to assume any local property on ∇U . Let us mention that the proof below, assuming that U ≥ 0, would be slightly simpler but less transparent.

Our proof is inspired by methods found in Ichihara [START_REF] Ichihara | Explosion problems for symmetric diffusion processes[END_REF], who uses Dirichlet forms, and Grigor'yan [START_REF] Grigor'yan | Stochastically complete manifolds[END_REF] and [START_REF] Grigor'yan | Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds[END_REF]Section 9], who studies manifold-valued diffusions. Both deal with the time-homogeneous case (β t = β 0 for all t ≥ 0). In [START_REF] Grigor'yan | Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds[END_REF], non-explosion is proved under some very weak conditions (allowing e.g. for some additional logarithmic factors in (15)), while [START_REF] Ichihara | Explosion problems for symmetric diffusion processes[END_REF] is more stringent (roughly, he treats only the case where U (x) ≥ -L(1 + |x|)).

We start with the following remark.

Remark 16. (i) To prove Theorem 15, one may assume additionally that [START_REF] Zitt | Annealing diffusions in a potential function with a slow growth[END_REF] there is t 0 > 0 such that β t = β t0 for all t ≥ t 0 .

(ii) For any x 0 ∈ R d , for (X t ) t∈[0,ζ) the solution to [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] and for t > 0, the measure f t defined by f t (A) = P(ζ > t, X t ∈ A) is absolutely continuous with respect to the Lebesgue measure on R d .

(iii) It suffices to prove Theorem 15 for a.e. x 0 ∈ R d .

Proof. (i) Assume that Theorem 15 holds under the additional condition [START_REF] Zitt | Annealing diffusions in a potential function with a slow growth[END_REF] and consider β : R + → (0, ∞) of class C ∞ . We fix T > 0, introduce β : R + → (0, ∞) of class C ∞ satisfying [START_REF] Zitt | Annealing diffusions in a potential function with a slow growth[END_REF], such that β t = βt on [0, T ] and we introduce the corresponding solution ( Xt ) t∈[0, ζ) . We have

(X t ) t∈[0,T ∧ζ) = ( Xt ) t∈[0,T ∧ ζ) , whence in particular {ζ ≤ T } = { ζ ≤ T }. Since ζ = ∞ a.
s., we conclude that P(ζ ≤ T ) = 0. Since T is arbitrarily large, this implies that ζ = ∞ a.s.

(ii) Fix a Lebesgue-null set A ∈ R d . Since ∇U is bounded on compact sets, we deduce from the Girsanov theorem that P(ζ n > t, X t ∈ A) = 0 for all n ≥ 1. By monotone convergence, we conclude that P(ζ > t, X t ∈ A) = 0 as desired.

(iii) Assume that for any β : R + → (0, ∞) of class C ∞ , P 0,x (ζ < ∞) = 0 for a.e. x ∈ R d . Then for a given β : R + → (0, ∞) of class C ∞ , for all t ≥ 0, P t,x (ζ < ∞) = 0 for a.e. x ∈ R d . Since ζ > 0 a.s. by continuity, we may write, for all x 0 ∈ R d ,

P 0,x0 (ζ < ∞) = lim t→0 P 0,x0 (t < ζ < ∞) = lim t→0 E 0,x0 [1 {ζ>t} P t,Xt (ζ < ∞)] = 0.
We used the Markov property and that 1 {ζ>t} P t,Xt (ζ < ∞) = 0 a.s. when t > 0 by point (ii) and since P t,x (ζ < ∞) = 0 for a.e. x ∈ R d .

Above and in the whole section, we denote by E t0,x0 the expectation concerning the process starting from x 0 ∈ R d at time t 0 ≥ 0: under E t0,x0 , the process (X t ) t≥0 solves (in law) the S.D.E.

X t = x 0 + B t -1 2 t 0 β t0+s ∇U (X s )ds.
In the whole section, we denote by v d the volume of the unit ball and, for r > 0, we set

B r = {x ∈ R d : |x| < r}, Br = {x ∈ R d : |x| ≤ r} and ∂B r = {x ∈ R d : |x| = r}.
We will study of the following Kolmogorov backward equation, which consists of a particular case of the Feynman-Kac formula.

Lemma 17. Adopt the assumptions of Theorem 15 and suppose [START_REF] Zitt | Annealing diffusions in a potential function with a slow growth[END_REF]. Fix n ≥ 1 and α > 0. There is a function u

n,α ∈ C 1,2 (R + × Bn ) such that u n,α = 1 on R + × ∂B n and (17) ∂ t u n,α (t, x) + L βt u n,α (t, x) = αu n,α (t, x) for (t, x) ∈ [0, ∞) × B n . For ϕ : R d → R of class C 2 , β > 0 and x ∈ R d , we have set L β ϕ(x) = 1 2 [∆ϕ(x) -β∇U (x) • ∇ϕ(x)].
For any t ≥ 0, any x ∈ Bn , it holds that u n,α (t

, x) = E t,x [exp(-αζ n )].
Proof. This relies one more time on classical results found in Friedman [START_REF] Friedman | Partial differential equations of parabolic type[END_REF]. We fix some t 0 ≥ 0 such that β t = β t0 for all t ≥ t 0 . All the coefficients of (17) are smooth and bounded, since restricted to Bn , whose boundary is smooth. Hence all the results cited below do indeed apply.

By [5, Chapter 3, Theorem 19], there exists a solution v n,α ∈ C 2 ( Bn ) to the elliptic boundary problem L βt 0 v n,α = αv n,α on B n and v n,α = 1 on ∂B n . By [5, Chapter 3, Theorem 7] (after time-reversing), there exists a solution w n,α belonging to C 1,2 ([0, t 0 ] × Bn ) to the parabolic problem ∂ t w n,α + L βt w n,α = αw n,α on (0, t 0 ) × B n , with boundary condition w n,α = 1 on [0, t 0 ] × ∂B n and terminal condition w n,α (t 0 , x) = v n,α (x) on B n .

The function u n,α defined by u n,α (t, x) = v n,α (x) if t ≥ t 0 and u n,α (t, x) = w n,α (t, x) if t ∈ [0, t 0 ] satisfies the conditions of the statement.

Finally, using the Itô formula and (17), one checks that for all t ≥ 0, all x ∈ B n , all T ≥ 0,

E t,x [u n,α (T ∧ ζ n , X T ∧ζn )e -α(T ∧ζn) ] = u n,α (t, x).
We let T → ∞ and find that E t,x [e -αζn ] = u n,α (t, x) by dominated convergence and since u n,α (ζ n , X ζn ) = 1 a.s.

Remark 18. By the Green formula, for all β > 0, for all r > 0, for all ϕ : Br → R of class C 1 and ψ : Br

→ R of class C 2 , 2 Br ϕ(x)L β ψ(x)e -βU (x) dx = - Br ∇ϕ(x) • ∇ψ(x)e -βU (x) dx + ∂Br ϕ(x)[∇ψ(x) • ν(x)]e -βU (x) dS,
where ν(x) = x/|x| is the unit vector normal to ∂B r and where dS is its surface element.

Although this is already known, see Grigor'yan [7, Section 9], we recall for the sake of completeness how to treat the homogeneous case. We use an approach closer to the one of Ichihara [START_REF] Ichihara | Explosion problems for symmetric diffusion processes[END_REF], who however assumes more than [START_REF] Royer | A remark on simulated annealing of diffusion processes[END_REF] and whose proof is more intricate and relies on the study of v

(x) = E x [e -σ1 ], where σ 1 = inf{t ≥ 0 : |X t | ≤ 1}.
Proposition 19. Assume that U : R d → R is C ∞ and satisfies [START_REF] Royer | A remark on simulated annealing of diffusion processes[END_REF]. If β t = β 0 > 0 for all t ≥ 0, then P x (ζ < ∞) = 0 for a.e. x ∈ R d .

Proof. For α > 0 and n ≥ 1, we set u α (x) = E x [e -αζ ] and u n,α (x) = E x [e -αζn ]. We divide the proof into 2 steps. We recall that L is defined in [START_REF] Royer | A remark on simulated annealing of diffusion processes[END_REF].

Step 1. Here we prove that for all r > 0, there is a constant C r > 0 such that (18) ∀α > 0, Br u 2 α (x)e -β0U (x) dx ≤ C r e -α/(2β0L) .

By Proposition 17, u n,α ∈ C 2 ( Bn ), u n,α = 1 on ∂B n and L β0 u n,α = αu n,α on B n . For any r ∈ (0, n], we have

Φ n,α (r) := Br (2αu 2 n,α (x) + |∇u n,α (x)| 2 )e -β0U (x) dx = ∂Br u n,α (x)[∇u n,α (x) • ν(x)]e -β0U (x) dS.
Indeed, it suffices to write 2αu 2 n,α = 2u n,α L β0 u n,α and to use Remark 18 with ϕ = ψ = u n,α . Hence for all r ∈ (0, n], since 2αa

2 + b 2 ≥ 2 √ 2αab, Φ n,α (r) = ∂Br (2αu 2 n,α (x) + |∇u n,α (x)| 2 )e -β0U (x) dS ≥2 √ 2α ∂Br u n,α (x)|∇u n,α (x)|e -β0U (x) dS ≥ 2 √ 2αΦ n,α (r). Thus Φ n,α (r) ≤ Φ n,α (n)e 2 √ 2α(r-n) for all r ∈ (0, n].
But, writing 2αu n,α = 2L β0 u n,α and using Remark 18 with ϕ = 1 and ψ = u n,α , x) dx ≤ 2αv d n d e β0L(1+n 2 ) by [START_REF] Royer | A remark on simulated annealing of diffusion processes[END_REF]. All this shows that for all α > 0, all n ≥ 1, all r ∈ (0, n],

2 Bn αu n,α (x)e -β0U (x) dx = ∂Bn [∇u n,α (x) • ν(x)]e -β0U (x) dS = Φ n,α (n), because u n,α = 1 on ∂B n . Hence Φ n,α (n) ≤ 2α Bn e -β0U (
Br u 2 n,α (x)e -β0U (x) dx ≤ 1 2α Φ n,α (r) ≤ v d n d e β0L(1+n 2 ) e 2 √ 2α(r-n) .
But u α (x) ≤ u n,α (x) for all n ≥ 1, all x ∈ R d . Hence for r > 0 fixed, with the choice n = r + √ α/(β 0 L), we conclude that, for some constant C r > 0 depending on r (and on L and d),

Br u 2 α (x)e -β0U (x) dx ≤ v d (r + √ α/(β 0 L)) d e β0L(1+2r 2 )+2α/(β0L) e -2 √ 2α/(β0L) ≤ C r e -α/(2β0L) .
Step 2. We now conclude. Assume by contradiction that With the choice η = 1/(8β 0 L), this contradicts (18).

Using in particular some clever ideas found in Grigor'yan [START_REF] Grigor'yan | Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds[END_REF]Section 9], who studies, in the homogeneous case, the P.D.E. satisfied by w(t, x) = P x [ζ < t], we can now give the Proof of Theorem 15. We consider U : R d → R and β : R + → (0, ∞) of class C ∞ . We recall that ζ = lim n ζ n , with ζ n = inf{t ≥ 0 : |X t | ≥ n}. We set u n (t, x) = E t,x [e -ζn ] and u(t, x) = E t,x [e -ζ ], omitting the subscript α since we now always work with α = 1. By Remark 16, we may moreover suppose that there is t 0 > 0 such that β t = β t0 for all t ≥ t 0 , and it suffices to prove that u(0, x) = 0 for a.e. x ∈ R d . Since u(t 0 , x) = 0 for a.e. x ∈ R d by Proposition 19, it is sufficient to prove that (19) ∃δ 0 > 0, ∀t 1 ∈ [0, t 0 ],

R d u(t 1 , x)dx = 0 ⇒ ∀t ∈ [(t 1 -δ 0 ) ∨ 0, t 1 ], R d u(t, x)dx = 0 .
Step 1. Here we check that for all a ∈ [0, 1], all b ≥ 0, all ε > 0, all η > 0, ab ≤ a ηε + e ηb-1/ε η .

We fix a ∈ [0, 1], ε > 0, and η > 0 and study f (b) = a ε +e ηb-1/ε -ηab. We have f (0) > 0, f (∞) = ∞ and f (b) = η[e ηb-1/ε -a]. If a ≤ e -1/ε , then f is non-decreasing, so that f is nonnegative on R + . If now a > e -1/ε , then f attains its minimum at b 0 = 1 η [log a + 1 ε ] and f (b 0 ) = a -a log a > 0.

Step 2. Here we prove that are some constants δ > 0 and κ 0 > 0 such that for all t 1 ∈ [0, t 0 ], all R > 1, there is a C 1 function α t1,R : [(t 1 -δ) ∨ 0, t 1 ] × R d → [0, 1] enjoying the properties that 2(2δ + t -t 1 ) , which equals 0 on [(t 1 -δ) ∨ 0, t 1 ] × B R and solves ∂ t ξ t1,R + 1 2 |∇ξ t1,R | 2 = 0 on [(t 1 -δ) ∨ 0, t 1 ] × B c R . If δ > 0 is small enough, the function α t1,R (t, x) = η R (x) exp(-ξ t1,R (t, x)/2) enjoys the desired properties: it is C 1 , [0, 1]-valued, we have α t1,R (t, x) = 1 if |x| ≤ R and α R (t, x) = 0 if |x| ≥ 4R, and we have

α t1,R = 1 on [(t 1 -δ) ∨ 0, t 1 ] × B R , α t1,R = 0 on [(t 1 -δ) ∨ 0, t 1 ] × B c
|∇α t1,R | 2 -∂ t [α 2 t1,R ] = ∇η t1,R - 1 2 η t1,R ∇ξ t1,R 2 e -ξ t 1 ,R + η 2 t1,R e -ξ t 1 ,R ∂ t ξ t1,R ≤ 2|∇η t1,R | 2 + 1 2 η 2 t1,R |∇ξ t1,R | 2 + η 2 t1,R ∂ t ξ t1,R e -ξ t 1 ,R = 2|∇η t1,R | 2 e -ξ t 1 ,R .
Since |∇η t1,R | ≤ R -1 1 B 4R \B 2R and since ξ t1,R ≥ R 2 4δ on [(t 1 -δ) ∨ 0, t 1 ] × B c 2R , we deduce from [START_REF] Royer | A remark on simulated annealing of diffusion processes[END_REF], that m t1,R (t) ≤2R -2 e -R 2 /(4δ) e βtL(1+16R 2 ) Vol(B 4R \ B 2R ).

Since β is bounded on [0, t 0 ], it indeed suffices to choose δ > 0 small enough to complete the step.

Step 3. We consider δ > 0 as in Step 2, fix t 1 ∈ [0, t 0 ] and set, for R > 1, n ≥ 5R and t ∈ [(t 1 -δ) ∨ 0, t 1 ],

ϕ n,t1,R (t) = R d u 2
n (t, x)α 2 t1,R (t, x)e -βtU (x) dx.

The goal of this step is to verify that there is a constant κ 1 > 0 such that (20) ∀t 1 ∈ [0, t 0 ], ∀R > 1, ∀n ≥ 5R, ∀t ∈ [(t 1 -δ)∨0, t 1 ], ϕ n,t1,R (t) ≥ -κ 1 R 2 ϕ n,t1,R (t)+e -R 2 .

By (17) (with α = 1), we know that ∂ t u n (t, x) = u n (t, x) -L βt u n (t, x) on [0, ∞) × B n . Since Supp α t1,R (t, •) ⊂ B 4R and since n ≥ 5R, we may write ϕ n,t1,R (t) = I n,t1,R (t) + J n,t1,R (t) -K n,t1,R (t), where But since u(t 1 , x) = 0 for a.e. x ∈ R d , it holds that ϕ t1,R (t 1 ) = 0 for all R > 1. Hence for all fixed t ∈ [(t 1 -δ 0 ) ∨ 0, t 1 ], all fixed R 0 > 0, all R > R 0 > 1, since α t1,R (t, •) ≥ 1 B R 0 , B R 0 u 2 (t, x)e -βtU (x) dx ≤ ϕ t1,R (t) ≤ e -R 2 /2 , whence B R 0 u 2 (t, x)e -βtU (x) dx = 0 and thus u(t, x) = 0 for a.e. x ∈ R d as desired.

I n,t1,R (t) = R d 2[u n (t,

  Fix a dimension d ≥ 1 and a function U :R d → R + of class C ∞ such that lim |x|→∞ U (x) = ∞ and min x∈R d U (x) = 0. For x, y ∈ R d , we set E(x, y) = inf max t∈[0,1] U (γ t ) -U (x) -U (y) : γ ∈ C([0, 1], R d ), γ 0 = x, γ 1 = yand we suppose that c * = sup{E(x, y) : x, y ∈ R d } < ∞.

  |x|→∞ |∇U (x)| = ∞ and ∀ x ∈ R d , ∆U (x) ≤ C + |∇U (x)| 2 .

1 . 5 .

 15 (f) Still in Section 5, we conclude the proof of Theorem 1: lim inf t→∞ |X t | < ∞ a.s. by (b), whence the success of the simulated annealing by (e). More comments. It is well-known that, even in the compact case, the condition c > c * is necessary, see Holley-Kusuoka-Stroock [8, Corollary 3.11].

Theorem 15 . 2 t 0 β

 1520 Assume that U : R d → R and β : R + → (0, ∞) are of class C ∞ . Fix x 0 ∈ R d and consider the pathwise unique maximal solution(X t ) t∈[0,ζ) to (14) X t = x 0 + B t -1 s ∇U (X s )ds, where ζ = lim n ζ n , with ζ n = inf{t ≥ 0 : |X t | ≥ n}. Assume that(15)there is L > 0 such that for all x ∈ R d , U (x) ≥ -L(1 + |x| 2 ).

  R d P x (ζ < ∞)dx > 0 and fix η > 0. It holds that R d P x (ζ ≤ η)dx > 0.Else we would have, by the Markov property,R d P x (ζ ≤ 2η)dx = R d P x (η < ζ ≤ 2η)dx = R d E x [1 {ζ>η} P Xη (ζ ≤ η)]dx = 0 thanks to Remark 16-(ii).Iterating the argument, we would find thatR d P x (ζ ≤ Kη)dx = 0 for all K ≥ 1, whence R d P x (ζ < ∞)dx = 0. Consequently, we can find r 0 > 0 such that q := Br 0 [P x (ζ ≤ η)] 2 e -β0U (x) dx > 0. We then have, since u α (x) = E x [e -αζ ] ≥ e -αη P x (ζ ≤ η), ∀α > 0,Br 0 u 2 α (x)e -β0U (x) dx ≥ qe -2αη .

4R 2 .

 2 and, for all t ∈ [(t 1 -δ) ∨ 0, t 1 ],m t1,R (t) := R d |∇α t1,R (t, x)| 2 -∂ t [α 2 t1,R (t, x)] + e -βtU (x) dx ≤ κ 0 e -RWe start with aC ∞ -function η R : R d → [0, 1] such that η R = 1 on B 2R , η R = 0 on B c 4R and |∇η R | ≤ 1/R.We also introduce theC 1 function on [(t 1 -δ) ∨ 0, t 1 ] × R d defined by ξ t1,R (t, x) = (|x| -R) 2 +

  applied to a large flat torus: the condition sup t≥0 |X t | < ∞ almost tells us that we are in a compact setting. (d) Still in Section 4, we check, although stated in slightly different words, see Proposition 8, that for any B ≥ 1, there are C B > B and t B > 0 such that

	inf |x0|≤B,t0≥t B	P t0,x0 sup t≥0

  x) -L βt u n (t, x)]u n (t, x)α 2 t1,R (t, x)e -βtU (x) dx,

	J n,t1,R (t) =	R d	u 2 n (t, x)∂ t [α 2 t1,R (t, x)]e -βtU (x) dx,
	K n,t1,R (t) =β t	R d	U (x)u 2 n (t, x)α 2 t1,R (t, x)e -βtU (x) dx.

Using Remark 18 with any r > 4R, since α t1,R (t, •) is supported in B 4R , we have

We finally used that a 2 -2ab ≥ -b 2 with a = |∇u n |α t1,R and and b = u n |∇α t1,R |. Thus

n,t1,R (t), where

by [START_REF] Royer | A remark on simulated annealing of diffusion processes[END_REF], and

n,t1,R ≤

. This ends the step.

Step 4. We now conclude that (19) holds true with δ 0 = min{δ, 1/(2κ 1 )}, where δ > 0 and κ 1 > 0 were introduced in Steps 2 and 3. We thus fix t 1 ∈ [0, t 0 ] and assume that R d u(t 1 , x)dx = 0. Integrating (20), we find that for all t ∈ [(t 1 -δ 0 ) ∨ 0, t 1 ], all R > 1 and all n ≥ 5R,

the last inequality following from the fact that t 1 -t ≤ δ 0 ≤ 1/(2κ 1 ).

Since lim n u n (t, x) = u(t, x) by dominated convergence and since α t1,R (t, •) is compactly supported, we have lim n ϕ n,t1,R (t) = ϕ t1,R (t), where ϕ R (t) = R d u 2 (t, x)α 2 t1,R (t, x)e -βtU (x) dx. We thus find, for all t ∈ [(t 1 -δ 0 ) ∨ 0, t 1 ], all R > 1 ϕ t1,R (t) ≤ e R 2 /2 ϕ t1,R (t 1 ) + e -R 2 /2 .


Yiwei Shan

Ying Guo

Yu Wang

Xiran Du

Jun Yu

Hao Luo

Hui Wu

Bruno Boury
email: bruno.boury@umontpellier.fr

He Xiao

Liulian Huang

Lihui Chen

Nanocellulose-derived carbon/g-C3N4 heterojunction with a hybrid electron transfer pathway for highly photocatalytic hydrogen peroxide production

Keywords: Cellulose nanofibers, Photocatalysis, Carbon nitride, Hydrogen peroxide production, Carbon fibers

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Hydrogen peroxide (H2O2) is a promising energy storage product widely used both as an oxidant and reductant in chemical synthesis, industrial bleaching, and medical sterilization [1,2]. Moreover, in line with the need for new answers to the increase in energy demand, H2O2 has been proposed as the oxidizing agent at the cathode of fuel cells with environmentally acceptable by-products, H2O and O2 [3].

Presently, the methods adopted to produce H2O2 on a large scale are anthraquinone autoxidation [4] and electrochemical synthesis [5]. These methods consume a large amount of energy and organic solvents; in addition, contamination by various organic impurities can occur and result in increased difficulties of purification [6,7].

Consequently, new production processes must succeed in satisfying the demand for H2O2, resolve and eliminate the shortcomings of traditional production methods, be efficient, and above all be sustainable and environmentally friendly with respect to the principles of green chemistry.

In recent decades, photocatalysis has been employed in light-driven production of H2O2 through photoreduction of O2 in H2O [8], a process governed by the basic photochemical reaction resulting from light-semiconductor interaction in the presence of water. In this approach, the semiconductor generates holes (h + ) in the valence band (VB) that oxidize H2O near the surface and release protons (H + ; Eq. 1).

Simultaneously, the electrons (e -) promoted in the conduction band (CB) can reduce O2 to generate different species, including superoxide radicals (•O2 -) and, subsequently, H2O2 by either a (ⅰ) single-electron indirect reduction and protonation (Eq. 2-4) or (ⅱ) one-step two-electron direct O2 reduction (Eq. 5).

O2 + e -→ •O2 -(Eq. 2)

•O2 -+ H + → HO2• (Eq. 3) HO2• + H + + e -→ H2O2 (Eq. 4)

O2 + 2 e -+ 2 H + → H2O2 (Eq. 5)

ZnO [9] and TiO2 [10][11][12][13][14] are the two major classical inorganic semiconductors used as photocatalysts [15,16], among many others. Recently, carbon nitride with graphitic structure (g-C3N4) has emerged as a highly promising photocatalyst.

Compared with classical metal-based semiconductors, it has a high efficiency and several advantages [17][18][19][20], the most attractive one being the absence of metal species known to decompose H2O2 via MOOH formation. Additionally, g-C3N4-based catalysts exhibit good photostability and photocatalytic activity under visible light.

Finally, g-C3N4 can be prepared by the polycondensation and pyrolysis (500-600°C) of nitrogen-and carbon-containing precursors such as melamine and urea, both being cheap, abundant, and non-toxic in comparison with ZnO and TiO2 [21]. However, pure carbon nitride has some bottlenecks limiting its photocatalytic efficiency, such as a fast electron-hole recombination of photogenerated species, insufficient absorption in visible light, a relatively large band gap (~2.7 eV), and a low specific surface area that limits the number of active sites for interfacial photoreactions. To address these issues, different strategies have been employed to enhance the photocatalytic activity, including non-metal doping (e.g., O [22,23], P [24], N [25], and S [26]), metal ion doping (K [7], Cu [27]), forming heterojunctions with other semiconductors (e.g., ZnO [28], MnO2 [29], and CeO2 [30]), composites with metal nanoparticles (e.g., Au [20] and Ag [31]), and composites with carbonaceous materials (e.g., carbon dots [32], graphene [33], and carbon nanotubes [34]). The carbonaceous materials are selected as an electron transfer agent and photosensitizer in the photocatalysis field to reduce the electron-hole (e -/h + ) recombination and to increase the absorption of incident light, respectively. Nowadays, the production of such functional carbon should avoid the use of non-renewable and fossil fuel-derived carbon precursors (e.g., polyacrylonitrile

[PAN]), and instead use biosourced carbon. Cellulose nanofibers (CNF) are one example, with 1-3 µm length and 5-10 nm width. They not only have the advantage of size but also possess adequate surface functional groups such as -OH and -COOH, facilitating regulation of the growth and chemical anchoring of chemical species such as precursors of inorganic photocatalysts [35]. Therefore, anchoring graphitic-like materials should be possible if they possess compatible functional groups. This implies the necessity of edge functional groups in g-C3N4 that possess H-bonding donor or acceptor ability.

Various research groups have already demonstrated the synergy of nanocellulose/g-C3N4 composites, with or without pyrolysis of the cellulose. It has led to the development of adsorbents or photocatalysts for the oxidation of dyes (methylene blue [36][37][38][39] or rhodamine [40,41]), formaldehyde [42], Cr(IV) [43,44], oil/water emulsion [45] or bacteria [46] and also the preparation of thermal insulator materials [47,48]. These are promising routes especially for pollutant remediation; however, the future of photocatalysis also involves the production of chemicals by clean and low-energy processes. To the best of our knowledge, only one previous study used g-C3N4 and nanocellulose to produce an efficient photocatalyst to produce chemicals, H2 in that case. However, nanocellulose was used only as a template to generate a chiral nematic structure in addition to the use of silica, with nanocellulose subsequently eliminated by calcination [49]. In our study, we used nanocellulose as both a template and a C-precursor to prepare a C/g-C3N4 composite by the simplest possible process. It is also worth mentioning the work by Jin et al., who used soluble cellulose acetate in combination with melamine as a g-C3N4 precursor to produce a C/g-C3N4 photocatalyst for the production of hydrogen [50].

In the present study, g-C3N4 nanosheets (CN2) were fabricated for the first time using a mixture of melamine and urea and a two-step calcination method, to generate a higher proportion of amino groups (-NHx) than that in g-C3N4 prepared with only melamine (CN1); see Scheme 1 for the summary of the general strategy. We found that the as-prepared g-C3N4 nanosheets can easily and strongly bind the CNF, possibly by hydrogen or even covalent bonds. The cellulose nanofiber/g-C3N4 composite is then transformed by pyrolysis into a covalent heterojunction C/g-C3N4 (CNCF1 or CNCF2) with consequences on the band structure and electron transfer process and an important improvement in the photocatalytic activity. As usual when designing a nanocomposite, apart from the nature of the interaction between the components, their mass ratio is an important factor. Prior experiments demonstrated that the optimum mass ratio of the cellulose nanofiber/g-C3N4 is ∼3-4 %; therefore, this ratio was implemented throughout this study.

Experimental

Materials and reagents

Melamine (C3H6N6; 99%), urea (CH4N2O; 99.5%), potassium dihydrogen phosphate (KH2PO4; 99.5%), potassium hydrogen phosphate (K2HPO4; 99%), N,N-Diethyl-p-phenylenediamine sulfate (DPD; 98%), horseradish peroxidase (POD; enzymatic activity >160 units/mg), anhydrous ethanol (AR), 5,5-dimethyl-1-pyrroline N-oxide (DMPO, 97%) were obtained from Aladdin Industrial Co., Ltd, China.

Cellulose nanofibers (1 wt% aqueous solution; average length: 1-3 um, average width: 5-10 nm) was purchased from Guilin Qihong Technology Co., Ltd., China.

Polyethersulfone (PES) membrane (∅ 13 mm × 0.22 µm) was obtained from Jin Teng Experimental Equipment Co., Ltd., Tianjin, China. All the chemical reagents mentioned above were of analytical grade and were utilized without further purification.

Preparation of photocatalysts

The g-C3N4 nanosheets were synthesized by a two-step pyrolysis treatment of melamine and urea [51]. First, melamine (3.2 g) and urea (0.8 g) were uniformly mixed and thoroughly ground in an agate mortar. The samples were placed into a muffle furnace and heated to 500°C with a heating rate of 2°C min -1 in static air and left for 2 h; then, the temperature was risen to 520°C for 2 h at the heating rate of 2°C min -1 . The residual yellow solids were ground into powder and calcined at 520°C for 4.5 h in air. Finally, the resultant light-yellow products with a productivity of 30% were denoted as CN2. For comparison, g-C3N4 nanosheets from a single precursor was also synthesized by calcining pure melamine through the same thermal treatment and denoted as CN1, with a productivity of 36%.

The g-C3N4 composited with nanocarbon fibers was prepared by a one-step thermal treatment of the compound of g-C3N4 and CNF. In brief, g-C3N4 nanosheets (CN1 or CN2; 0.10 g) was added into 10 mL of deionized water containing a suspension of CNF (0.33 g; CNF concentration 1 wt%) and ultrasonicated for 30 min, as shown in Figure S1. Subsequently, the mixtures were vacuum-dried at 60°C for 12 h. Then, the residual white yellow solids were directly pyrolyzed at 500°C for 1 h at the heating rate of 5°C min -1 in argon atmosphere, marked as CNCFx (x = 1, 2). The final products (CNCF1 and CNCF2) obtained also retained the productivity of 80% and 70%, respectively. The pure CNF were carbonized into cellulosic carbon fibers, denoted as CF, under the same calcination conditions.

Characterization

The crystal structures of as-prepared samples were characterized using an X-ray diffractometer (Rigaku, Ultima Ⅳ, Japan) with Cu Ka radiation in the range between 5° and 80°. Fourier transform infrared (FTIR) spectra were recorded on a Bruker VERTEX 70 spectrophotometer at the wavelength ranging from 400 to 4000 cm -1 .

The porosimetry measurements and BET-specific surface area of the samples was detected by nitrogen adsorption/desorption on an adsorption apparatus (Micromeritics, ASAP 2020, USA). The morphologies of the samples were observed using a scanning electron microscope (Hitachi, SU8010, Japan) and transmission electron microscope (JEOL, JEM-2100, Japan). The X-ray photoelectron spectrometer (XPS, ESCALAB 250, Thermo Scientific, USA) equipped with Al Kα radiation monochromatic source was used to detect chemical composition and elemental states. The ultraviolet (UV)-vis diffuse reflectance spectra (UV-vis DRS) were obtained using a UV-vis spectrophotometer (Cary 500, Varian, USA) in the range of 300-800 nm with BaSO4 as a reflectance standard. Photoluminescence spectra were recorded on a LS55 spectrophotometer (Perkin-Elmer, USA) under the excitation wavelength of 365 nm at room temperature. Time-resolved photoluminescence (TRPL) spectra were obtained on a fluorescence spectrophotometer (FLS980, Edinburgh Instruments, UK).

Photo-generated •O2 -radical species in the photocatalytic process were detected with DMPO as a trapping agent. The absorbance of hydrogen peroxide solution was determined by UV-visible spectrophotometer (US-Vis, Agilent 8453, USA) at the wavelength 551 nm.

Photoelectrochemical and electrochemical measurement

The transient photocurrent curves and electrochemical impedance spectroscopy were obtained on CHI660-E electrochemical workstation (Chenhua Instrument Company, Shanghai, China) based on a three-electrode cell composed of Pt wire as the counter electrode, Ag/AgCl electrode as the reference electrode, and fluorine-doped tin oxide (FTO) glass (1 × 1 cm 2 ) as the working electrode. Na2SO4 aqueous solution (0.2 mol/L) was used as the electrolyte. Five milligrams of the sample (CNx or CNCFx) was added to 1 mL of Nafion solution (0.5%) under ultrasonication for 30 min, and then deposited onto the FTO glass. The as-prepared working electrode was dried at 80°C for 3 h. The photoelectrochemical measurement was performed under visible light irradiation (Xe arc lamp: 300 W, the distance between the surface of the solution and light filter: 10 cm, filter wavelength: λ > 420 nm). The transient photocurrent was collected with the light on and off. The electrochemical impedance spectroscopy (EIS) was tested at the frequency of 0.01 and 10 6 Hz with the alternating current amplitude of 5 mV.

Rotating disk electrode (RDE) tests were also taken on a CHI660-E electrochemical workstation with a modulated speed rotator and a glassy carbon disk electrode (working electrode). The Ag/AgCl electrode and Pt wire electrode were used as the reference electrode and counter electrode, respectively. Two milligrams of the photocatalyst was dispersed into 1 mL of Nafion solution (10%) under ultrasonication for 30 min. Then, 10 µL of the suspension was dropped on a glassy carbon electrode with a diameter of 5 mm and air-dried at room temperature as the working electrode. The linear sweep voltammetry (LSV) curves were obtained using a phosphate buffer solution (0.1 mmol/L; pH 7) with O2-saturated purging at a scanning rate of 10 mV/s and the rotating speed between 0 and 2500 rpm.

Photocatalytic activity measurements

For evaluating the photocatalytic H2O2 generation, the catalyst powder (50 mg) was dispersed in a solution of deionized water (45 mL) and ethanol (5 mL). Then, the suspension was ultrasonicated for 5 min and purged with O2 gas for 30 min prior to visible irradiation. Then, the photocatalytic performance was examined under visible light irradiation (Xe arc lamp: 300 W, the distance between surface f the solution and light filter: 10 cm, filter wavelength: λ > 420 nm) with continuous O2 purging and stirring for 6 h. Every hour, an aliquot of the suspension (4 mL) was taken and filtered to remove the catalysts. The concentration of H2O2 was measured by a colorimetric method. To this end, 5 mL of the filtered solution was immediately mixed with 0.5 mL of phosphate buffer (0.5 M K2HPO4 and 0.5 M KH2PO4), DPD solution (50 µL), and peroxidase (50 µL), with vigorous shaking for 30 s. Then, the absorbance of the aqueous solution was measured at 551 nm using a UV-visible spectrophotometer [52].

The decomposition behavior of H2O2 with photocatalysts was investigated by adding 50 mg of the samples to 50 mL of H2O2 solution (1 mmol/L) with continuous stirring under visible light irradiation for 1 h. The stability of the as-prepared samples was evaluated by performing three consecutive tests. When finishing each run, the photocatalyst was collected using membrane filtration and washed with deionized water, then vacuum-dried at 50°C overnight, finally reused in the next cycle under the same conditions. Photocatalytic H2O2 concentration was also investigated by measuring the absorbance of the filtrate according to the Beer-Lambert law [52].

Scheme 1. Procedures for sample synthesis

Results and discussion

Photocatalyst preparation and characterization

Specific surface area is important for the activity of a photocatalyst; therefore, a bi-component approach was selected to prepare g-C3N4 with the highest possible surface area [53,54]. Urea was used as an additive to melamine owing to the release of large quantities of volatile gas during thermal treatment, resulting in high porosity [55,56]. It also can generate a high quantity of residual amino groups, generally on the edges of the nanosheets. Because CNF have surface -OH and -COOH groups, H-bonding and formation of amide or ester groups can occur through reactions with amino-rich carbon nitride. For comparison, a g-C3N4 was prepared by the mono-component approach using only melamine, which is referred to as CN1. A comparison between the dispersal behaviors of CN1 and CN2 with CNF in aqueous phase is shown in Figure S1, suggesting a higher stabilization of the suspension in the case of CNF/CN2 mixture.

Carbonization at 500°C of the CNF/CN1 and CNF/CN2 mixtures transforms their CNF into carbon, generating porosity throughout the materials and modifying the hydrophilic/hydrophobic balance of the surface in the corresponding CNCF1 and CNCF2, as shown in Figure S2. The results show that the CN2 (from dual precursors) was more hydrophilic than CN1 (from single precursor) owing to the presence of more hydrophilic groups (amino groups) in CN2. Moreover, the contact angles of CN1 (24.5 o ) and CN2 (20.5 o ) respectively decreased to 17.6 o (CNCF1) and 11 o (CNCF2) when g-C3N4 assembled with the carbon nanofibers pyrolyzed from nanocellulose was used, because of the increase in the stacking distance in g-C3N4.

The optimized mass ratio of CNF/g-C3N4 is ∼3.3 wt%, which is a miniscule quantity of CNF, but as shown below, it has a substantial effect on the photocatalytic performance compared with pure carbon nitride. Moreover, the sample yield is ∼70-80% and with cheap and renewable products. This weight loss is attributed to the decomposition of cellulose and also the condensation reaction between the surface functional groups of cellulose and g-C3N4. The generation of carbon results in an important modification of the color of the material from slightly yellow for pure carbon nitride (CN1 and CN2) to gray for CNCF1 and CNCF2, as shown in Figure S3. Analyzed by FT-IR (Figure 1a), all spectra of CNx and CNCFx (x = 1, 2) are dominated by the signals related to the presence of g-C3N4. Several strong bands in the 1200-1600 cm -1 region were typical for the stretching modes of the CN heterocycles [57]. The sharp peak at 810 cm -1 corresponds to the characteristic breathing vibration of tri-s-triazine units [58]. The broad bands ranging from 3000 cm -1 to 3300 cm -1 correspond to the NH and OH stretching vibrations [59]. The peak intensity of -NHx between 3000 cm -1 and 3300 cm -1 in CNCF1 and CNCF2 is slightly lower than that of CN1 and CN2, respectively; it implies a decrease in the number of amino groups [60].

The presence of g-C3N4 was also clearly detected in the signals observed in the X-ray diffraction powder patterns of the CNx and CNCFx samples, as shown in Figure 1b. The two characteristic peaks at 2θ = 12.9° and 27.5° are ascribed to the (100) and (002) diffraction planes of the in-plane repeat tri-s-triazine units and the interlayer stacking reflection of conjugated aromatic segments, respectively [61][62][63].

A slight downshift in the position of the (002) crystal plane was detected when comparing CN1 and CN2 and CNCFx. It corresponds to a very small increase in the stacking distance in g-C3N4 when assembled with the carbon nanofibers [64], possibly owing to the intercalation of organic species during pyrolysis. A weaker peak intensity of (002) was noted for CN2 and CNCF2 than for CN1 and CNCF1, respectively, which may be ascribed to the effective exfoliation of g-C3N4 into sheet-like structures by overcoming the weak van der Waals forces between layers in the calcination process; this finding is consistent with other reports on g-C3N4 [65].

Porosity of the materials was determined by N2 adsorption-desorption with the isotherms shown in Figure 1c. All samples exhibited typical type IV isotherms with H3 type hysteresis loops, indicating that the samples were micro-, meso-, and slightly macroporous with a specific surface area of 50 m 2 g -1 (CN1), 55 m 2 g -1 (CNCF1), 83 m 2 g -1 (CN2), and 120 m 2 g -1 (CNCF2). The data emphasize the effect of CNF in addition to the preparation method of g-C3N4. CN2 and CNCF2 exhibited 1.5-2.4 times higher BET SSA than CN1 and CNCF1 (Table S1). An increase was also observed for the average pore volume, calculated to be 0.34 m 3 g -1 (CN1), 0.39 m 3 g -1 (CNCF1), 0.44 m 3 g -1 (CN2), and 0.85 m 3 g -1 (CNCF2; Table S1). Clearly, the bi-component route results in materials with enhanced porosity that is ascribed to a higher proportion of CO2 and NH3 release during calcination [66]. Notably, the assembly of CNF with CN1 or CN2 leads to a decrease in the number of small mesopores and an increase in the number of large mesopores increased, which is expected when considering the formation of carbonaceous residue upon pyrolysis of CNF (Figure 1d). Scanning electron microscopy (SEM) and transmission electron microscopy (TEM) revealed the morphology and microstructure of CNx and CNCFx (x = 1 or 2).

As shown in Figure 2, CN1 and CN2 are mainly composed of stacked irregular nanosheets. Compared with CN1, a thinner sheet-like structure was obtained for CN2 upon calcination. For samples after pyrolysis (CNCF1 and CNCF2), carbon nanofibers with an average length of 0.1-1 µm were identified on top of or between micron-sized layers of carbon nitride. Sheet-like carbon nitride particles of 8-12 µm width were observed by TEM (Figure 2), suggesting that the assembly also occurs at the nanometer scale because isolated carbon nanofibers ∅ ≈ 10-20 nm were identified. This clearly showcases the integration of the CNF within the g-C3N4 during the assembly prior to pyrolysis. Elemental mapping by TEM shows that C and N, the two main elements of carbon nitride, are uniformly spread in CNCF1 and CNCF2, and only traces of oxygen are likewise distributed (Figure 2). X-ray photoelectron spectroscopy (XPS) allowed us to identify the composition and chemical status of the main elements in the samples (Figure 3). As shown in Figure 3a, similar X-ray photoelectron spectra were recorded for the four samples exhibiting the two peaks expected for g-C3N4, namely N 1s and C 1s and a weak signal of O 1s [67]. For N 1s spectra in Figure 3b, the three peaks at 398.9, 400.4, and To investigate the optical absorption and band structures, UV-visible diffuse reflectance spectra (UV-vis DRS) of the samples were recorded (Figure 4a). All samples presented absorption in the blue region typical of the 2D structure of carbon nitride [74]. However, compared with CN1 and CN2, the absorption intensity of both CNCF1 and CNCF2 was significantly increased owing to the presence of carbon fibers and multiple reflection of incident ray light across porous structures and band structure regulation [75]. The corresponding band gaps determined from Tauc's plots were 2.60, 2.50, 1.84, and 1.55 eV for CN1, CN2, CNCF1, and CNCF2, respectively (Figure 4b). Therefore, the CNCF1 and CNCF2 samples exhibited a narrower band gap than other samples and consequently higher absorption of visible light useful in photogenerating carriers. The band structures of different samples were further investigated by VB XPS spectra as shown in Figure 4c. An additional effect of the presence of carbon fibers is the variation of the VB potential; values for CN1 (1.98 eV) and CN2 (1.90 eV) were decreased for CNCF1 (1.69 eV) and CNCF2 (1.45 eV). A decrease in the VB potential is beneficial in narrowing the band gap and decreasing the generation of •OH because of the decreased oxidation ability. Combining the VB XPS spectra (Figure 4c), the location of VB and CB bands was determined using the formula Eg = EVB -ECB (where Eg: band gap, EVB: VB value, ECB: CB value; Figure 4d). The Eg of CNCF1 and CNCF2 was smaller than that of CN1 and CN2, respectively. Importantly, the Eg is the smallest for CNCF2, which also has a more positive CB level (-0.1 eV vs NHE) than other samples, which is 0.79 eV more negative than the reduction potential of O2/H2O2 (0.69 eV) and large enough to reduce O2 to H2O2 [76]. Moreover, the positive shift of CB could easily trigger photogenerated electrons from CB, accepting electrons from carbon fibers and enhancing the two-electron reduction of O2 to H2O2 [77,78]. Electrochemical impedance spectroscopy (EIS) was further employed to understand the charge separation and their transfer. As shown in Figure 5a, the Nyquist plots revealed that under visible light irradiation, the observed arc radius of CNCFx is smaller than the corresponding CNx, which indicates that the charge transfer resistance of CNCFx is significantly decreased. This is ascribed to the introduction of carbon nanofibers and carbonaceous residue that enhance electron transfer. The smaller arc radius of CNCF2 than CNCF1 indicates a faster charge transfer and a better efficiency of electron-hole separation. The transient photocurrent curves under visible light irradiation of the as-prepared samples are shown in Figure 5b. CNCF1 and CNCF2 displayed much higher photocurrent than that of CN1 and CN2, confirming that carbon fibers derived from nanocellulose accelerate electron transfer. The recombination/separation of photogenerated species was investigated by photoluminescence (PL) spectra of the samples (Figure 5c). As a general trend, the lower the photoluminescence peak intensity, the higher the efficiency of photogenerated electron-hole pairs. The strong fluorescence emission peaks of CN1 (462 nm) and CN2 (466 nm) were slightly red shifted to 470 nm and 469 nm for CNCF1 and CNCF2, respectively, which further confirms the band gap decrease [79].

CNCF1 and CNCF2 samples had considerably lower photoluminescence intensities than CN1 and CN2, respectively, suggesting better separation rate of electrons and holes. Moreover, it is worth noting that the photoluminescence quenching efficiency of CNCF2 was lower than CNCF1, suggesting more effective electron transfer from photogenerated g-C3N4 to nanocarbon fibers [74]. To obtain the average radiative lifetime of the recombining charge carriers, time-resolved photoluminescence (TRPL) characterization was investigated (Figure 5d). The average radiative lifetime (τave) was further calculated according to the following equation: τave = (B1τ1 2 + B2τ2 2 )/(B1τ1+ B2τ2). The value of τave for CN1 and CN2 was 1.89 and 2.35 ns, respectively, whereas that for CNCF1 and CNCF2 was 2.51 and 2.71 ns, respectively. This result indicated that, with the addition of carbon fibers derived from nanocellulose, the markedly extended lifetime of photogenerated charge carriers accelerates the charge separation efficiency. In summary, the presence of cellulose-derived carbon fibers assembled with g-C3N4 highly improved the efficiency of the photocatalysts in promoting charge transfer and carrier separation in g-C3N4. Photocatalytic production of H2O2 under visible light irradiation was evaluated by measuring the quantity of H2O2 versus time of reaction under atmospheric pressure at 25 ± 0.1°C using a catalyst concentration of 1.00 ± 0.01 g L -1 (Figure 6a). Because ethanol is known to have a good effect on photocatalytic H2O2 production when used as a proton donor (Figure S4), 5 mL (85.9 mmol) ethanol was used in this photocatalytic system with an ethanol-to-water ratio of 1: 9 (v/v). With CN1, only 0.31 mmol L -1 of H2O2 is formed after 1 h. Although the productivity is improved with CN2, it remains low: 0.47 mmol L -1 of H2O2 after 1 h. CNCF1 and CNCF2 presented a considerably higher H2O2 productivity after 1 h, 0.7 mmol L -1 and 1.18 mmol L -1 , respectively. Therefore, the kinetic study presented in Figure 6c suggests that the order of photocatalytic efficiency is as follows: CNCF2 > CNCF1 > CN2 > CN1.

Photocatalytic production of H2O2

The decomposition behavior of H2O2 was investigated under visible light irradiation with photocatalysts and a H2O2 concentration =1 mmol L -1 . As shown in Figure 6b, the level of H2O2 decomposition was ∼8% and ∼5% with CNCF1 and CNCF2, respectively, after 1 h of visible light irradiation, which was lower than that with CN1 and CN2 (∼10%). The H2O2 decomposition rate constant (Kd) was calculated with the following equation: Kd = -ln(Ct/C0)/t, where Ct is the H2O2 concentration at time t, and C0 is the initial concentration of H2O2 (1 mmol L -1 ).

Figure 6c shows that the Kd of CN1 (1.335 × 10 -3 min -1 ) and CN2 (1.095 × 10 -3 min -1 ) was higher than that of CNCF1 (0.868 × 10 -3 min -1 ) and CNCF2 (0.676 × 10 -3 min -1 ).

These results confirm that carbon fibers derived from nanocellulose effectively inhibit the decomposition of H2O2 on carbon nitride.

Compared with previously reported values [7,75,[80][81][82][83][84][85][86], the photocatalyst CNCF2 presented a significantly improved activity (Figure 6d). To investigate the recyclability of the as-prepared photocatalysts, after 6 h of reaction time, the catalyst was separated and washed from the suspension by filtration and then used again in a new batch of H2O2 production under the same conditions (Figure 6e). After three cycles, the photocatalyst was almost unchanged and the photocatalytic activity was To further investigate the O2 reduction pathway, a rotating disk electrode (RDE) analysis of the O2 reduction reaction (ORR) was performed. The linear sweep voltammetry (LSV) curves in Figure 7a-d for the as-prepared samples were measured by an RDE in phosphate buffer solution (0.1 mol L -1 , pH = 7) at different rotating speeds. The Koutecky-Levich plots derived from the LSV data are presented in Figure 7e. The average number of transfer electrons was calculated using the following equations:

j -1 = jk -1 +B -1 ω -1/2 B = 0.2nFν -1/6 CD 2/3
, where j is the current density, jk is the kinetic current density, ω is the rotating speed, F is the Faraday constant (96485 C mol -1 ), ν is the kinetic viscosity of water (0.01 cm 2 s -1 ), and C is the O2 concentration in water (1.26 × 10 -5 cm 2 s -1 ). The average number of electrons (n) was 1.11 and 1.14 for CN1 and CN2, respectively, and 1.57 and 1.74 for CNCF1 and CNCF2, respectively. These results suggest that single-electron O2 reduction occurs on CN1 and CN2 because the n value of CN1 and CN2 was approximately 1. In contrast, the average number of electrons is above 1.5 when nanocellulose-derived carbon is present, especially for CNCF2, for which an O2 reduction by two-electron process is likely. Thus, the ORR for CNCF1 and CNCF2 had a hybrid single/two-electron function, whereas that for CN1 and CN2 showed only single-electron function.

Electron paramagnetic resonance (EPR) analysis with DMPO as a trapping agent of •O2 -was performed to identify the pathway of conversion of molecular oxygen to H2O2. As shown in Figure 7f, the characteristic peaks of DMPO-•O2 -were observed in all samples, indicating that •O2 -was generated as an intermediate in the formation of H2O2. As seen in Figure 7f, CN1 and CN2 both presented higher-intensity EPR signals than CNCF1 and CNCF2, indicating that CN1 and CN2 generate more •O2 - than CNCF1 and CNCF2. The finding implies that reduction of O2 into •O2 -occurs on the four samples by a single-electron pathway leading to H2O2 formation. It is interesting that, CNCF1 and CNCF2 present partial O2 directly reduction into H2O2 by a two-electron pathway [87]. Scheme 2 illustrates photocatalytic H2O2 production by nanocellulose-derived C/g-C3N4 nano-heterojunction. Cellulose nanofibers with -OH and -COOH groups can bind to carbon nitride with intrinsic amino groups (-NH, -NH2) and thermally transform into an effective heterojunction, increasing the linking between g-C3N4 nanosheets and carbon fibers. Furthermore, the carbon fibers reduce the band gap and greatly increase the electron transfer during the photocatalytic reaction. The photocatalytic efficiency also depends on the capacity and strength of adsorption of molecular oxygen onto the surface of carbon nitride. By porosimetrie, BET SSA of CN2 is 2.2 times higher than that of CN1 (Table S1), which provides more space and active sites to adsorb oxygen molecules [88]. The adsorption strength of molecular oxygen is also potentially reinforced by the presence of N atoms of amino groups in CN2, which are able to bind to oxygen as a Lewis acid [89]. As previously reported, such adsorption of oxygen on the surface of the photocatalyst catalyst is highly beneficial to the electron transfer [90]. An exciting point is to observe that with the presence of carbon nanofiber, the EPR intensity of •O2 -of CNCF1 is higher than CNCF2. The data demonstrate the interest of the bi-component synthetic route chosen for the preparation of the g-C3N4. This is ascribed to both the larger BET specific surface area and the specific photophysical properties.

Scheme 2. Role of cellulosic carbon fibers for photocatalytic H2O2 production photocatalyzed by a nanocellulose-derived carbon/g-C3N4 nanocomposite.

Conclusions

The high improvement of the photocatalytic production of hydrogen peroxide (H2O2) over carbon nitride (g-C3N4) that we report is ascribable of main major phenomenon. First, we show that the presence of a very low content of nanocellulose-derived carbon deeply modify the photoactivity of the g-C3N4 assembled with it. The result is an outstanding photocatalytic H2O2 production of 1.10 mmol L -1 h -1 under visible light. The second critical point is to achieve a high level of interaction between the g-C3N4 nanosheets and CNF when assembling these two nanocomponents. Third, a high level of interaction can be achieved through the preparation of amino-rich g-C3N4 using a bi-component mixture of melamine and urea. When carbon fibers are closely incorporated into the stacked structure of g-C3N4, the nanofibers not only improve the porosity of the materials but also have a strong effect on the photophysical properties of the g-C3N4 in narrowing the band gap; improving the visible light absorption; increasing the number of photogenerated carriers, especially excitable electrons; and accelerating the electron transfer and further promoting a direct two-electron O2 reduction reaction to generate hydrogen peroxide. In subsequent studies, this approach will be tested for other materials for the photoproduction of other chemicals.

Figure 1 .

 1 Figure 1. Fourier transform infrared spectroscopy curves (a), X-ray diffraction patterns (b), N2 adsorption-desorption isotherms (c) and pore size distribution derived from desorption isotherm (d) of CN1, CN2, CNCF1, and CNCF2.

Figure 2 .

 2 Figure 2. Scanning electron microscopy (SEM) images of CN1 and CNCF1; transmission electron microscopy (TEM) and TEM mapping of CNCF1 (a), SEM images of CN2 and CNCF2; TEM and TEM mapping of CNCF2 (b).

Figure 3 .

 3 Figure 3. X-ray photoelectron spectroscopy data of CN1, CN2, CNCF1, and CNCF2 samples: survey scan (a), core spectra with deconvolution for N1s (b), C1s (c), and O1s (d).

Figure 4 .

 4 Figure 4. For CN1, CN2, CNCF1, and CNCF2: characterization by UV-vis DRS spectra (a), determination of the band gaps (b), determination of VB XPS spectra (c), and energy band structures (d)

Figure 5 .

 5 Figure 5. Electrochemical impedance spectra: Nyquist plots (a), transient photocurrent response curves (b), photoluminescence spectra (c), time-resolved photoluminescence decay spectra (d) of the samples.

Figure 6 .

 6 Figure 6. Photocatalytic production of H2O2 as a function of time under visible light with CN1, CN2, CNCF1, and CNCF2 as photocatalysts (a); photocatalytic decomposition of H2O2 under visible light irradiation (b); formation rate (Kf) and decomposition rate constant (Kd) for H2O2 photocatalytic production (c); comparison of reported photocatalytic performances (photocatalytic production rate of H2O2 within 3h and H2O2 concentration after 3h reaction) (d); reusability assessment of the different photocatalysts (e); X-ray diffraction patterns of CNCF2 before and after photocatalytic H2O2 production (f)

Figure 7 .

 7 Figure 7. Linear sweep voltammetry curves of CN1 (a), CN2 (b), CNCF1 (c) and CNCF2 (d) measured on a rotating disk electrode at different rotating speeds; Koutecky-Levich plots of the data obtained at a constant electrode potential (-1.0 V vs. Ag/AgCl) (e), Electron paramagnetic resonance spectra of the DMPO-•O2 -adduct (f)

Acknowledgments

This work was supported by the Project Funded by China Postdoctoral Science Foundation (2020T130215), Key Laboratory of State Forestry and Grassland Ministry (2019KFJJ16), Distinguished Young Talents of Fujian Agriculture and Forestry University (XJQ201806), Science and Technology Innovation Fund of Fujian Agriculture and Forestry University (CXZX2018013), and Fujian Innovation and Entrepreneurship Training Program (S202010389078).

and C-NHx, respectively [41]. An estimate of the relative proportion of the different species was obtained by deconvolution (Table S2). It appears that the percentage of C-NHx either in CN2 or CNCF2 prepared by the bi-component route is considerably higher than that for the materials prepared by the mono-component route (CN1 and CNCF1). This is an additional indication of the effect of the urea precursor during the formation of the materials, and it is consistent with the FT-IR data [68] (Figure 1a).

Moreover, the content of C-NHx decreased after g-C3N4 was assembled and pyrolyzed with CNF, a possible explanation being the reaction of this amino group with the functional groups of cellulose (hydroxyl and carboxylic group) during assembly or with gaseous products issued from the decomposition of cellulose during pyrolysis.

In Figure 3c, C 1s spectrum could be deconvoluted into two peaks at 284.8 and 288.2 eV. The peak at 288.2 eV represents by far the highest proportion of C species and was attributed to sp 2 -hybridized carbon covalently bound by N atom (N=C-N), as expected for g-C3N4. The peak at 284.8 eV was assigned to C-C carbon species [69], both routes of preparation led to a similar quantity of this species, indicating the presence of carbonaceous residue. With the assembly and pyrolysis of CNF, the content of N-C=N decreased by a few percent and that of C-C increased by 10% for CNCF1, which is expected because the pyrolysis of CNF leads to the formation of carbonaceous char that can adsorb on the surface of g-C3N4. However, this increase was not observed when comparing CN2 and CNCF2 and is attributable to a higher integration of the carbon fiber into the sheet-like structure of g-C3N4. C 1s XPS data also revealed a peak at 286.1 eV for CNCF1 and CNCF2, which was ascribed to C-O [70] that likely results from the assembly and pyrolysis with CNF, which generates O-containing carbon as observed in carbon-derived polysaccharides [71]. However, different routes of preparation resulted in different C-O content: 1-2% in CNCF1 versus 4-5 % for CNCF2. Concerning oxygen, the high-resolution O 1s spectra was fitted based on two reference peaks centered at binding energies of 533.4 eV and 532.5 eV for adsorbed O2 and adsorbed H2O, respectively [72]. Compared with the as-prepared g-C3N4 samples CN1 and CN2, the new peak at 530.9 eV in