Thin shell implies spectral gap up to polylog via a stochastic localization scheme

Introduction

is the $\varepsilon$-extension of $A$. The main point of this paper is to find an upper bound for the constant,

Where $\mu$ runs over all isotropic log-concave measures and $\varphi$ runs over all smooth enough functions with $\int\varphi d\mu=0$ and $C>0$ is some universal constant.

In this note we will show that, up to a small correction, the above is implied by a seemingly weaker hypothesis.

Next, we would like to formulate the thin-shell conjecture. Let $\sigma_{n}\geq 0$ satisfy

There exists a universal constant $C$ such that,

An application of (3) with the function $\varphi(x)=|x|^{2}$ shows that the thin-shell conjecture is weaker than the KLS conjecture.

The first nontrivial bound for $\sigma_{n}$ was given by Klartag in [K1], who showed that $\sigma_{n}\leq C\frac{n^{1/2}}{\log(n+1)}$. Several improvements have been introduced around the same method, see e.g [K2] and [Fl1]. The best known bound for $\sigma_{n}$ at the time of this note is due to Guedon and E. Milman, in [Gu-M], extending previous works of Klartag, Fleury and Paouris, who show that $\sigma_{n}\leq Cn^{\frac{1}{3}}$. The thin-shell conjecture was shown to be true for several specific classes of convex bodies, such as bodies with a symmetry for coordinate reflections (Klartag, [K3]) and certain random bodies (Fleury, [Fl2]).

It was found by Sudakov, [Sud], that the parameter $\sigma_{n}$ is highly related to almost-gaussian behaviour of certain marginals of a convex body, a fact now known as the central limit theorem for convex sets [K1]. This theorem asserts that most of the one-dimensional marginals of an isotropic, log-concave random vector are approximately gaussian in the sense that the Kolmogorov distance to the standard gaussian distribution of a typical marginal has roughly the order of magnitude of $\sigma_{n}/\sqrt{n}$. Therefore the conjectured bound (5) actually concerns the quality of the gaussian approximation to the marginals of high-dimensional log-concave measures. The first theorem of this note reads,

There exists a constant $C>0$ such that for all $n\geq 2$,

Plugging the results of this paper into the currently best known bound for $\sigma_{n}$ (proven in [Gu-M]), $\sigma_{n}\leq Cn^{1/3}$, it follows that

This slightly improves the previous bound, $G_{n}\leq Cn^{5/12}$, which is a corollary of [Gu-M] and [Bo].

In [EK1], B. Klartag and the author have found a connection between the thin-shell hypothesis and another well known conjecture related to convex bodies, known as the hyperplane conjecture. The methods of this paper share some common lines with the methods in [EK1]. In a very recent paper of K.Ball and V.H. Nguyen, [BN], a connection between the KLS conjecture and the hyperplane conjecture that applies for individual log-concave measures has also been established. They show that the isotropic constant of a log concave measure which attains a spectral gap is bounded by a constant which depends exponentially on the spectral gap.

Compare this result with the result in [Bo]. Bobkov’s theorem states that for any log-concave random vector $X$ and any smooth function $\varphi$, one has

Under the thin-shell hypothesis, Bobkov’s theorem gives $G_{n}\leq Cn^{1/4}$.

The bound in theorem 1.1 will rely on the following intermediate constant which corresponds to a slightly stronger thin shell bound. Define,

There exists a constant $C>0$ such that for all $n\geq 2$,

Theorem 1.1 will be a consequence of the above lemma along with,

There exists a constant $C>0$ such that for all $n\geq 2$,

The constant $K_{n}$ satisfies the following bound:

We move on to the second result of this paper, a stability result for the Brunn-Minkowski Inequality. The Brunn-Minkowski inequality states, in one of its normalizations, that

When there is an almost-equality in (7), $K$ and $T$ are almost translates of each other in a certain sense (which varies between different estimates). Estimates of this form, often referred to as stability estimates, appear in Diskant [Dis], in Groemer [Groe], and in Figalli, Maggi and Pratelli [FMP1, FMP2], Segal [Seg].

The result [FMP2], which is essentially the strongest result in its category, and other existing stability estimates share a common thing: the bounds become worse as the dimension increases. In a recent paper, [EK2], Klartag and the author suggested that the correct bounds might actually become better as the dimension increases, as demonstrated by certain results. The estimates presented here may be viewed as a continuation of this line of research.

In order to formulate our result, we define the two constants

so that $\sigma_{n}\leq\tau_{n}n^{\kappa}$. Note that the thin-shell conjecture implies $\kappa=0$ and $\tau_{n}<C$. Our main estimate reads,

For every $\epsilon>0$ there exists a constant $C(\epsilon)$ such that the following holds: Let $K,T$ be convex bodies whose volume is $1$ and whose barycenters lie at the origin. Suppose that the covariance matrix of the uniform measure on $K$ is equal to $L_{K}Id$ for a constant $L_{K}>0$. Denote,

It follows from theorem 1.4 in [EK2] that the above estimate is true with

If $\kappa\geq 1/4$, then the result we prove here weaker than the one in [EK2]. However, under the thin shell hypothesis, the result of this paper becomes stronger, and is in fact tight up to the term $C(\epsilon)n^{\epsilon}$. This tightness is demonstrated, for instance, by taking $K$ and $T$ to be the unit cube and a unit cube truncated by a ball of radius $\sqrt{n}$ and normalized to be isotropic.

Using the bound in [Gu-M], the theorem gives

Note that if the assumption (9) is dropped, even if the covariance matrices of $K$ and $T$ are assumed to be equal, the best corresponding bound would be $\delta=C\sqrt{n}L_{K}$ as demonstrated, for example, by a cube and a ball.

The above bound complements, in some sense, the result proven in [FMP1], which reads,

for some choice of $x_{0}$, where $\Delta$ denotes the symmetric difference between the sets. Unlike the result presented in this paper, the result in [FMP1] gives much more information as the expression $Vol_{n}((K+T)/2)-1$ approaches zero. On the other hand the result presented here already gives some information when $Vol_{n}((K+T)/2)=10$.

The structure of this paper is as follows: In section 2, we construct a stochastic localization scheme which will be the main ingredient our proofs. In section 3, we establish a bound for the covariance matrix of the measure throughout the localization process, which will be essential for its applications. In section 4, we prove theorem 1.1 and in section 5 we prove theorem 1.2 and its corollaries. In section 6 we tie some loose ends.

Acknowledgements I owe this work to countless useful discussions I have had with my supervisor, Bo’az Klartag, through which I learnt the vast part of what I know about the subject, as well as about related topics, and for which I am grateful. I would also like to thank Vitali and Emanuel Milman and Boris Tsirelson for inspiring discussions and for their useful remarks on a preliminary version of this note. Finally, I would like to thank the anonymous referee for doing a tremendous job reviewing a preliminary version of this paper, thanks to his/her ideas the proofs are significantly simpler, shorter and more comprehensible.

A stochastic localization scheme

The assumption (10) ensures that $V_{f}$, $a_{f}$ and $A_{f}$ are smooth functions of $c,B$. Let $W_{t}$ be a standard Wiener process and consider the following system of stochastic differential equations:

Taking into account the fact that the functions $A_{f},a_{f}$ are smooth and that $A_{f}(c,B)$ is positive definite for all $c,B$, we can use a standard existence and uniqueness theorem (see e.g., [Ok], section 5.2) to ensure the existence and uniqueness of a solution in some interval $0\leq t\leq t_{0}$, where $t_{0}$ is an almost-surely positive random variable. Next, we construct a 1-parameter family of functions $\Gamma_{t}(f)$ by defining,

so that $a_{t}$ and $A_{t}$ are the barycenter and the covariance matrix of the function $f_{t}$.

The following lemma may shed some light on this construction.

The function $F_{t}$ satisfies the following set of equations:

Equation (11) clearly implies that $[B]_{t}=0$. Let $Q_{t}(x)$ denote the quadratic variation of the process $\langle x,c_{t}\rangle$. We have,

Applying Itô’s formula one last time yields,

In view of the above lemma it can be seen that, in some sense, the above is just the continuous version of the following iterative process: at every time step, multiply the function by a linear function equal to $1$ at the barycenter, whose gradient has a random direction distributed uniformly on the ellipsoid of inertia. This construction may also be thought of as a variant of the Brownian motion on the Riemannian manifold constructed in [EK1].

Rather than defining the process $F_{t}$ through equations (11) and (12), one may alternatively define it directly with the infinite system of stochastic differential equations in formula (13). In this case, the existence and uniqueness of the solution can be shown using [KX, Theorem 5.2.2, page 159] (however, some extra work is needed in order to show that the conditions of this theorem hold).

In the remainder of this note, most of the calculations involving the process $f_{t}$ will use the formula (13) rather than the formulas (11) and (12). The remaining part of this section is dedicated to analyzing some basic properties of $\Gamma_{t}(f)$. We begin with:

In order to prove (i), we will first need the following technical lemma:

For every dimension $n$, there exists a constant $c(n)>0$ such that,

The proof of this lemma is postponed section 6. Proof of lemma 2.4: To prove (i), we have to make sure that $A_{t}^{-1/2}$ does not blow up. To this end, define $t_{0}=\inf\{t|~{}~{}\det A_{t}=0\}$. By continuity, $t_{0}>0$. Equation (12) suggests that $f_{t}$ is log-concave for all $t<t_{0}$. The fact that $t_{0}=\infty$ will be proven below. We start by showing that both (ii) and (iii) hold for any $t<t_{0}$. We first calculate, using (13),

with probability 1. The last equality follows from the definition of $a_{t}$ as the barycenter of the measure $f(x)F_{t}(x)dx$. We conclude (ii). We continue with proving (iii). To do this, fix some $0<s<t_{0}-t$ and write,

which is clearly an isotropic probability density. Let us inspect $\Gamma_{t}(g(x))$. We have, using (13),

which proves (iii). We are left with showing that $t_{0}=\infty$. To see this, write,

where $c(n)$ is the constant from lemma 2.5. Note that, by continuity, $s_{1}$ is well-defined and almost-surely positive. When time $s$ comes, we may define $L_{1}$ as in (15), and continue running the process on the function $f\circ L_{1}^{-1}$ as above. We repeat this every time $||A_{t}^{-1}||_{OP}$ hits the value $c^{-1}(n)$, thus generating the hitting times $s_{1},s_{2},...$. Lemma 2.5 suggests that,

which implies that, almost surely, $s_{i+1}-s_{i}>c(n)$ for infinitely many values of $i$. Thus, $\lim_{n\to\infty}s_{n}=\infty$ almost surely, and so $t_{0}=+\infty$. Part (iv) follows immediately from formula (13). The lemma is proven.

where the third equality follows from the defition of $a_{t}$, which implies,

One of the crucial points, when using this localization scheme, will be to show that the barycenter of the measure does not move too much throughout the process. For this, we would like to attain upper bounds on the eigenvalues of the matrix $A_{t}$. We start with a simple observation: Equation (12) shows that the measure $f_{t}$ is log-concave with respect to the measure $e^{-\frac{1}{2}|B_{t}^{1/2}x|^{2}}$. The following result, which is well-known to experts, shows that measures which possess this property attain certain concentration inequalities.

where $A_{K\Theta}$ is the $K\Theta$-extension of $A$, defined in the previous section. (ii) For all $\theta\in S^{n-1}$,

Proof: Denote the density of $\mu$ by $\rho(x)$. Let $B$ be the complement of $A_{K\Theta}$, where the constant $\Theta$ will be chosen later on. Define,

Note that for $x\in A$ and $y\in B$, we have $|x-y|>K\Theta$. Thus, by the parallelogram law,

Since the function $\phi$ is assumed to be convex, we obtain

Now, using the Prekopa-Leindler theorem, we attain

Clearly, a large enough choice of the constant $\Theta$ gives (i). To prove (ii), we define,

and take $A=\{x;\langle x,\theta\rangle<g^{-1}(0.5)\}$. An application on (i) on the set $A$ gives,

Part (ii) of the proposition is a direct consequence of the last two equations.

Plugging (12) into part (ii) of this theorem gives,

By our assumption (10) we deduce that $A_{t}$ is bounded by $n^{2}Id$, which immediately gives

The bound (18) will be far from sufficient for our needs, and the next section is dedicated to attaining a better upper bound. However, it is good enough to show that the barycenter, $a_{t}$, converges in distribution to the density $f(x)$. Indeed, (18) implies that

It is interesting to compare this construction with the construction by Lehec in [Leh]. In both cases, a certain Itô process converges to a given log-concave measure. In the result of Lehec, the convergence is ensured by applying a certain adapted drift, while here, it is ensured by adjusting the covariance matrix of the process.

We end this section with a simple calculation in which we analyze the process $\Gamma_{t}(f)$ in the simple case that $f$ is the standard Gaussian measure. While the calculation will not be necessary for our proofs, it may provide the reader a better understanding of the process. Define,

According to formula (12), the function $f_{t}$ takes the form,

Recall that $B_{t}=\int_{0}^{t}A_{s}^{-1}ds$. It follows that,

In the previous section we saw that the covariance matrix of the $f_{t}$, $A_{t}$, satisfies (18). The goal of this section is to give a better bound, which holds also for small $t$. Namely, we want to prove:

Before we move on to the proof, we will establish some simple properties of the matrix $A_{t}$. Our first task is to find the differential of process $A_{t}$. We have, using Itô’s formula with equation (13),

Let us try to understand each of this terms. The second term is,

Recall that by (16), $da_{t}=A_{t}^{1/2}dW_{t}$, which gives,

Plugging equations (23), (24) and (25) together gives,

The vectors $\xi_{i,j}$ satisfy the following bounds: (i) For all $1\leq i\leq n$, $|\xi_{i,i}|<C$ for some universal constant $C>0$. (ii) For all $1\leq i\leq n$, $\sum_{j=1}^{n}|\xi_{i,j}|^{2}\leq K_{n}^{2}$.

Proof: Since $A_{t}^{1/2}v_{i}=\sqrt{\alpha_{i,i}}v_{i}$ for all $1\leq i\leq n$, we have

Using this with (30) establishes (i). Next, by the definition of $K_{n}$, we have for all $1\leq i\leq n$,

We are now ready to prove the main proposition of the section. Proof of proposition 3.1: We fix a positive integer $p$ whose value will be chosen later, and define,

Since $S_{t}$ is a smooth function of the coefficients $\{\alpha_{i,j}\}$, which are Itô processes (assuming that the basis $v_{1},...,v_{n}$ is fixed), $S_{t}$ itself is also an Itô process. Fix some $t>0$. Our next goal will be to find $dS_{t}$. To that end, define $\Gamma$ to be the set of $(p+1)$-tuples, $(j_{1},,..,j_{p+1})$, such that $j_{i}\in\{1,...,n\}$ for all $1\leq i\leq p+1$ and such that $j_{1}=j_{p+1}$. It is easy to verify that,

The second type of term will contain exactly two off-diagonal entries, and due to the symmetry of the matrix and the constraint $j_{1}=j_{p+1}$, it has the form:

where $i\neq j$ and $0\leq k\leq p-2$. Keeping in mind that $\alpha_{i,j}=0$, we calculate,

We may clearly assume $\alpha_{1,1}\geq\alpha_{2,2}\geq...\geq\alpha_{n,n}$, which implies that for $i<j$ and for all values of $k$, one has

Inspect the equation (32). For every $1\leq i\leq n$, the expansion on the right hand side contains exactly one term of the first type, and for every distinct $i,j$ with $i\neq j$, it contains $\frac{p(p-1)}{2}$ terms of the second type (or otherwise, for all choices such that $i<j$, it contains $p(p-1)$ terms of this type). Using (33) and (34), we conclude

where in the last inequality we used the part (ii) of lemma 3.2. A well-known property of Itô processes is existence and uniqueness of the decomposition $S_{t}=M_{t}+E_{t}$, where $M_{t}$ is a local martingale and $E_{t}$ is an adapted process of locally bounded variation. In the last equation, we attained,

Next, we use the unique decomposition $\log S_{t}=Y_{t}+Z_{t}$ where $Y_{t}$ is a local martingale, $Z_{t}$ is an adapted process of locally bounded variation and $Y_{0}=0$. According to Itô’s formula and formula (36),

Choosing $t$ to be a large enough universal constant, $C_{1}$, yields

(where we used the fact that $p\geq 1$). Using (37), we attain

for some universal constant $C_{2}>0$. We now use Itô’s formula again, this time with formula (35), to get

The last two equations and the legitimate assumption that $K_{n}\geq 1$ give,

for some universal constant $C^{\prime}>0$. Define the event $F$ as the complement of the event in the equation above,

Clearly, whenever the event $F$ holds, we have,

Our next task is to bound the norm for larger values of $t$. To this end, recall the bound (17). Recalling that $B_{t}=\int_{0}^{t}A_{s}^{-1}ds$, and applying (17) gives,

By the definition of $B_{t}$ and by (38), it follows that whenever $F$ holds one has,

where $\delta^{2}=\frac{1}{K_{n}^{2}\log n}$. Equations (39) and (40) imply,

Proposition 2.6 gives an immediate corollary to part (iii) of proposition 3.1:

where $E_{\Theta/\delta}$ is the $\frac{\Theta}{\delta}$-extension of $E$, defined in the introduction.

Thin shell implies spectral gap

Our goal in this section is to show that,

for some universal constants $c,\Theta>0$, where $\delta=\frac{1}{K_{n}\sqrt{\log n}}$ and $E_{\Theta/\delta}$ is the $\frac{\Theta}{\delta}$-extension of $E$. The idea is quite simple. Define $f_{t}:=\Gamma_{t}(f)$, the localization of $f$ constructed in section 2, and fix $t>0$. By the martingale property of the localization, we have,

Corollary 3.3 suggests that if $t$ is large enough, the right term can be bounded from below if we only manage to bound the integral $\int_{E}f_{t}(x)dx$ away from 0 and from 1. Define,

In view of the above, we would like to prove:

There exists a universal constant $T>0$ such that,

Define $h(t)=(g(t)-0.5)^{2}$. By Itô’s formula,

Plugging the last two equations together gives,

The lemma follows from an application of Chebyshev’s inequality.

The last ingredient needed for our proof is a theorem of E. Milman, [Mil2, Theorem 2.1]. The following is a weaker formulation of this theorem which will be suitable for us:

Note that equation (48) is the exact type of inequality defining the constant $G_{n}$ in equation (2). We are now ready to prove the main proposition of this section. Proof of proposition 1.7: Let $T$ be the constant from lemma 4.1. Denote,

The result now follows directly from an application of theorem 4.2.

In the above proof, we used E. Milman’s result in order to reduce the theorem to the case where $\int_{E}f(x)dx$ is exactly $\frac{1}{2}$, as well as to attain an isoperimetric inequality from a certain concentration inequality for distance functions. Alternatively, we may have replaced propsition 2.6 with an essentially stronger result due to Bakry-Emery, proven in [BE] (see also Gross, [Gros1]). Their result, which relies on the hypercontractivity principle, asserts that a density of the form (12) actually possesses a respective Cheeger constant. Using this fact, we may have directly bounded from below the surface area of any set with respect to the measure whose density is $f_{t}$.

The proof of lemma 1.6 is in section 6. Along with this lemma, we have established theorem 1.1.

Stability of the Brunn-Minkowski inequality

The following lemma is a variant of lemma 6.5 from [EK2].

and let $\{\delta_{i}\}_{i=1}^{n}$ be the eigenvalues of $A$ such that the order of $|\delta_{i}-1|$ is decreasing. Then,

where $C,C_{1}>0$ are universal constants.

Our main ideas in this section are contained in the following lemma:

Proof: As explained in the beginning of the section, we will couple between the measures $f$ and $g$ in means of coupling between the processes $\Gamma_{t}(f)$ and $\Gamma_{t}(g)$. To that end, we define, as in (13),

is the covariance matrix of $fF_{t}$. As usual denote $f_{t}=F_{t}f$. Next, we define,

and denote $g_{t}(x)=g(x)G_{t}(x)$. Finally, we ”interpolate” between the two processes by defining,

where $C$ is the same constant as in (20). Finally, denote $E_{t}=G_{t}\cap F_{t}$. By proposition 3.1 and equation (54), $P(E_{t})>1-\epsilon$ for all $t>0$. Define a stopping time by the equation,

Now, thanks to formula (19), we can take $T$ large enough (and deterministic) such that,

is the covariance matrix of $g_{t}$. Therefore,

The first two terms are martingale. We use the unique decomposition

where $M_{t}$ is a local martingale and $N_{t}$ is an adapted process of locally bounded variation. We get,

where $D_{t}=A_{t}^{1/2}(I-A_{t}^{-1/2}C_{t}A_{t}^{-1/2})$. Our next task is to use lemma 5.1 to bound $||D_{t}||_{HS}$ under the assumption f$t<\tau$. We start by denoting the eigenvalues of the matrix $I-A_{t}^{-1/2}C_{t}A_{t}^{-1/2}$ by $\delta_{i}$, in decreasing order, and the eigenvalues of the matrix $A_{t}$ by $\lambda_{i}$, also in decreasing order. By theorem 1 in [T],

Fix some constant $(1-2\kappa)<\alpha<1$, whose value will be chosen later. For now, we assume that $\kappa>0$. Using Hölder’s inequality, we calculate,

where $\beta=\frac{1}{n}\sum_{j=1}^{n}\lambda_{j}$. Recall that $\alpha>(1-2\kappa)$, which gives,

Take $\alpha$ such that $\epsilon=\alpha-(1-2\kappa)$. Equations (61) and (62) give,

Finally, using equations (56) and (58), we conclude,

In the above lemma, if we replace the assumption that $f$ is isotropic by the assumption that $f,g$ are log-concave with respect to the Gaussian measure, then following the same lines of proof while using proposition 2.6, one may improve the bound (52) and get,

We move on to the proof of theorem 1.2. Proof of theorem 1.2: Let $K,T$ be convex bodies of volume $1$ such that the covariance matrix of $K$ is $L_{k}^{2}Id$. Fix $\epsilon>0$. Define,

so both $f$ and $g$ are probability measures and $f$ is isotropic. We have,

where $d(x,T/L_{T})=\inf_{y\in(T/L_{T})}|x-y|$. Denote,

It follows from Markov’s inequality and from (64) and (65) that,

Finally, taking $\delta=L_{K}\Theta/\sqrt{\epsilon}$ gives

Tying up loose ends

Let $k\leq n$ and let $E_{k}$ be a subspace of dimension $k$. Denote $P(X)=Proj_{E_{k}}(X)$ and $Y=|P(X)|-\sqrt{k}$. By definition of $\sigma_{k}$,

Using the last inequality and applying Cauchy-Schwartz gives,

Next, in order to provide the reader with a better understanding of the constant $K_{n}$, we introduce two new constants. First, define

where the supremum runs over all isotropic log-concave random vectors, $X$, and all quadratic forms $Q(x)$. Next, define

where $\mu$ runs over all isotropic log-concave measures and $E$ runs over all ellipsoids with $\mu(E)\leq 1/2$.

There exist universal constants $C_{1},C_{2}$ such that

where $B$ runs over all symmetric matrices. Let $B$ be a symmetric matrix. Fix coordinates under which $B$ is diagonal, and write $X=(X_{1},...,X_{n})$ and $B=diag\{a_{1},..,a_{n}\}$. Define $Q(x)=\langle Bx,x\rangle$. We have,

We suspect that there exists a universal constant $C>0$ such that $K_{n}\leq C\sigma_{n}$, but we are unable to prove that assertion.

We move on to the proof of lemma 2.5. Proof of lemma 2.5: Throughout the proof, all the constants $c,c_{1},c_{2},...$ may depend only on the dimension $n$. Recall that $f(x)$ is assumed to be isotropic and log-concave. It is well-known that there exist two constants $c_{1},c_{2}>0$, such that

(see for example [LV, Theorem 5.14]). Define $g(x)=c_{1}\mathbf{1}_{\{|x|\leq c_{2}\}}$. It is also well-known (see for example [LV, Lemma 5.7]) that there exist two constants $c_{3},c_{4}>0$ such that

which implies that whenever $|c|<c_{4}$ and $B$ is positive semi-definite,

It follows that for all $|c|<c_{4}$ and $B\leq Id$ (in the sense of positive matrices), one has

for some constant $c_{5}>0$. Define the stopping times,

so the lemma would be concluded if we manage to show that

for some constant $c>0$. Define the event $E=\{T_{2}\leq T_{1}\}$. Whenever $E$ holds, we have the following: First, using (67),

Recall that $\frac{d}{dt}B_{t}=A_{t}^{-1}$. It follows that

By taking $t=T_{2}$ in the last equation, we learn that $T=T_{2}\geq c_{5}$ whenever $E$ holds, so

Therefore, it is enough to prove that $P(T_{1}>c)>c$ for some $c>0$. Furthermore, in the following we are able to assume that $P(E)\leq 0.1$. To that end, consider the defining equation (11) and use Itô’s formula to attain

Define the process $e_{t}$ by the equations,

Using (68), we deduce that whenever $t<T$, one has

Define $\delta=\min\left(T,\frac{c_{6}}{4c_{4}^{2}c_{5}^{-1}}\right)$. Note that, by (71),

Another application of (68), this time with the assumption (10) gives,

for all $t<T$, and for a constant $c_{7}$. By plugging (73) and (74) into (70), we learn that whenever $F$ holds, one has

References