EL PLAZO RAZONABLE

The first step in the proof of the convergence theorems of this chapter is to use Kuratowski convergence to make rigorous the intuition that the zeroes of the vector field−E_ε′(t,·) “fill up” the stable setS(ℓ(t)):

Lemma 3.5.3. Let Sε(ℓ) denote the fixed-point set of the dynamics for (3.2.3) at scale ε withℓ_∈R∗ fixed, i.e.

Sε(ℓ) :={x∈R|Eε′(t, x)≡ −V′(x) +gε(x) +ℓ= 0}, (3.5.1)

and assume that the rescaling (3.2.1) holds, i.e. gε(x) =g(x/ε). Then

g has property (z_{) =⇒ K-lim inf}

ε→0 Sε(ℓ) =S(ℓ).

Proof. Since obviously Sε(ℓ) ⊆ S(ℓ) for every ε >0 and S(ℓ) is closed, it follows that

K-lim inf

ε→0 Sε(ℓ)⊆ S(ℓ).

Since the left-hand side is closed, it suffices to show that

gsatisfies (z_{) =⇒ S}˚_{(ℓ)⊆K-lim inf}

ε→0 Sε(ℓ).

Let x be an interior point of S(ℓ). Since V is strictly convex, this is equivalent to the assumption that −V′(x) +ℓ ∈ E˚_{. Fix any r > 0. It is} required to show that property (z_{) is a sufficient condition for it to hold}

3.5. PROOFS AND SUPPORTING RESULTS 43 This will follow from the intermediate value theorem if it can be shown that, for all sufficiently smallε >0, (x₋r, x+r) contains both a maximum and a minimum ofx7→gε(x)≡g(x/ε).

Define the distances D±

n as in the definition of (z); without loss of generality, assume that x > 0 and work with D_n+. If g only attains its extremes within some bounded subset of R, then Sε(ℓ)∩(x−r, x+r) =∅ for small enoughε; thus, conditions (a) and (b) are necessary. The strategy also fails if, having takenεr, nr>0 such that

εr "_nr X i=0 D_i+, nr+1 X i=0 D_i+ # ⊆(x₋r, x+r),

there exists some “later” interval that is “too big” in the sense that the rescaled interval doesnot fit within (x₋r, x+r), i.e., for somen > nr,

Pn+1 i=0 D+i Pn i=0Di+ = 1 + D + n+1 Pn i=0Di+ ≥ x_x+r −r.

Condition (c) of (z_{) ensures that there is only a finite number of such “bad”}

intervals, and so property (z_{) implies the claimed Kuratowski convergence}

on the positive half-line. The case x < 0, using D−_n, is similar; the case

x= 0 uses bothD_n±.

Lemma 3.5.4 (Almost-stability for positive time). For all r > 0, for all

τ >0, and all x0 ∈R, it holds true that for all small enough ε >0,

dist(zε(t),S(ℓ(t)))≤r for allt∈[τ, T]([0, T].

Furthermore, ifx0 ∈ S(ℓ(0)), then the same conclusion holds for τ ≥0. Proof. The proof has two parts: showing that zε must get close to the stable “sausage”S(ℓ(·)); and showing that, once close, it cannot later go far away. For æsthetic reasons, the second part will be shown first. For brevity, denote bySr(ℓ) the closedr-neighbourhood of S(ℓ):

Sr(ℓ) := [

x∈S(ℓ)

Br(x) ={x∈R|dist(x,S(ℓ))≤r}.

It is claimed that, for sufficiently small ε, the vector field ₋ε−1_E′

ε is inward-pointing on ∂Sr([0, T]), thus ruling out the possibility that zε may

escape Sr([0, T]). The strict convexity of V implies that, on ∂Sr(ℓ(t)),

|E′_{(t,·)| > σ; furthermore, since E}′ _{is continuous, the extreme value the-}

orem implies that |E′| is uniformly bounded away from σ on ∂Sr([0, T]): there exists σ0 > σ such that

t_∈[0, T], x_∈∂_Sr(ℓ(t)) =⇒ |E′(t, x)| ≥σ0

=_{⇒ |}E_ε′(t, x)_{| ≥}σ0−σ.

The two curves that comprise the two connected components of∂_Sr([0, T]) are uniformly Lipschitz with some Lipschitz constant L > 0; denote the upper/lower components by ∂±Sr([0, T]) in the natural way. The Lipschitz condition implies that if x _∈ ∂±Sr(ℓ(t)), then every vector of the form (1,∓y)∈R1+1 withy > Lpoints into Sr([0, T]) at (t, x).

Ifzεwere to escapeSr([0, T]) at some timet∗, then it would be impossible

for _dd_t(t∗, zε(t∗)) to be an inward-pointing vector. However, if

0< ε <(σ0−σ)/Land zε(t∗)∈∂Sr(t∗),

then the above arguments imply that d dt(t∗, zε(t∗)) = 1,−1 εE ′ ε(t∗, zε(t∗))

points intoSr([0, T]). Hence,zε cannot escapeSr([0, T]) once inside it. Thus, if the initial condition x0 ∈ Sr(ℓ(0)), then, for small enough ε,

zε(t) ∈ Sr(ℓ(t)) for allt∈[0, T]. Therefore, it remains only to consider the case x0 6∈ Sr(ℓ(0)). Fix r >0 and, using the Lipschitz bounds onV and ℓ, choose ˜τ >0 small enough that

t_∈[0,τ˜] =_{⇒ S}(ℓ(t))_{⊆ Sr/2}(ℓ(˜τ)).

Thus, _|E′_ε| is locally uniformly bounded away from 0: that is, there exists

σ1 > σ such that t∈[0,τ˜], x6∈ Sr(ℓ(˜τ)) =⇒ |E′(t, x)| ≥σ1 =⇒ 1 ε|E ′ ε(t, x)| ≥ σ1−σ ε .

3.5. PROOFS AND SUPPORTING RESULTS 45 Thus,zε must lie in Sr(ℓ(˜τ)) after a time of at most

εdist(x0,Sr(ℓ(˜τ))) σ1−σ

has elapsed. In particular, for everyτ >0,zε(τ)∈ Sr(ℓ(τ)) for small enough

ε, and hence zε(t)∈ Sr(ℓ(t)) for allt≥τ.

Since _S([0, T]) is closed, the almost-stability lemma has the following immediate consequence for every cluster point of the familyzε, whether the limit is taken in the uniform topology or even just the topology of pointwise convergence:

Corollary 3.5.5. If any subsequence of zε converges pointwise to some

z0: [0, T] → R, then z0(t) ∈ S(t) for all t ∈ (0, T]. If x0 ∈ S(0), then z0(t)∈ S(t) for allt∈[0, T].

Of course, the existence of such cluster points has yet to be established, and to do so will require some compactness arguments. The main tool for compactness in this context is, of course, the Arzel`a–Ascoli theorem. The almost-stability lemma will aid in producing the required estimates for the modulus of continuity.