ε -geodesic stars

We finally prove Lemma19. In these proofs, up to consideringB2₁

2ε

instead ofB2(ε), we will replace the condition that min_e∈E(s)W_e−2ε by the similar bound with −ε without impacting the result.

Lemma36. There exists some constant C >0 such that CLM⁽⁴⁾₁

s∈/P, min

e∈E(f0)W_e−ε

Cε⁵.

Proof. As in Lemma34, a scaling argument shows that it suffices to prove the same bound with the measureCLM⁽⁴⁾(e^−σ·) replacingCLM⁽⁴⁾₁ . LetE_O,1(s) be the set of edges

a1 a2

Figure 10. The four dominant, non-predominant schemata with four faces, f0 being the external face, considered up to obvious symmetries. Blank vertices indicate elements ofVN(s).

The labelsa1,a2,a3 anda4 are the integration variables used in the proof of Lemma36.

By Lemmas30and31, we obtain CLM^(k+1) Now let us focus again on the casek=3. The predominant schemata are the ones that are obtained from the first pre-schema of Figure3, by adding three null-vertices inside each edge incident tof₀, and then labeling the three “interior” faces by f₁, f₂ andf₃, and choosing a root.

All other (dominant) schemata are not predominant, and are indicated in Figure10.

Let us consider the first schema in this figure. In this case, we have #V_I(s)=4, #E_I(s)=1 and #E_O,1(s)=0, and taking β_e=¹₂, wheree is the unique edge of #E_I(s), the contri-bution to the upper bound (48) is bounded by

ε⁵

R⁴₊

da₁da₂da₃da₄

(a₁+1)⁴(a₂+1)⁴(a₃+1)^3/2(a₄+1)^3/2Cε⁵.

Let us turn to the dominant schemata corresponding to the third pre-schema of Figure3.

In this case, one has #V_I(s)=4, #E_I(s)=1 and #E_O,1(s)=1, and taking β_e=¹₃, the contribution to (48) is bounded by

ε⁶

R⁴₊

da₁da₂da₃da₄

(a₁+1)⁴(a₂+1)^4/3(a₃+1)^4/3(a₄+1)^10/3Cε⁶.

For the dominant schemata corresponding to the fourth pre-schema of Figure 3 one has #V_I(s)=4, #E_I(s)=1 and #E_O,1(s)=2, and taking alternativelyβ_e∈₁

3,¹₂ for the three edges inE_O,1(s)∪E_I(s), the contribution to (48) is at most

ε⁷

R⁴₊

da₁da₂da₃da₄

(a₁+1)^10/3(a₂+1)^7/6(a₃+1)^7/6(a₄+1)^10/3Cε⁷.

For the dominant schemata corresponding to the fifth pre-schema of Figure 3 one has

#V_I(s)=4, #E_I(s)=0 and #E_O,1(s)=3, and taking β_e=−1 for every edge in E_O,1(s), the contribution to (48) is bounded by

ε⁷

R⁴+

da₁da₂da₃da₄

(a₁+1)²(a₂+1)²(a₃+1)²(a₄+1)³Cε⁷. This entails the result.

Lemma37. There exist finite C, χ>0 such that, for every ε>0, CLM⁽⁴⁾₁ (B2(ε))Cε^4+χ.

Proof. In all of this proof, the mention of (1), (2), (3) and (4) will refer to the four points in the definition ofB₂(ε) (page379), for someε>0.

Using scaling as in (23), it suffices to prove a similar bound forCLM⁽⁴⁾(e^−σ1_B2(ε)).

Let us introduce some notation. We let T_z^e13=inf{s0:M_e13(s)=z}∈[0,∞] for every z∈R. We defineT_z^e23 in an analogous way. Let

Ξ =−

min

e∈E(f3):¯e∈E(f0)W_e∧inf{W^(s)_e

13,0sT₀^e13}∧inf{W^(s)_e

23,0sT₀^e23} and ¯Δ_y=sup_0zyΔ_z, where

Δ_z=−z−(inf{W^(s)_e

13, T_(−z)^e13 ₋sT_−z^e13}∧inf{W^(s)_e

23, T_(−z)^e23 ₋sT_−z^e23}).

Then, by takingt in (37) to be of the form T_−y^e13, as long asy >0 is such thatT_−y^e13<∞, and by choosingt=T_−y^e23 in a similar way, we obtain that

(−Ξ)∧(−Δ¯_y−y)(−Ξ)∧ inf

0zy(−Δ_z−z)−2y.

Letα, η,η andη be positive numbers, all strictly larger thanε, and such that η >η. Their values will be fixed later to be appropriate powers of ε. We observe thatB2(ε) is contained in the union of the three eventsB₂(ε),B₂(ε) andB₂(ε), which are defined by points (1), (2) and (3) in the definition ofB2(ε), together with (4), (4) and (4), respectively, where

(4) eitherr_e13η orr_e23ηor T_−α^e13>η or T_−α^e23>η; (4) r_e13∧r_e23>η,T_−α^e13∨T_−α^e23η and Ξ>η;

(4) r_e13∧r_e23>η,T_−α^e13∨T_−α^e23η andη∨Δ¯_yy for everyy∈[0, α].

It remains to estimate separately the quantities CLM⁽⁴⁾(e^−σ1_B

2(ε)), CLM⁽⁴⁾(e^−σ1_B

2(ε)) and CLM⁽⁴⁾(e^−σ1_B 2 (ε)).

For this, it suffices to restrict our attention to the event that s is the predominant schema of Figure8, since the other predominant schemata are the same up to symmetries.

Moreover, let us observe that the points (4), (4) and (4) only involve the snakesW_e whereeor its reversal is incident tof₃, and not the others. Therefore, when writing the above three quantities according to the definition (23) ofCLM⁽⁴⁾, there are going to be a certain number of common factors, namely, those which correspond to the contribution of points (2) and (3) to the five edges of sthat are not incident to f₃. Renaming the labels_v, v∈V(s)\V_N(s), asa₁,a₂, a₃ anda₄ as indicated in Figure8, we obtain that these common factors are

• a factor

E^(0,∞)_a4 [QX[e^−σ(W)]QX[e^−σ(W)1_{W−ε}]]² ε a₄+ε

₄

corresponding to the contribution to (2) of the two edges incident tof₀ and ending at the vertex with labela₄, where we used (42);

• a factor

E^(0,∞)_a1 [QX[e^−σ(W)]QX[e^−σ(W)1_{W−ε}]]E^(0,∞)_a2 [QX[e^−σ(W)]QX[e^−σ(W)1_{W−ε}]]

ε

a₁+ε ₂

ε a₂+ε

₂ , corresponding to the contribution to (2) of the two edges incident tof₁andf₂that end at the vertices with labelsa₁anda₂, where we used (42) again;

• a factor

Ba4!a3[QX[e^−σ(W)]²1_{X0}]2a₄,

corresponding to the contribution to (3) of the edgee₁₂, and where we used the fact that Q_X[e^−σ(W)]=e^−ζ(X)√2 and (40).

Let us now bound CLM⁽⁴⁾(e^−σ1_B which is bounded by (39), whilee₁₃ contributes by a factor

Ba1!a3[QX[e^−σ(W)]²1_{ζ(X)η or_T−α>η}]

Now, on the one hand, sincep_r(x, y) is a probability density, we have

From the Markovian bridge construction of [13] we have, for everyr>η, P^r_a1!a3(T_−α> η) =E_a1 Now we have, by symmetry and scaling of Brownian motion, and sinceT₁ has the same distribution asX₁⁻²under P0,

η). By putting together all the factors, recalling thatα>ε, we have obtained

Let us now turn to the estimation of CLM⁽⁴⁾(e^−σ1_B

2(ε)). We first observe the fol-lowing absolute continuity-type bound: For any non-negative measurable functionF and λ>0 we have

R

dyBx!y[e^−λζ(X)F(X(s),0sT_−α)1{ζ(X)>η,T−αη}]1

λE^(−α,∞)_x [F(X)]. (51) Indeed, note that, by using again the Markovian bridge description of [13],

2(ε)), note that there are exactly two edges incident to f₃with reversal incident tof₀, and we lete₁ be the one that is linked to the vertex with labela₁, ande₂the one linked to the vertex with label a₂. Note that the event{Ξ>η} is then included in the union

{W_e1<−η}∪{W_e2<−η}∪ For symmetry reasons, the first two have the same contribution (after integrating with respect toa₁, a₂, a₃ and a₄). Similarly, the last two have the same contribution. The

On the other hand, the edgese₁ande₂ do not contribute to {inf{W^(s)_e

13,0sT₀^e13}<−η},

and involve only, via (2), a factor (ε/(a₁+ε))²(ε/(a₂+ε))². The contribution ofe₁₃ and e₂₃, integrated with respect toa₃, is bounded by

where we used (49) in the second step, and Lemma32in the third step: Here, underP, (Δ_t, t0) is a Poisson process with intensity 2da/a², and ¯Δ is its supremum process.

These estimations, together with our preliminary remarks, entail that CLM⁽⁴⁾(e^−σ1_B

2 (ε)). Points (2) and (3) induce contributions of the edges incident tof₀, as well as the edgee₁₂, that are bounded by

Now, point (4) involves only the edgese₁₃ande₂₃, and contributes by a factor bounded above by

whereIwas defined in Lemma32, ¯I_y=sup_0zyI_y, andI and ¯Iare defined in a similar way from the trajectoryW. Now we use again a bound with the same spirit as (49).

Namely, for everyλ>0, everyx, y∈Rand for non-negative measurableF, we have

This is obtained by first checking this forF of a product form and using the Markovian description of bridges, and then applying a monotone class argument. We then use the boundp_r−η(X(η), z)(2π(η−η))^−1/2, valid forrη >η, and use Fubini’s theorem to integratep_r−η(X(η), z) with respect toz, as in the derivation of (51). Therefore, after integrating with respect to the variablesa₃, we obtain that the edgese₁₃ande₂₃together contribute by by an application of the Markov property, noticing that ¯I_y and ¯I_y only involve the processesW^(s)forT₀sT_−y, and similarly for ¯I_y (we skip the details). By Lemma32 and standard properties of Poisson measures, the process (I_y∨I_y,0yα) under the lawE^(−α,∞)₀ ⊗E^(−α,∞)₀ [Q_X(dW)Q_X(dW)] is a Poisson process on the time-interval [0, α]

with intensity 4da/a². By Lemma33, we obtain that the last displayed quantity is less than (η/α)^e−4, and we conclude that

This, together with (50) and (52), finally entails that CLM⁽⁴⁾(e^−σ1_B2(ε))Cε⁴ η∨ α

√η+ ε η−ε

₂ + ε

η+ 1

√η−η η

α _e⁻⁴

.

Let us now chooseα,η,η andη of the form

α=ε^β, η=ε^ν, η=ε^ν and η=ε^ν,

withβ, ν, ν, ν∈(0,1), and let us assume for the time being thatε<1. Then the condition η >η amounts to ν <ν, and by picking ε even smaller if necessary (depending on the choice ofν) we may assume that η−η>¹₂η and η−ε>¹₂η. The above bound then becomes

Cε⁴(ε^ν+ε^β−ν/2+ε^1−ν+ε^{(ν−β)e−4−ν/2}).

Therefore, it suffices to choose ν, β and ν so that 0<ν<2β <2ν<2, and thenν so that 0<ν <ν∧2(ν−β)e⁻⁴, to obtain a bound of the form Cε^4+χ with positive χ, as wanted. Once the choice is made, this bound remains obviously valid without restriction onε, by takingClarger if necessary.

The combination of (38) and Lemmas 36and37finally entail Lemma19.

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