Tube estimates for diffusions under a local strong Hörmander condition

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Tube estimates for diffusions under a local strong Hörmander condition

Vlad Bally, Lucia Caramellino, Paolo Pigato

To cite this version:

Vlad Bally, Lucia Caramellino, Paolo Pigato. Tube estimates for diffusions under a local strong Hörmander condition. Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2019, 55 (4), pp.2320–2369. �10.1214/18-AIHP950�. �hal-01413546�

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arXiv:1607.04542v2 [math.PR] 18 Jul 2016

Diffusions under a local strong H¨ ormander condition.

Part I: density estimates

Vlad Bally^∗ Lucia Caramellino^†

Paolo Pigato^‡ July 19, 2016

Abstract

We study lower and upper bounds for the density of a diffusion process in Rⁿ in a small (but not asymptotic) time, say δ. We assume that the diffusion coefficients σ1, . . . , σd may degenerate at the starting time 0 and pointx0 but they satisfy a strong H¨ormander condition involving the first order Lie brackets. The density estimates are written in terms of a norm which accounts for the non-isotropic structure of the problem:

in a small timeδ, the diffusion process propagates with speed√

δin the direction of the diffusion vector fields σj and with speed δ = √

δ×√

δ in the direction of [σi, σj]. In the second part of this paper, such estimates will be used in order to study lower and upper bounds for the probability that the diffusion process remains in a tube around a skeleton path up to a fixed time.

1 Introduction

In this paper we study bounds for the density of a diffusion process at a small time under a local strong H¨ormander condition. To be more precise, let X denote the process in Rⁿ solution to

dXt= Xd

j=1

σj(t, Xt)◦dW_t^j+b(t, Xt)dt, X0=x0. (1.1) whereW = (W¹, ..., W^d) is a standard Brownian motion and◦dW_t^jdenotes the Stratonovich integral. We assume nice differentiability and boundedness or sublinearity properties for the diffusion coefficientsb and σ_j,j= 1, . . . , d, and we consider a degenerate case:

dimσ(0, x₀) = dim Span{σ₁(0, x₀), . . . , σ_d(0, x₀)}< n, (1.2) dimS denoting the dimension of the vector space S. Our aim is to study lower and upper bounds for the density of the solution to (1.1) at a small (but not asymptotic) time, sayδ, under the following local strong H¨ormander condition:

Span{σ_i(0, x₀),[σ_p, σ_j](0, x₀), i, p, j= 1, . . . , d}=Rⁿ (1.3) in which [·,·] denotes the standard Lie bracket vector field. Notice that we ask for a H¨ormander condition at time 0. Our estimates are written in terms of a norm which reflects the non-isotropic structure of the problem: roughly speaking, in a small time interval of length δ, the diffusion process moves with speed √

δ in the direction of the diffusion vector fields σ_j and with speed δ =√

δ×√

δ in the direction of [σ_i, σ_j]. In order to catch this behavior we introduce the following norms. LetA_δ(0, x₀) denote the n×d² matrix

A_δ(0, x₀) = [A_1,δ(0, x₀), . . . , A_d2,δ(0, x₀)]

where the general columnA_l,δ(0, x0),l= 1, . . . , d², is defined as follows:

• forl= (p−1)d+iwithp, i∈ {1, . . . , d}and p6=ithen A_l,δ(0, x₀) = [√

δσ_i,√

δσ_p](0, x₀) =δ[σ_i, σ_p](0, x₀);

• forl= (p−1)d+iwithp, i∈ {1, . . . , d}and p=ithen A_l,δ(0, x₀) =√

δ σ_i(0, x₀).

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Under (1.3), the rank ofA_δ(0, x₀) is equal ton, hence the following norm is well defined:

|ξ|A_δ(0,x0)=h(A_δ(0, x₀)A_δ(0, x₀)^T)⁻¹ξ, ξi^1/2, ξ ∈Rⁿ,

where the supscriptT denotes the transpose andh·,·istands for the standard scalar product.

We prove in [3] that the metric given by this norm is locally equivalent with the control distanced_c (the Carath´eodory distance) which is usually used in this framework. We denote byp_δ(x₀,·) the density of the solution to (1.1) at time δ. Under (1.3) and assuming suitable hypotheses on the boundedness and sublinearity of the coefficients b and σ_j, j = 1, . . . , d (see Assumption 2.1 for details), we prove the following result (recall dimσ(0, x₀) given in (1.2)):

[lower bound]there exist positive constants r, δ_∗, C such that for every δ ≤ δ_∗ and for every y with|y−x₀−b(0, x₀)δ|Aδ(0,x0)≤r one has

p_δ(x₀, y)≥ 1 Cδⁿ⁻^dim^σ(0,x² ⁰⁾

;

[upper bound]for anyp >1, there exists a positive constant C such that for everyδ ≤1 and for every y∈Rⁿ one has

p_δ(x0, y)≤ 1 δⁿ⁻^dim^σ(0,x² ⁰⁾

C

1 +|y−x₀|^p_A_δ_(0,x₀₎.

This is stated in Theorem 2.4, where an exponential upper bound is achieved as well, provided that stronger boundedness assumptions on the diffusion coefficients hold (see Assump- tion 2.3).

In the context of a degenerate diffusion coefficient which fulfills a strong H¨ormander condition, the main result in this direction is due to Kusuoka and Stroock. In the celebrated paper [12], they prove the following two-sided Gaussian bounds: there exists a constant M ≥ 1 such that

1

M|B_d_c(x₀, δ^1/2)|exp

−M d_c(x₀, y)² δ

≤p_δ(x₀, y)≤ M

|B_d_c(x₀, δ^1/2)|exp

−d_c(x₀, y)² M δ

(1.4)

where δ ∈ (0,1], x₀, y ∈ Rⁿ, B_d_c(x, r) = {y ∈ Rⁿ : d_c(x, y) < r}, d_c denoting the control (Carath´eodory) distance, and |B_d_c(x, r)| stands for the Lebesgue measure of B_d_c(x, r). It is worth to be said that (1.4) holds under special hypotheses: in [12] it is assumed that the coefficients do not depend on the time variable and that b(x) = Pd

j=1α_iσ_i(x), with αi ∈C_b^∞(Rⁿ) (i.e. the drift is generated by the vector fields of the diffusive part, which is a quite restrictive hypothesis). Other celebrated estimates for the heat kernel under strong H¨ormander condition are provided in [4, 5]. The subject has also been widely studied by analytical methods - see for example [10] and [17]. We stress that these are asymptotic results, whereas we prove estimate for a finite, positive and fixed time. In [15], [8], non- isotropic norms similar to| · |Aδ(0,x0) are used to provide density estimates for SDEs under H¨ormander conditions of weak type. We also refer to [7], which considers the existence of the density for SDEs with time dependent coefficients, under very weak regularity assumption.

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The paper [3] will follow the present work, considering related results from a “control”

point of view, discussing in particular tube estimates and the connection with the control- Carath´eodory distance. Tube estimates are estimates on the probability that an Itˆo process remains around a deterministic path up to a given time. These will be obtained from a concatenation of the short-time density estimates presented here. Then we will consider, as in [12],b(t, x) =b(x) and σ(t, x) =σ(x). Defining the semi distance d via: d(x, y) <√

δ if and only if |x−y|Aδ(x) <1, we will prove in [3] the local equivalence of d and d_c. This will give a rewriting of the upper/lower estimates of the density in terms of the control distance as well.

The paper is organized as follows. In Section 2 we set-up the framework and give the precise statement of our main result (Theorem 2.4). The proof is split in two sections: Section 3, which is devoted to the lower bound, and Section 4, in which we deal with the upper bound.

The main tool we are going to use is given by the estimates of localized densities which have been developed in [2]. These need to use techniques from Malliavin calculus, so we briefly report in Appendix D all these arguments. But in order to set-up our program we also require some other facts, which have been collected in other appendices. First, we use a key decomposition of the solutionX_δto (1.1) at a small (but not asymptotic) timeδ(see Section 3.1), and we postpone the proof in Appendix A. This decomposition allows us to work with a random variable whose law, conditional to a suitableσ-algebra, is Gaussian, and in Appendix B we study some useful support properties that are applied to our case. Moreover, since the key-decomposition brings to handle a perturbed Gaussian random variable, in Appendix C we prove density estimates via local inversion for such kind of random variables.

2 Notations and main results

We need to recall some notations. Forf, g:R⁺×Rⁿ→Rⁿwe define the directional derivative (w.r.t. the space variable x) ∂_gf(t, x) = Pn

i=1gⁱ(t, x)∂_x_if(t, x), and we recall that the Lie bracket (again w.r.t. the space variable) is defined as [g, f](t, x) =∂_gf(t, x)−∂_fg(t, x). Let M ∈ Mn×m be a matrix with full row rank. We writeM^T for the transposed matrix, and M M^T is invertible. We denote by λ_∗(M) (respectively λ^∗(M)) the smallest (respectively the largest) singular value of M. We recall that singular values are the square roots of the eigenvalues of M M^T, and that, when M is symmetric, singular values coincide with the absolute values of the eigenvalues ofM. In particular, whenM is a covariance matrix, λ_∗(M) and λ^∗(M) are the smallest and the largest eigenvalues ofM.

We consider the following norm onRⁿ:

|y|_M =q

h(M M^T)⁻¹y, yi. (2.1)

Hereafter, α = (α₁, ..., α_k) ∈ {1, ..., n}^k represents a multi-index with length |α| = k and

∂_x^α = ∂xα1...∂x_αk. We allow the case k = 0, giving α =∅, and ∂_x^αf = f. Finally, for given vectorsv₁, . . . , v_n∈R^m, we definehv₁, . . . , v_ni ⊂R^m the vector space spanned byv₁, . . . , v_n. LetX denote the process inRⁿ already introduced in (1.1), that is

dX_t= Xd

j=1

σ_j(t, X_t)◦dW_t^j+b(t, X_t)dt, X₀=x₀, (2.2)

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W being a standard Brownian motion in R^d. We suppose the diffusion coefficients fulfill the following requests:

Assumption 2.1. There exists a constant κ >0 such that, ∀t∈[0,1], ∀x∈Rⁿ: Xd

j=1

|σ_j(t, x)|+|b(t, x)|+ Xd

j=1

X

0≤|α|≤2

|∂_x^α∂_tσ_j(t, x)| ≤κ(1 +|x|) Xd

j=1

X

1≤|α|≤4

|∂^α_xσ_j(t, x)|+ X

1≤|α|≤3

|∂_x^αb(t, x)| ≤κ

Remark that Assumption 2.1 ensures the strong existence and uniqueness of the solution to (2.2). We do not assume here ellipticity but a non degeneracy of H¨ormander type. In order to do this, we need to introduce then×d² matrixA(t, x) defined as follows. We setm=d² and define the function

l(i, p) = (p−1)d+i∈ {1, . . . , m}, p, i∈ {1, . . . , d}. (2.3) Notice thatl(i, p) is invertible. For l= 1, . . . , m, we set the (column) vector fieldA_l(t, x) in Rⁿ as follows:

A_l(t, x) = [σ_i, σ_p](t, x) if l=l(i, p) with i6=p,

=σ_i(t, x) if l=l(i, p) with i=p (2.4) and we setA(t, x) to be then×mmatrix whose columns are given byA₁(t, x), . . . , A_m(t, x):

A(t, x) = [A₁(t, x), . . . , A_m(t, x)]. (2.5) A(t, x) can be interpreted as a directional matrix. We denote byλ(t, x) the smallest singular value ofA(t, x), i.e.

λ(t, x)²=λ_∗(A(t, x))² = inf

|ξ|=1

Xm

i=1

hA_i(t, x), ξi². (2.6) In this paper, we assume the following non degeneracy condition. We write it in a “time dependent way” because this is useful in [3], which represents the second part of the present article. In fact, we use here just A(0, x₀) and λ(0, x₀), whereas in [3] we consider A(t, x_t) andλ(t, x_t), x_t denoting a skeleton path.

Assumption 2.2. Let x₀ denote the starting point of the diffusion X solving (2.2). We suppose that

λ(0, x0)>0.

Notice that Assumption 2.2 is actually equivalent to require that the first order H¨ormander condition holds in the starting point x₀, i.e. the vector fields σ_i(0, x₀), [σ_j, σ_p](0, x₀), as i, j, p= 1, ..., d, span the wholeRⁿ.

We define now them×mdiagonal scaling matrix Dδ as

(Dδ)_l,l =δ if l=l(i, p) with i6=p,

=√

δ if l=l(i, p) with i=p

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and the scaled directional matrix

A_δ(t, x) =A(t, x)Dδ. (2.7)

Notice that the lth column of the matrix A_δ(t, x) is given by √

δσ_i(t, x) if l = l(i, p) with i= p and by δ[σ_i, σ_p](t, x) = [√

δσ_i,√

δσ_p](t, x) if i 6= p. Therefore, A_δ(t, x) is the matrix given in (2.5) when the original diffusion coefficients σ_j(t, x), j = 1, . . . , d, are replaced by

√δσ_j(t, x),j= 1, . . . , d.

This matrix and the associated norm | · |A_δ(0,x0) are the tools that allow us to account of the different speeds of propagation of the diffusion: √

δ (diffusive scaling) in the direction of σ and δ in the direction of the first order Lie Brackets. In particuar, straightforward computations easily give that

√ 1

δλ^∗(A(t, x))|y| ≤ |y|_A_δ_(t,x)≤ 1

δλ_∗(A(t, x))|y|, (2.8) We also consider the following assumption, as a stronger version of Assumption 2.1 (morally we ask for boundedness instead of sublinearity of the coefficients, in the spirit of Kusuoka- Stroock estimates in [12]).

Assumption 2.3. There exists a constant κ >0 such that for every t∈ [0,1] and x ∈Rⁿ one has

X

0≤|α|≤4

hX^d

j=1

|∂_x^ασ_j(t, x)|+|∂^α_xb(t, x)|+|∂_x^α∂_tσ_j(t, x)|i

≤κ.

The aim of this paper is to prove the following result:

Theorem 2.4. Let Assumption 2.1 and 2.2 hold. Let p_X_t denote the density of X_t, t >0, with the starting condition X₀=x₀. Then the following holds.

(1) There exist positive constants r, δ_∗, C such that for every δ ≤δ_∗ and for every y such that |y−x₀−b(0, x₀)δ|Aδ(0,x0)≤r,

1

Cδⁿ⁻^dimhσ(0,x² ⁰⁾ⁱ ≤p_X_δ(y).

(2) For any p > 1, there exists a positive constant C such that for every δ ≤ 1 and for every y ∈Rⁿ

p_X_δ(y)≤ 1 δⁿ⁻^dimhσ(0,x² ⁰⁾ⁱ

C

1 +|y−x₀|^p_A_δ_(0,x₀₎.

(3) If also Assumption 2.3 holds (boundedness of coefficients) there exists a constant C such that for every δ ≤1 and for every y∈Rⁿ.

p_X_δ(y)≤ C δⁿ⁻^dimhσ(0,x² ^{0 )}ⁱ

exp − 1

C|y−x₀|Aδ(0,x0)

.

Here dimhσ(0, x0)i denotes the dimension of the vector space spanned by σ1(0, x0), . . . , σ_d(0, x₀).

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Remark 2.5. It might appear contradictory that the lower estimate (1) in Theorem 2.4 is centered in x₀+δb(x₀), whereas the upper estimates are centered in x₀. In fact, this is important only for the lower bound, the upper bounds (2) and (3) holding true either if we write|y−x₀−δb(x₀)|Aδ(0,x0) or |y−x₀|Aδ(0,x0) (see next Remark 4.8).

Remark 2.6. As already mentioned, the two sided bound (1.4) by Kusuoka and Stroock [12] is proved under a strong H¨ormander condition of any order, but the drift coefficient must be generated by the vector fields of the diffusive part, and the diffusion coefficientsb andσ_j,j = 1, . . . , d, must not depend on time. Here, on the contrary, we allow for a general drift and time dependence in the coefficients, but we consider only first order Lie Brackets.

Moreover, in assumption 2.1, we also relax the hypothesis of bounded coefficients. Anyways, the two estimates are strictly related, since our matrix norm is locally equivalent to the Carath´eodory distance – this is proved [3] (see Section 4 therein).

Remark 2.7. Our main application is developed in [3], which is the second part of this paper and concerns tube estimates. To this aim, we are mostly interested in the diagonal estimates, that is, aroundx0+δb(0, x0). In particular, what we need is the precise exponent n−dimhσ(0, x₀)i/2, which accounts for the time-scale of the heat kernel when δ goes to zero. However, our results are not asymptotic, but hold uniformly forδ small enough. This is crucial for our application to tube estimates, and this is also a main difference with the estimates in [4, 5].

Remark 2.8. The upper bounds in (2) and (3) of Theorem 2.4 give also the tail estimates, which are exponential if we assume the boundedness of the coefficients, polynomial otherwise.

The proof of Theorem 2.4 is long, also different according to the lower or upper estimate, and we proceed by organizing two sections where such results will be separately proved.

So, the lower estimate will be discussed in Section 3 and proved in Theorem 3.7, whereas Section 4 and Theorem 4.6 will be devoted to the upper estimate.

3 Lower bound

We study here the lower bound for the density ofX_δ. 3.1 The key-decomposition

We start with the decomposition of the process that will allow us to produce the lower bound in short (but not asymptotic) time.

We first use a development in stochastic Taylor series of order two of the diffusion process X defined through (2.2). This gives

X_t=x₀+Z_t+b(0, x₀)t+R_t (3.1) where

Z_t= Xd

i=1

a_iW_tⁱ+ Xd

i,j=1

a_i,j Z t

0

W_sⁱ◦dW_s^j witha_i=σ_i(0, x₀), a_i,j =∂_σ_iσ_j(0, x₀)

(3.2)

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and

R_t= Xd

j,i=1

Z t

0

Z s

0

(∂_σ_iσ_j(u, X_u)−∂_σ_iσ_j(0, x₀))◦dW_uⁱ◦dW_s^j (3.3)

+ Xd

i=1

Z t

0

Z s

0

∂_bσ_i(u, X_u)du◦dW_sⁱ+ Xd

i=1

Z t

0

Z s

0

∂_uσ_j(u, X_u)du◦dW_sⁱ

+ Xd

i=1

Z t

0

Z s

0

∂_σ_ib(u, X_u)◦dW_uⁱds+ Z t

0

Z s

0

∂_bb(u, X_u)duds.

Since R_t = O(t^3/2), we expect the behavior of X_t and Z_t to be somehow close for small values of t. Our first goal is to give a decomposition for Z_t in (3.2). We start introducing some notation. We fixδ >0 and set

s_k(δ) = k

dδ, k= 1, . . . , d.

We now consider the following random variables: fori, k = 1, . . . , d,

∆ⁱ_k(δ, W) =W_sⁱ

k(δ)−W_sⁱ

k−1(δ), ∆^i,j_k (δ, W) = Z s_k(δ)

sk−1(δ)

(W_sⁱ−W_sⁱ_k−1)◦dW_s^j. (3.4) Notice that ∆^i,j_k (δ, W) is the Stratonovich integral, but for i 6= j it coincides with the Itˆo integral. When no confusion is possible we use the short notation s_k = s_k(δ),∆ⁱ_k =

∆ⁱ_k(δ, W),∆^i,j_k = ∆^i,j_k (δ, W). We also denote the random vector ∆(δ, W) in R^m

∆_l(δ, W) = ∆^i,p_p (δ, W) if l=l(i, p) withi6=p,

= ∆^p_p(δ, W) if l=l(i, p) with i=p. (3.5) (recall l(i, p) in (2.3)). Moreover, with Pd

l>p =Pd p=1

Pd

l=p+1, we define V(δ, W) =

Xd

p=1

h X

i6=p

∆ⁱ_p+ X

i6=j,i6=p,j6=p

a_i,j∆^i,j_p + Xd

l=p+1

X

i6=p

X

j6=l

a_i,j∆^j_l∆ⁱ_p+1 2

X

i6=p

a_i,i∆ⁱ_p²i

; εp(δ, W) =

Xd

l>p

X

j6=l

ap,j∆^j_l + Xd

p>l

X

j6=l

aj,p∆^j_l +X

j6=p

ap,j∆^j_p, p= 1, ..., d;

η_p(δ, W) = 1

2a_p,p∆^p_p²+ Xd

l>p

a_p,l∆^l_l∆^p_p+ ∆^p_pε_p(δ, W), p= 1, ..., d.

(3.6) We have the following decomposition:

Lemma 3.1. Let ∆(δ, W) and A(0, x₀) be given in (3.5) and (2.5)respectively. One has Z_δ =V(δ, W) +A(0, x₀)∆(δ, W) +η(δ, W), (3.7) whereV(δ, W) is given in (3.6)andη(δ, W) =Pd

p=1η_p(δ, W), η_p(δ, W) being given in (3.6).

The proof of Lemma 3.1 is quite long, so it is postponed to Appendix A.

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Remark 3.2. The reason of this decomposition is the following. We split the time interval (0, δ) in dsub intervals of length δ/d.We also split the Brownian motion in corresponding increments: (Ws^p −Ws^p_k−1)_s_k−1_≤_s_≤_s_k, p = 1, ..., d. Let us fix p. For s ∈ (s_p₋₁, s_p) we have the processes (W_sⁱ−W_sⁱ_p−1)_s_p−1_≤_s_≤_s_p, i = 1, ..., d. Our idea is to settle a calculus which is based on W^p and to take conditional expectation with respect to Wⁱ, i 6= p. So (W_sⁱ − W_sⁱ_p−1)_s_p−1_≤_s_≤_s_p, i6= p will appear as parameters (or controls) which we may choose in an appropriate way. The random variables on which the calculus is based are ∆^p_p =W_s^p_p−W_s^p_p−1 and ∆^i,p_p =Rsp

sp−1(W_sⁱ−W_sⁱ_p−1)dW_s^p, j6=p.These are the r.v. that we have emphasized in the decomposition ofZ_δ.Notice that, conditionally to the controls (W_sⁱ−W_sⁱ_p−1)sp−1≤s≤sp, i6=p, this is a centered Gaussian vector and, under appropriate hypothesis on the controls this Gaussian vector is non degenerate (we treat in section B the problem of the choice of the controls). In order to handle the term ∆^p,ip = Rsp

sp−1(Ws^p−Ws^pp−1)dW_sⁱ. we use the identity

∆^p,i_p = ∆ⁱ_p∆^p_p−∆^i,p_p .

We now emphasize the scaling inδ in the random vector ∆(δ, W). We defineB_t=δ⁻^1/2W_tδ and denote

Θ_l= 1 δ∆^i,p_p =

Z ^p_d

p−1 d

(B_sⁱ−Bⁱp−1 d

)dB_s^p if l=l(i, p) withi6=p,

= 1

√δ∆^p_p =B^pp

d −B^pp−1 d

if l=l(i, p) with i=p,

(3.8)

l(i, p) being given in (2.3). For p = 1, ..., d we denote with Θ_(p) the pth block of Θ with lengthd, that is

Θ_(p)= (Θ_(p₋_1)d+1, ...,Θ_pd), so that Θ = (Θ₍₁₎, . . . ,Θ_(d)). We will also denote

l(p) =l(p, p) = (p−1)d+p and Θ_l(p)= 1

√δ∆^p_p. (3.9) Consider now theσ field

G:=σ(W_s^j−W_s^j

p−1(δ), s_p₋₁(δ)≤s≤s_p(δ), p= 1, ...d, j 6=p). (3.10) Then conditionally to G the random variables Θ_(p), p = 1, ..., d are independent centered Gaussianddimensional vectors and the covariance matrix Q_p of Θ_(p) is given by

Q^p,j_p =Q^j,p_p = Z ^p_d

p−1 d

B^j_s−B^jp−1 d

ds, j6=p, Q^i,j_p =

Z ^p_d

p−1 d

B_s^j−B^jp−1 d

B_sⁱ−Bⁱp−1 d

ds, j 6=p, i6=p, Q^p,p_p = ¹_d.

(3.11)

It is easy to see that detQ_p 6= 0 almost surely. It follows that conditionally toG the random variable Θ = (Θ₍₁₎, ...,Θ_(d)) is a centeredm=d²dimensional Gaussian vector. Its covariance

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matrixQis a block-diagonal matrix built withQ_p, p= 1, . . . , d:

Q=



 Q₁

. ..

Q_d



 (3.12)

In particular detQ= Qd

p=1detQ_p 6= 0 almost surely, and λ_∗(Q) = min_p=1,...,dλ_∗(Q_p). We also have λ^∗(Q) = max_p=1,...,dλ^∗(Q_p). We will need to work on subsets where we have a quantitative control of this quantities, so we will come back soon on these covariance matrices. But let us show now how one can rewrite decomposition (3.7) in terms of the random vector Θ. As a consequence, the scaled matrixA_δ =A_δ(0, x₀) in (2.7) will appear.

We denote by Aⁱ_δ ∈ R^m, i = 1, ..., n the rows of the matrix A_δ. We also denote S = hA¹_δ, ..., Aⁿ_δi ⊂ R^m and S^⊥ its orthogonal. Under Assumption 2.2 the columns of A_δ span Rⁿ so the rows A¹_δ, ..., Aⁿ_δ are linearly independent and S^⊥ has dimension m−n. We take Γⁱ_δ, i = n+ 1, ..., m to be an orthonormal basis in S^⊥ and we denote Γⁱ_δ = Aⁱ_δ(0, x0) for i = 1, ..., n. We also denote Γ_δ the (m−n) ×m matrix with rows Γⁱ_δ, i = n+ 1, . . . , m.

Finally we denote by Γ_δ the m×m dimensional matrix with rows Γⁱ_δ, i = 1, ..., m. Notice that

Γ_δΓ^T_δ =

A_δA^T_δ(0, x₀) 0

0 Id_m₋_n

(3.13) where Id_m₋_n is the identity matrix inR^m⁻ⁿ.It follows that for a pointy= (y₍₁₎, y₍₂₎)∈R^m withy₍₁₎∈Rⁿ, y₍₂₎ ∈R^m⁻ⁿ we have

|y|²_Γ_δ =y₍₁₎²

Aδ(0,x0)+y₍₂₎² (3.14) where we recall that|y|²_Γ_δ =

(Γ_δΓ^T_δ)⁻¹y, y

. For a∈R^m we define the immersion J_a:Rⁿ→R^m, (J_a(z))_i =z_i, i= 1, ..., n and (J_a(z))_i =

Γⁱ_δ, a

, i=n+ 1, ..., m.

(3.15) In particular J₀(z) = (z,0, ...,0) and

|J₀z|_Γ_δ =|z|_A_δ_(0,x₀₎. (3.16) Finally we denote

V_ω = V(δ, W) η_ω(Θ) =

Xd

p=1

a_p,p

2 δΘ²_l(p)+δ^1/2Θ_l(p)ε_p(δ, W) + Xd

q>p

a_p,qδΘ_l(q)Θ_l(p)

! (3.17)

whereV(δ, W) and ε_p(δ, W) are defined in (3.6) and Θ_l(p) is given in (3.9). We notice that η_ω(Θ) =Pd

p=1η_p(δ, W),η_p(δ, W) being defined in (3.6). We also remark that bothV(δ, W) andε_p(δ, W) are G-measurable, so (3.17) stresses a dependence on ωwhich is G-measurable and a dependence on the random vector Θ whose conditional law w.r.t.G is Gaussian.

Now the decomposition (3.7) may be written as

Z_δ =V_ω+A_δ(0, x₀)Θ +η_ω(Θ).

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We embed this relation in R^m and obtain

J_Θ(Z_δ) =J₀(V_ω) + Γ_δΘ +J₀(η_ω(Θ)).

We now multiply with Γ⁻_δ¹: setting

Ze= Γ⁻_δ¹J_Θ(Z_δ), Ve_ω= Γ⁻_δ¹J₀(V_ω), ηe_ω(Θ) = Γ⁻_δ¹J₀(η_ω(Θ)) (3.18) and

G= Θ +ηe_ω(Θ), (3.19)

we get

Ze=Ve_ω+G. (3.20)

Notice that, conditionally to G, Ze is a translation of the random variable G = Θ +eη_ω(Θ) which is a perturbation of a centred Gaussian random variable. Thanks to this fact, we can to apply the results in Appendix C: we use a local inversion argument in order to give bounds for the conditional density ofZe, which will be used in order to get bounds for the non conditional density. As a consequence, we will get density estimates forZ_δ.

3.2 Localized density for the principal term Ze

We study here the density of Ze in (3.20), “around” (that is, localized on) a suitable set of Brownian trajectories, where we have a quantitative control on the “non-degeneracy”

(conditionally toG) of the main Gaussian Θ.

We denote

q_p(B) =X

j6=p

B^jp

d −B^jp−1 d

+ X

j6=p,i6=p,i6=j

Z ^p_d

p−1 d

(B_s^j−B^ji−1 d

)dB_sⁱ

. (3.21)

For fixed ε, ρ >0, we define Λ_ρ,ε,p=n

detQ_p≥ε^ρ,supp−1 d ≤t≤^p_d

P

j6=p|B_t^j−B^j_p−1

d | ≤ε⁻^ρ, q_p(B)≤εo

, p= 1, . . . , d Λ_ρ,ε=∩^d_p=1Λ_ρ,ε,p.

(3.22) Notice that Λρ,ε,p ∈ G for every p = 1, . . . , d. By using some results in Appendix B, we get the following.

Lemma 3.3. Let Λ_ρ,ε be as in (3.22). There exist c and ε_∗ such that for every ε≤ε_∗ one has

P Λ_ρ,ε

≥c×ε¹²^m(d+1). (3.23)

Proof. We apply here Proposition B.3. Letp∈ {1, . . . d}be fixed and consider the Brownian motion Bb_t =√

d(Bp−1+t

d −Bp−1

d ). Let Q(Bb) be the matrix in (B.1). Up to a permutation of the components ofBb, we easily get Q^p,p(B) =b d×Q^p,p_p ,Q^p,j(Bb) =d^3/2×Q^p,j_p forj6=p, Q^i,j(Bb) =d²×Q^i,j_p fori6=pand j6=p. Therefore,

detQ_p =d^2d⁻¹detQ(B)b ≥detQ(B).b

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Let nowq(B) be the quantity defined in (B.3). Withb q_p(B) as in (3.21), it easily follows that qp(B)≤q(B).b

Moreover, sup_t_≤₁|Bb_t|=√

dsupp−1

d ≤s≤_d^p|B_s−Bp−1

d | ≥ supp−1

d ≤s≤^p_d|B_s−Bp−1

d |. As a consequence, with Υ_ρ,ε the set defined in (B.4), we get

Υ_ρ,ε(B)b ⊂Λ_ρ,ε,p

and by using (B.5), we may find some constantscandε_∗such thatP(Λ_ρ,ε,p)≥cε¹²^d(d+1), for ε≤ε_∗. This holds for everyp. Since Λρ,ε =∩^dp=1Λρ,ε,p, by using the independence property we get (3.23).

Let Q be the matrix in (3.12). On the set Λρ,ε ∈ G we have detQ = Qd

p=1detQp ≥ ε^dρ. Remark that

λ^∗(Q)

√m ≤ |Q|l :=



 1 m

X

1≤i,j≤m

Q²_i,j





1/2

≤λ^∗(Q). (3.24)

Fora >0 we introduce the following function, ψ_a(x) = 1_|_x_|≤_a+ exp

1− a²

a²−(x−a)²

1_a<_|_x_|_<2a,

which is a mollified version of 1_[0,a]. We can now define ourlocalization variables.

Ue_ε= (ψ_a₁(1/detQ))ψ_a₂(|Q|l)ψ_a₃(q(B)), with a₁=ε⁻^dρ, a₂ =ε⁻^2ρ, a₃=dε (3.25) in which we have set

q(B) = Xd

p=1

q_p(B).

Remark that Ue_ε is measurable w.r.t. G. The following inclusions hold: for every ε small enough,

Λ_ρ,ε⊂n

detQ≥ε^dρ,|Q|l≤ε⁻^2ρ, q(B)≤dεo

={Ue_ε = 1} ⊂ {Ue_ε 6= 0}. (3.26) We can considerUeε as a smooth version of the indicator function of Λρ,ε. We also define, for fixedr >0,

U¯_r= Yn

i=1

ψ_r(Θ_i). (3.27)

In order to state a lower estimate for the (localized) density of Ze in (3.20), we define the following set of constants:

C =n

C >0 : C= exp c κ

λ(0, x₀) q

, ∃c, q >0o

(3.28) and we set

1/C={C >0 : 1/C∈ C}.

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Lemma 3.4. Suppose Assumption 2.1 and 2.2 both hold. Let U_ε,r =Ue_εU¯_r,Ue_ε and U¯_r being defined in (3.25) and (3.27) respectively, with ρ = _8m¹ . Set dP_U_ε,r =U_ε,rdP and let p_Z,U_e

ε,r

denote the density ofZein (3.20)when we endowΩwith the measureP_U

ε,r. Then there exist C∈ C, ε, r∈1/C such that for |z| ≤r/2,

p_Z,U_e

ε,r(z)≥ 1

C. (3.29)

Proof. STEP 1: lower bound for the localized conditional density given G.

Letp_Z,_e_U_¯_r_|G denote the localized density w.r.t. the localization ¯Ur ofZeconditioned toG, i.e.

E[f(Z) ¯e Ur|G] = Z

f(z)p_Z,_e_U_¯

r|G(z)dz, (3.30)

forf positive, measurable, with support included inB(0, r/2). We start proving that there existC ∈ C,ε, r∈1/C such that, on Ue_ε6= 0, for|z| ≤r/2

p_Z,_e_U_¯

r|G(z)≥ 1 C.

We recall (3.20):Ze=Ve_ω+ Θ +ηe_ω(Θ), whereω7→Ve_ω andω 7→ηe_ω(·) are bothG-measurable and the conditional law of Θ given G is Gaussian. This allows us to use the results in Appendix C. In particular, we are interested in working on the set{Ue_ε6= 0} ∈ G, so one has to keep in mind thatω∈ {Ue_ε 6= 0}.

OnUe_ε6= 0, by (3.25) and (3.24) one has λ^∗(Q)≤2√

mε⁻^2ρ, and ε^dρ

2 ≤detQ≤λ_∗(Q)λ^∗(Q)^m⁻¹ ≤λ_∗(Q)(2√

m)^m⁻¹ε⁻^2ρ(m⁻¹⁾, and this givesλ_∗(Q)≥ ₍₂^ε^√^3mρ_m)m. So, fixingρ= 1/(8m), for ε≤ε^∗,

1 16m²

λ_∗(Q)

λ^∗(Q) ≥C_mε^3mρ+2ρ≥ε. (3.31)

To apply (C.8) toG= Θ +ηe_ω(Θ) we need to check the hypothesis of Lemma C.3. We are going to use the notation of Appendix C, in particular for c_∗(ηe_ω, r) in (C.5) and c_i(ηe_ω), i= 2,3, in (C.1). Recall that ηe_ω is defined in (3.18) through η_ω given in (3.17). Since the third order derivatives ofη_ω are null, we havec₃(eη_ω) = 0. Also, fori=l(p) andj=l(q) we have ∂_i,jη_ω(Θ) = δa_ij, otherwise we get ∂_i,jη_ω(Θ) = 0. So |∂_i,jη_ω(Θ)| ≤ δP

i,j|a_i,j|. Using (2.8) we obtain

|∂i,jeηω(Θ)|=|J0(∂i,jηω(Θ))|_Γ_δ =|∂i,jηω(Θ)|_A_δ ≤ P

i,j|a_i,j|

λ(0, x₀) ≤C∈ C. So, withh_η_ω as in (C.2), we get

h_η_ω = 1

16m²(c₂(ηe_ω) +p

c₃(eη_ω)) ≥ 1 C1

, ∃C₁ ∈ C (3.32)

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We compute now the first order derivatives. For j /∈ {l(p) : p = 1, ..., d} we have ∂_jη_ω = 0 and forj =l(p) we have

∂_jη_ω(Θ) =δ Xd

q=p

a_p_∧_q,p_∨_qΘ_l(q)+√

δε_j(δ, W).

So, as above, we obtain|∂_jηe_ω(Θ)| ≤C(|Θ|+|ε_j(δ, W)|/√

δ). Remark now that on{U¯_r 6= 0} we have|Θ| ≤Cr, and on{Ue_ε 6= 0} we have q(B)≤2dε, so

Xd

j=1

|εj(δ, W)| ≤C√

δq(B)≤C√

δε. (3.33)

Therefore

c_∗(ηe_ω,16r) ≤C₂(r+ε), ∃C₂ ∈ C. (3.34) We also consider the following estimate of|Ve_ω|=|V_ω|Aδ. First, we rewriteV_ω as follows:

Vω=X

p

apµp(δ, W) +X

p

ψp(δ, W), with µ_p(δ, W) =X

i6=p

∆^p_i and ψ_p(δ, W) = X

i6=j,i6=p,j6=p

a_i,j∆^i,j_p + Xd

l=p+1

X

i6=p

X

j6=l

a_i,j∆^j_l∆ⁱ_p+1 2

X

i6=p

a_i,i∆ⁱ_p²

Using again (2.8) we have

Xd

p=1

a_pµ_p(δ, W)

Aδ

= 1

√δ

A_δJ₀X^d

p=1

µ_p(δ, W)

Aδ ≤ Xd

p=1

√1

δ|µ_p(δ, W))| ≤Cq(B) and

|ψ(δ, W)|_A_δ ≤ |ψ(δ, W)| δp

λ(0, x₀) ≤Cq(B).

Sinceω ∈ {Ueε 6= 0} we get

|Ve_ω| ≤Cq(B)≤C3ε, ∃C3 ∈ C. (3.35) We consider (3.35), and fix ^r_ε = 2C₃ ∈ C, so |Ve_ω| ≤ r/2. Then we consider (3.34) and we obtain

c_∗(eη_ω,4r)≤C₂(2C₃+ 1)ε≤ε^1/2, for ε≤ 1

(4C₂C₃)² ∈ 1 C. Moreover, looking at (3.32)

r= 2C₃ε≤ 1

C₁ for ε≤ 1

2C₁C₃ ∈ 1 C. So, with

ε=ε^∗∧ 1

(4C2C3)² ∧ 1

2C1C3 ∈ 1 C,

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andr = 2C₃εwe have

|Ve_ω| ≤ r

2, c_∗(ηe_ω,4r)≤ε^1/2, r≤ 1 C₁.

Now, by using also (3.31) and (3.32), it follows that (C.6) holds, and we can apply Lemma C.3. We obtain

1

KdetQ^1/2exp

− K λ_∗(Q)|z|²

≤p_G,U¯r|G(z)

for|z| ≤r, whereK does not depend on σ, b. Remark that, usingλ_∗(Q)≥ ₍₂^ε^√^3mρ_m)m,ρ= _8m¹ , r/ε= 2C₁ and ε≤1/(4C₂C₁)²,

|z|²

λ_∗(Q) ≤ (2√ m)^mr²

ε^3mρ ≤(2√ m)^mr²

ε ≤(2√ m)^mr²

ε²ε≤(2√

p_Z,_e_U_¯

r|G(z)≥ 1

C, for|z| ≤r/2 on the set {Ue_ε 6= 0}. (3.37) STEP 2: we get rid of the conditioning onG, to have non-conditional bound for p_Z,U_e

ε,r. Since Ueε is G measurable, for every non-negative and measurable function f with support included inB(0, r/2) we have

E(f(Z)Ue ε,r) =E UeεE(f(Z) ¯e Ur|G) . By (3.30) and (3.37), we obtain

E(f(Ze)Uε,r)≥ 1 CE(Ueε)

Z

f(z)dz

Since Λρ,ε ⊂ {Ueε = 1},E(Ueε) ≥P(Λρ,ε), so by using (3.23) and ε∈1/C we finally get that E(Ue_ε)≥ _C¹, so (3.29) is proved.

3.3 Lower bound for the density of Xδ

We study here a lower bound for the density of X_δ, X being the solution to (2.2). Recall decomposition (3.1):

X_δ−x₀−b(0, x₀)δ=Z_δ+R_δ.

Our aim is to “transfer” the lower bound forZe= Γ⁻_δ¹J_Θ(Z_δ) already studied in Lemma 3.4 to a lower bound for X_δ. In order to set up this program, we use results on the distance between probability densities which have been developed in [1]. In particular, we are going to use now Malliavin calculus. Appendix D is devoted to a recall of all the results and notation the present section is based on. In particular, we denote with D the Malliavin derivative with respect toW, the Brownian motion driving the original equation (2.2).

But first of all, we need some properties of the matrix Γ_δ, which can be resumed as follows.

We set SO(d) the set of the d×d orthogonal matrices and we denote with Id_d the d×d identity matrix.

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Lemma 3.5. Set for simplicity A_δ = A_δ(0, x₀) and let Γ_δ be as in (3.13). There exist U ∈SO(n), U ∈SO(m−n) and V ∈SO(m) such that

Γ_δ=

U 0 0^T U

Σ¯ 0 0^T Idm−n

V^T

where 0 denotes a nulln×(m−n) matrix and Σ = ¯¯ Σ = Diag(λ1(A_δ), . . . , λn(A_δ)), λi(A_δ), i= 1, . . . , n, being the singular values of A_δ (which are strictly positive because A_δ has full rank).

Proof. We recall that

Γ_δ= A_δ

Γ_δ

,

where Γ_δ is a (m−n)×nmatrix whose rows are vectors of R^m which are orthonormal and orthogonal with the rows ofA_δ. We take a singular value decomposition for A_δ and for Γ_δ. So, there existU ∈SO(n) and ¯V ∈SO(m) such that

A_δ =U Σ 0¯ V¯^T,

0 denoting then×(m−n) null matrix. Similarly, there existU ∈SO(m−n) andV ∈SO(m) such that

Γ_δ=U 0^T Id_m₋_n V^T,

the diagonal matrix being Id_m₋_nbecause the rows of Γ_δ are orthonormal. Therefore, we get Γ_δ=

U 0 0^T U

Σ¯ 0 0^T Idm−n

V^T

whereV is a m×mmatrix whose first ncolumns are given by the firstncolumns ofV and the remaining m−n columns are given by the last m−n columns of V. Moreover, since each row ofA_δ is orthogonal to any row of Γ_δ, it immediately follows that all columns ofV are orthogonal. This proves thatV ∈SO(m), and the statement follows.

Then we have

Lemma 3.6. Suppose Assumption 2.1 and 2.2 both hold. LetUε,r denote the localization in Lemma 3.4 and let U andΣ¯ be the matrices in Lemma 3.5. Set

α=UΣ¯ and Xb_δ=α⁻¹(X_δ−x₀−b(0, x₀)δ).

Then there existC ∈ C,δ_∗, r∈1/C such that forδ ≤δ_∗, |z| ≤r/2, p_X_b

δ,Uε,r(z)≥ 1

C, (3.38)

p_X_b

δ,Uε,r denoting the density of Xb_δ with respect to the measure P_U

ε,r.

Proof. We set Zb_δ = α⁻¹Z_δ and we use Proposition D.1, with the localization U = U_ε,r, applied to F =Xb_δ and G=Zb_δ. Recall that the requests in (1) of Proposition D.1 involve several quantities: the lowest singular value (that in this case coincides with the lowest eigenvalue)λ_∗(γ_X_b

δ) andλ_∗(γ_Z_b

δ) of the Malliavin covariance matrix ofXb_δandZb_δrespectively,

Tube estimates for diffusions under a local strong Hörmander condition

HAL Id: hal-01413546

https://hal-upec-upem.archives-ouvertes.fr/hal-01413546

Tube estimates for diffusions under a local strong Hörmander condition

Vlad Bally, Lucia Caramellino, Paolo Pigato

To cite this version:

arXiv:1607.04542v2 [math.PR] 18 Jul 2016

Diffusions under a local strong H¨ ormander condition.

Part I: density estimates

Contents

1 Introduction

2 Notations and main results

3 Lower bound