## Abstract

In this paper, we consider diffusion semigroups generated by second-order differential operators of degenerate type. The operators that we consider *do not*, in general, satisfy the Hörmander condition and are *not* hypoelliptic. In particular, instead of working under the Hörmander paradigm, we consider the so-called UFG (uniformly finitely generated) condition, introduced by Kusuoka and Strook in the 1980s. The UFG condition is weaker than the uniform Hörmander condition, the smoothing effect taking place only in certain directions (rather than in every direction, as it is the case when the Hörmander condition is assumed). Under the UFG condition, Kusuoka and Strook deduced sharp *small* time asymptotic bounds for the derivatives of the semigroup in the directions where smoothing occurs. In this paper, we study the *large* time asymptotics for the gradients of the diffusion semigroup in the same set of directions and under the same UFG condition. In particular, we identify conditions under which the derivatives of the diffusion semigroup in the smoothing directions decay exponentially in time. This paper constitutes, therefore, a stepping stone in the analysis of the long-time behaviour of diffusions which do not satisfy the Hörmander condition.

### 1. Introduction

Consider the stochastic differential equation (SDE) in ${\mathbb{R}}^{N}$

*V*

_{0},…,

*V*

_{d}are smooth vector fields on ${\mathbb{R}}^{N}$, ° denotes Stratonovich integration and, for each

*i*,

*W*

^{i}(

*t*) is an

*N*-dimensional standard Brownian motion. The Markov semigroup ${\{{\mathcal{P}}_{t}\}}_{t\ge 0}$ associated with the SDE (1.1) is defined on the set

*C*

_{b}of continuous and bounded functions, as

*V*on ${\mathbb{R}}^{N}$, we can think of

*V*both as a vector-valued function on ${\mathbb{R}}^{N}$ and as a first-order differential operator on ${\mathbb{R}}^{N}$:

*V*

_{0},…,

*V*

_{d}satisfy the

*Hörmander condition*[1], in one of its many forms. As is well known, under the (parabolic) Hörmander condition, the transition probabilities of the semigroup ${\mathcal{P}}_{t}$ have a smooth density; furthermore, ${\mathcal{P}}_{t}f$ is differentiable in every direction and $u(t,x):=({\mathcal{P}}_{t}f)(x)$ is a classical solution of the Cauchy problem

In this paper, we relax the hypoellipticity assumption and work in the setting in which the vector fields *V*_{0},…,*V*_{d} satisfy a weaker condition, the so-called UFG condition. The acronym UFG stands for *uniformly finitely generated*. Informally, denoting by ${C}_{\mathrm{b}}^{\mathrm{\infty}}({\mathbb{R}}^{N})$ the set of smooth bounded functions with bounded derivatives, this condition states that the ${C}_{\mathrm{b}}^{\mathrm{\infty}}({\mathbb{R}}^{N})$-module $\mathcal{W}$ generated by the vector fields {*V*_{i},*i*=1,…,*d*} within the Lie algebra generated by {*V*_{i},*i*=0,1,…,*d*} is finite dimensional. In particular, we emphasize that the UFG condition does not require that the vector space $\{W(x)|W\in \mathcal{W}\}$ is homeomorphic to ${\mathbb{R}}^{N}$ for any $x\in {\mathbb{R}}^{N}$; indeed, the dimension of the space $\{W(x)|W\in \mathcal{W}\}$ is not even required to be constant over ${\mathbb{R}}^{N}$. Hence, in this sense, the UFG condition is weaker than the Hörmander condition. We give a precise (and easier to check) statement of the UFG condition in §2, see definition 2.1.

In a series of papers [2–5], Kusuoka and Stroock have analysed the smoothness properties of diffusion semigroups ${\{{\mathcal{P}}_{t}\}}_{t\ge 0}$ associated with the stochastic dynamics (1.1) when the vector fields {*V*_{i},*i*=0,1,…,*d*} satisfy the UFG condition. In particular they showed that, under the UFG condition, the semigroup ${\mathcal{P}}_{t}$ is no longer differentiable in the direction *V*_{0}; however, it is still differentiable in the direction $\mathcal{V}:={\mathrm{\partial}}_{t}-{V}_{0}$ and, therefore, a rigourous PDE analysis can still be built starting from the stochastic dynamics (1.1). In this case, one can indeed prove that for every *f*∈*C*_{b}, the function $u(t,x):=({\mathcal{P}}_{t}f)(x)$ is a classical solution of the Cauchy problem

*u*is twice continuously differentiable in the directions of the vector fields

*V*

_{i},

*i*=1,…,

*d*and once continuously differentiable in the direction ${\mathcal{V}}_{0}={\mathrm{\partial}}_{t}-{V}_{0}$, when viewed as a function (

*t*,

*x*)↦

*u*(

*t*,

*x*) over the product space $(0,\mathrm{\infty})\times {\mathbb{R}}^{d}$ (the notion of classical solution for the PDE (1.5) and further background material are gathered in appendix A).

This fundamental result was obtained by using probabilistic methods based on the use of the Malliavin calculus [6,7]. The small time asymptotics of $({\mathcal{P}}_{t}f)(x)$ constitutes the theoretical backbone for the development of a new class of algorithms, termed *cubature methods*, introduced by Kusuoka, Lyons, Ninomiya and Victoir in the last 10 years [8–10]. Such algorithms, which work under the UFG condition, provide high-order approximations of the law of the solutions of SDEs (and therefore can be used to compute statistical quantities of interest) and are faster than their classical counterparts, see [6]. The study of UFG diffusions has, therefore, opened interesting and promising research avenues both in the field of PDE theory and in the field of stochastic simulations.

The papers [2–4] introduce the UFG condition in the context of the theory of diffusion semigroups. However, related conditions had already independently appeared, in a completely different setting, in the work of Hermann [11], Lobry [12] and Sussman [13]. In these works, such a condition was considered for control theoretical purposes. More details on the nature of the UFG condition is given in §2.

Under the UFG condition, Kusuoka and Strook proved sharp estimates on *short-time* behaviour of the semigroup ${\mathcal{P}}_{t}$. Further work on the subject was carried out in [7], where a wealth of results regarding the short-time asymptotics are derived. To the best of our knowledge, nothing is known so far about the *long-time* behaviour of the semigroup under the UFG condition. In this paper, we provide the first step towards understanding the long-time asymptotics of this class of (possibly) degenerate diffusions; in particular, we obtain *pointwise* estimates on the time-behaviour of the (space) derivatives of the function $u(t,x)=({\mathcal{P}}_{t}f)(x)$. This is the first result concerning the long-time behaviour of UFG semigroups. The main result of the paper can be informally stated as follows (see theorem 4.2 for a precise statement).

### Theorem 1.1

*If the vector fields* {*V*_{i}*,i*=0,1,…,*d*} *satisfy both the UFG condition and some quantitative assumption* (*the ‘obtuse angle condition’* (*4.8*)) *then, for any bounded continuous function f* (*not necessarily smooth*), *any t*_{0}∈(0,1) *and any vector field, V, belonging to the* ${C}_{\mathrm{b}}^{\mathrm{\infty}}({\mathbb{R}}^{N})$*-module* $\mathcal{W}$*, there exist constants c*_{t0},λ>0 *such that*

We emphasize that the UFG condition alone does not suffice to ensure the exponential decay of the coefficients. For a simple counterexample, take the one-dimensional Ornstein–Uhlenbeck process with positive drift constant *a*>0. Then the semigroup is uniformly elliptic (hence it satisfies the UFG condition) but one has ${\mathrm{\partial}}_{x}{\mathcal{P}}_{t}f={\mathrm{e}}^{at}{\mathcal{P}}_{t}({\mathrm{\partial}}_{x}f)$ (see also note 4.3 on this point).

From a technical point of view, the methods we use in this paper are analytic; indeed, the strategy we use to prove our main result, theorem 4.2, is a variation of the classic approach established by Bakry and colleagues [14,15] to deduce exponential decay estimates and is similar to the approach adopted by Dragoni *et al.* [16]. We defer to note 3.3 a more careful comparison with this strand of the literature. Here, we just emphasize the pointwise nature of the above inequality. It is indeed customary to obtain bounds for the derivatives of semigroups in *L*^{p} spaces weighted by an appropriate invariant measure. This is not possible here, in the absence of an obvious invariant measure to exploit.

To summarize, the aim of this paper is twofold: (i) first, we move another step forward in the Kusuoka–Stroock programme and we produce results that are applicable to the study of cubature methods; (ii) second, we extend the classic semigroup approach of Bakry, which was introduced in the context of elliptic diffusions and then applied to hypoelliptic processes, to semigroups which are more general than hypoelliptic. In particular, regarding the latter point, the estimates obtained in this paper, together with the mentioned control-theoretical results of Herrmann, Lobry and Sussman [11–13], will form the stepping stone for future work on the ergodic theory for SDEs with generator which does not necessarily satisfy the Hörmander condition (corollary 4.10 is a simple example in this spirit). On a related note, we would like to emphasize that some commonly used diffusion processes do not satisfy the Hörmander condition, but satisfy the UFG condition; the simplest of such examples is Geometric Brownian motion. Another important motivation for the current work is to provide the basis of the asymptotic (in time) analysis of the error incurred by the high-order numerical approximations produced by cubature methods.

The paper is organized as follows: in §2, we introduce the UFG condition and the necessary notation. In §3, we present a version of the classical Bakry technique, adapted to our context. In §4, we present our main results concerning the exponential decay of the derivatives of the semigroup and explain how such estimates can be obtained by employing the techniques presented in §3. In §4a, we show one way of using our estimates to obtain information on the behaviour of the semigroup itself. More detailed results in this direction will be the object of future work. In §5, we gather all the proofs of the results of §4.

### 2. The UFG condition and notation

Fix $d\in \mathbb{N}$ and let $\mathcal{A}$ be the set of all *n*-tuples, of any size *n*≥1, of integers of the following form:

*n*-tuples of any length

*n*≥1 are allowed in $\mathcal{A}$, except the trivial one,

*α*=(0) (however,

*α*=(

*j*) belongs to $\mathcal{A}$ if

*j*∈{1,…,

*d*}). We endow $\mathcal{A}$ with the product

*α*=(

*α*

^{1},…,

*α*

^{h}) and

*β*=(

*β*

^{1},…,

*β*

^{ℓ}) in $\mathcal{A}$. If

*α*is an element of $\mathcal{A}$, we define the

*length*of

*α*, denoted by ∥

*α*∥, the integer

*B*is any set, |

*B*| denote the cardinality of the set

*B*.

^{1}

Given a vector field (or, equivalently, a first-order differential operator) *V* =(*V* ^{1}(*x*),*V* ^{2}(*x*),…, *V* ^{N}(*x*)) on ${\mathbb{R}}^{N}$, we refer to the functions {*V* ^{j}(*x*)}_{1≤j≤N} as to the *components* or *coefficients* of the vector field. We say that a vector field on ${\mathbb{R}}^{N}$ is smooth or that it is ${C}^{\mathrm{\infty}}$ if all the components *V* ^{j}(*x*), *j*=1,…,*N*, are ${C}^{\mathrm{\infty}}$ functions. Given two differential operators *V* and *W*, the commutator between *V* and *W* is defined as

*V*

_{i}:

*i*=0,…,

*d*} be a collection of vector fields on ${\mathbb{R}}^{N}$ and let us define the following ‘hierarchy’ of operators:

*α*∥=

*h*, then ∥

*α**

*i*∥=

*h*+1 if

*i*∈{1,…,

*d*} and ∥

*α**

*i*∥=

*h*+2 if

*i*=0. If $\alpha \in \mathcal{A}$ is a multi-index of length

*h*, with abuse of nomenclature we will say that

*V*

_{[α]}is a differential operator of length

*h*. We can then define the space ${\mathfrak{R}}_{m}$ to be the space containing all the operators of the above hierarchy, up to and including the operators of length

*m*(but excluding

*V*

_{0}):

*φ*

_{α,β}in the above belong to the set ${C}_{V}^{\mathrm{\infty}}({\mathbb{R}}^{N})$ of bounded smooth functions, ${\phi}_{\alpha ,\beta}={\phi}_{\alpha ,\beta}(x):{\mathbb{R}}^{N}\to \mathbb{R}$, such that

*n*and all

*γ*

_{(1)},…,

*γ*

_{(n)},

*α*and

*β*in ${\mathcal{A}}_{m}$. With this notation in place, we can now introduce the definition that will be central in this paper.

### Definition 2.1 (UFG Condition).

Let {*V*_{i}:*i*=0,…,*d*} be a collection of smooth vector fields on ${\mathbb{R}}^{N}$ and assume that the coefficients of such vector fields have bounded partial derivatives (of any order). We say that the fields {*V*_{i}:*i*=0,…,*d*} satisfy the UFG condition if there exists $m\in \mathbb{N}$ such that for any $\alpha \in \mathcal{A}$ of the form

We emphasize that the set of vector fields appearing in the linear combination on the right-hand side of the above identity does not include *V*_{0}.

### Example 2.2

Consider the following first-order differential operators on ${\mathbb{R}}^{2}$

*V*

_{0},

*V*

_{1}} do not satisfy the Hörmander condition (e.g. there is always a degeneracy at

*x*=0) but they do satisfy the UFG condition with

*m*=4. If the role of the fields is exchanged, i.e. if we set

*V*

_{0},

*V*

_{1}} still satisfy the UFG condition, this time with

*m*=1 (indeed, $[{V}_{0},{V}_{1}]=\mathrm{cos}x{V}_{1}$).

### Note 2.3

Under assumption (2.2) on the functions *φ*, if the UFG condition holds for some $m\in \mathbb{N}$, then it also holds for any $n\ge m,n\in \mathbb{N}$. In other words, if the UFG condition holds for some *m* in $\mathbb{N}$, then for any *V*_{[γ]} with ∥*γ*∥>*m* one has

*φ*

_{γ,β}. For this reason, it is appropriate to remark that in the remainder of the paper, when we assume that ‘the UFG condition is satisfied for some

*m*’, we mean the smallest such

*m*.

In this paper, we consider diffusion semigroups ${\{{\mathcal{P}}_{t}\}}_{t\ge 0}$ of the form (1.2); that is, we consider Markov semigroups associated with the stochastic dynamics (1.1). In particular, we will be interested in studying the semigroup ${\mathcal{P}}_{t}$ when the vector fields {*V*_{0},*V*_{1},…,*V*_{d}} satisfy the UFG condition. We recall that a semigroup ${\mathcal{P}}_{t}$ of bounded operators is Markov if

The UFG condition is strictly weaker than the uniform Hörmander condition [17]. However, one can still prove that, when such a condition is satisfied by the vector fields {*V*_{0},*V*_{1},…,*V*_{d}} appearing in the generator (1.4), the semigroup ${\mathcal{P}}_{t}$ still enjoys good smoothing properties: if *f*(*x*) is continuous, then $({\mathcal{P}}_{t}f)(x)$ is differentiable in all the directions spanned by the vector fields contained in ${\mathfrak{R}}_{m}$ (we recall that the set ${\mathfrak{R}}_{m}$ is defined in (2.1)).^{2}

Moreover, while the function $u(t,x):=({\mathcal{P}}_{t}f)(x)$ may not be differentiable in the direction *V*_{0} or in the time variable, it is still differentiable in the direction $\mathcal{V}:={\mathrm{\partial}}_{t}-{V}_{0}$, and it is the unique classical solution of the Cauchy problem

*f*is continuous and bounded. For the reader’s convenience, we include in appendix A the definition of classical solution for the PDE (2.3).

Suppose now, and for the remainder of this section, that the operators {*V*_{0},*V*_{1},…,*V*_{d}} satisfy the UFG condition for some *m*>0. We can then construct the vector field **V**, containing all the vector fields (operators) *V*_{[α]}, $\alpha \in {\mathcal{A}}_{m}$:

**V**has $|{\mathcal{A}}_{m}|$ entries; using the notation (1.3), each entry (i.e. each vector

*V*

_{[α]}, $\alpha \in {\mathcal{A}}_{m}$) can be expressed as follows:

**V**can be rewritten as

**V**as a function from ${\mathbb{R}}^{N}$ to ${\mathbb{R}}^{N|{\mathcal{A}}_{m}|}$. However, we will most often think of

**V**as a vector of operators rather than as a vector of vectors and, therefore, we will adopt the notation (2.4). More in general, the space of vectors with $|{\mathcal{A}}_{m}|$ entries, where each entry is an operator in span$\{{\mathfrak{R}}_{m}\}$, will be denoted by ${\mathfrak{R}}^{|{\mathcal{A}}_{m}|}$. Clearly, $\mathbf{\text{V}}\in {\mathfrak{R}}^{|{\mathcal{A}}_{m}|}$.

We emphasize that if $\mathbf{\text{X}}\in {\mathfrak{R}}^{|{\mathcal{A}}_{m}|}$, then **X** will always be denoted in bold font while the component of **X** corresponding to the multi-index *α* is simply a differential operator and it is, therefore, denoted by *X*_{[α]}. If *V*_{j} is any first-order differential operator, we also write

*α*is of length

*k*, we will denote it by

*α*

_{k}(when we want to emphasize its length) and

*V*

_{[αk]}will be the corresponding first-order operator of length

*k*(obviously, for a given $k\in \mathbb{N}$, there are many multi-indices of length

*k*and, correspondingly, many operators of length

*k*). With this more detailed notation, the quadratic form

*Γ*can equivalently be expressed as

*f*is any smooth enough function, then the quadratic form

*Γ*can be rewritten as

*j*≥0, we also define the linear mappings ${\mathit{\Lambda}}_{j},\mathit{\Lambda}:\mathrm{span}\{{\mathfrak{R}}_{m}\}\to \mathrm{span}\{{\mathfrak{R}}_{m}\}$ as follows:

*Λ*

_{j}also the linear mapping ${\mathit{\Lambda}}_{j}:{\mathfrak{R}}^{|{\mathcal{A}}_{m}|}\to {\mathfrak{R}}^{|{\mathcal{A}}_{m}|}$ that acts on the component [

*α*] of the vector

**V**as follows:

*Λ*as well.

### Note 2.4

In the view of proposition 3.1, we remark that all the objects defined so far, in particular the quadratic form *Γ* and the maps *Λ* and *Λ*_{j}, make sense, at least formally, irrespective of whether the UFG condition holds. In other words, the integer *m* appearing in the definitions of such objects could be any integer. Obviously, when the UFG condition holds with *m*, then all such definitions become meaningful for our purposes.

If the UFG condition holds with *m*, then we will denote by ‘Pol’ the set of functions *f* which are differentiable in the directions ${V}_{[\alpha ]},\alpha \in {\mathcal{A}}_{m},$ (but not necessarily in other directions) and such that

*κ*,

*q*>0. When, given a function

*f*∈

*Pol*, we want to stress the value of the constant

*κ*such that the above holds, we write

*f*∈

*Pol*(

*κ*).

We conclude this section by gathering some preliminary basic facts that we will repeatedly use in the remainder of the paper and by presenting a simple example to illustrate the notation introduced so far.

— If

*X*,*Y*and*Z*are any three first-order differential operators, then$$[X,YZ]=[X,Y]Z+Y[X,Z].$$— If $\mathcal{L}$ is the operator (1.4), using the above we find that for any vector field

*V*_{[α]}:$$\begin{array}{rl}[{V}_{[\alpha ]},\mathcal{L}]& ={V}_{[\alpha \ast 0]}+\sum _{j=1}^{d}{V}_{[\alpha \ast j\ast j]}+2\sum _{j=1}^{d}{V}_{j}{V}_{[\alpha \ast j]}\\ & =\mathit{\Lambda}{V}_{[\alpha ]}+2\sum _{j=1}^{d}{V}_{j}{\mathit{\Lambda}}_{j}{V}_{[\alpha ]}.\end{array}$$2.11

### Example 2.5 (UFG–Heisenberg Lie algebra).

We call this example the UFG–Heisenberg Lie algebra, as it is obtained by a modification of the so-called Heisenberg Lie algebra (which is the Lie algebra of vector fields that are invariant with respect to the action of the Heisenberg group on ${\mathbb{R}}^{3}$, see [18]). More precisely, set *d*=2 and *N*=3 and consider the operators

*X*

_{0},

*X*

_{1},

*X*

_{2}} is usually referred to as the Heisenberg Lie algebra. The vector fields {

*X*

_{0},

*X*

_{1},

*X*

_{2}} satisfy the Hörmander condition hence the operator $\mathcal{L}={X}_{0}+{X}_{1}^{2}+{X}_{2}^{2}$ is hypoelliptic on ${\mathbb{R}}^{3}$. If the above fields are slightly modified, we obtain new vector fields, {

*V*

_{0},

*V*

_{1},

*V*

_{2}}, that no longer satisfy the Hörmander condition, but satisfy the UFG condition instead. Indeed, let again

*d*=2 and

*N*=3 and consider the operators

*V*

_{0},

*V*

_{1},

*V*

_{2}} satisfy the UFG condition with

*m*=2, as

*V*

_{21}:=

*V*

_{[2*1]}=−

*V*

_{12},

*V*

_{21}does not need to be in the list of the base fields of ${\mathfrak{R}}_{2}$ ( for the same reason, it can also be omitted in the definition of

*Γ*below, as the constants

*a*

_{1},

*a*

_{2},

*a*

_{12}are anyway arbitrary). Using definition (2.5), the quadratic form

*Γ*associated with the UFG-Heisenberg group is

**V**is

**V**=(

*V*

_{1},

*V*

_{2},

*V*

_{12}) and the mappings

*Λ*

_{1}and

*Λ*

_{2}give

*Λ*

_{0}we have

### 3. Preliminary results: a Bakry-type approach

In this section, we consider Markov semigroups associated with operators $\mathcal{L}$ of the form (1.4), for a given set {*V*_{0},*V*_{1},…,*V*_{d}} of vector fields. We recall that the class of functions ‘Pol’ has been defined immediately after note 2.4.

### Proposition 3.1

*Let* ${f}_{t}:={\mathcal{P}}_{t}{f}_{0}$ *be the diffusion semigroup defined in* (*1.2*).

(*a*) *Let m be any positive integer and assume the initial datum f*_{0} *is a bounded smooth* (*in every direction*) *function such that* $\parallel {V}_{[\alpha ]}\hspace{0.17em}{f}_{0}{\parallel}_{\mathrm{\infty}}<\mathrm{\infty}$ *for all* $\alpha \in {\mathcal{A}}_{m}$. *Consider the quadratic form Γ defined in* (*2.5*):

*for some strictly positive constants*${\{{a}_{[\alpha ]}\}}_{\{\alpha \in {\mathcal{A}}_{m}\}}$ (

*to be chosen*).

*Suppose there exists*λ>0

*such that the following inequality holds*:

*Then*

*therefore*,

(*b*) *Suppose, in addition, that the vector fields* {*V*_{0},…,*V*_{d}} *satisfy the UFG condition* (*for some m*). *In this case, if* (*3.1*) *holds when f*_{0} *is smooth* (*and* $\parallel {V}_{[\alpha ]}\hspace{0.17em}{f}_{0}{\parallel}_{\mathrm{\infty}}<\mathrm{\infty}$), *then the following holds for any f*_{0}∈*Pol*(*κ*): *for every open ball* $\mathbb{B}(0,K)$ *of radius K and for all* $\alpha \in {\mathcal{A}}_{m}$,

*where*

*c*

_{K}>0

*is a constant dependent on*

*K*.

(*c*) *If the vector fields* {*V*_{0},…,*V*_{d}} *satisfy the UFG condition* (*for some m*) *and* (*3.1*) *is satisfied for any smooth initial datum, then then following holds when* *f*_{0} *is only continuous and bounded* (*but not necessarily smooth*): *for any* *t*_{0}∈(0,1) *and any K*>0 *there exists a constant* *c*_{t0,K}>0 *such that*

*Moreover, if the coefficients of the vector fields*{

*V*

_{0},…,

*V*

_{d}}

*are bounded, then the constant*

*c*

_{t0,K}

*does not depend on*

*K and we have the uniform bound*

Before proving the above result, we make the following remark, which we will use in the proof of proposition 3.1. We will make several comments on the above statement in note 3.3.

### Note 3.2

If the initial datum *f*_{0} is bounded and continuous and the UFG condition holds, then $({\mathcal{P}}_{t}{f}_{0})$ is differentiable in the directions ${V}_{[\alpha ]},\hspace{0.17em}\alpha \in {\mathcal{A}}_{m}$ [2–5]. Because we are assuming that the vector fields {*V*_{0},…,*V*_{d}} are smooth, the semigroup is differentiable an arbitrary number of times in such directions. Moreover, the following short-time asymptotic holds: for any ball of radius *K*, $\mathbb{B}(0,K)$ and for any $\alpha \in {\mathcal{A}}_{m}$,

*x*,

*t*or

*f*

_{0}). Details about the above short-time asymptotics (and many other results of this type) can be found in [7] (see in particular [7], pp. 68–80). Furthermore, when the vector fields

*V*

_{[α]}have bounded coefficients, the following holds:

### Proof of proposition 3.1.

(a) This is completely standard: By applying Gronwall’s lemma, from (3.1) we deduce

$${\mathcal{P}}_{t-s}\mathit{\Gamma}(\hspace{0.17em}{f}_{s})\le {\mathrm{e}}^{-\mathrm{\lambda}s}{\mathcal{P}}_{t}\mathit{\Gamma}(\hspace{0.17em}{f}_{0}),\phantom{\rule{1em}{0ex}}\mathit{for\; all}0s\le t.$$3.8Therefore, using (3.8) for*s*=*t*and the contractivity of the semigroup ${\mathcal{P}}_{t}$ in the supremum norm gives the result. Note, in particular, that if*f*_{0}is smooth in every direction, then also $({\mathcal{P}}_{t}{f}_{0})(x)$ is smooth in every direction (see appendix A); in particular, it is smooth in*t*as well, hence all of the above is justified.(b) We prove this statement in appendix A, see lemma A.5.

(c) Using note 3.2, note that for any

*t*_{0}∈(0,1) the function ${\mathcal{P}}_{t}{f}_{0}$ belongs to the set*Pol*(*κ*); in particular, by (3.6), the constant*κ*appearing in (2.10) is, for this function, $\kappa ={t}_{0}^{-m}\mathfrak{c}\parallel {f}_{0}(x){\parallel}_{\mathrm{\infty}}$. Therefore, by part (b), for any fixed 0<*t*_{0}<1 and for any*t*≥*t*_{0}, we can write$$|{V}_{[\alpha ]}({\mathcal{P}}_{t}{f}_{0}(x)){|}^{2}=|{V}_{[\alpha ]}({\mathcal{P}}_{t-{t}_{0}}{\mathcal{P}}_{{t}_{0}}{f}_{0}(x)){|}^{2}\le {\mathrm{e}}^{-\mathrm{\lambda}(t-{t}_{0})}{c}_{{t}_{0},K}\parallel {f}_{0}(x){\parallel}_{\mathrm{\infty}}.$$If the coefficients of the*V*_{[α]}’s are bounded, then (3.7) gives (3.5) by acting analogously to what we have just done. ▪

### Note 3.3

Proposition 3.1 part (a) provides a general framework to deduce the exponential decay for the derivatives of diffusion semigroups; part (a) is just the classic Bakry approach [14,15], readapted to our purposes. In particular:

— Proposition 3.1 part (a) is

*not*a smoothing result, it is just a long-time asymptotics. Indeed in the statement of part (a), we assumed that the initial datum*f*_{0}(*x*) is a smooth function with bounded derivatives. This is to make sense of the expression (*Γf*_{0})(*x*) and to be able to take time-derivatives in (3.1). Such a result is quite general and it is independent of whether the UFG condition holds (see also note 2.4 in this respect).— Once the exponential decay (3.2) is obtained for smooth initial data, one can use the semigroup property and the smoothing effects which are guaranteed to hold under the UFG condition (and quantified by the estimates (3.6) and (3.7)) in order to prove exponential decay of the derivatives of the semigroup for any initial datum

*f*_{0}(*x*) which is just continuous and bounded. This is the content of part ((b) and) (c) of proposition 3.1. Therefore, in the proof of our main results, we just need to focus on showing exponential decay for smooth initial data.— The analysis used here is based on the adaption of the Bakry technique used in [16]. The difference between the quadratic forms

*Γ*that we use here and those considered in [16] is the appearance of the constants*a*_{[αk]}. That is, the quadratic form used in [16] can be obtained from ours by just setting*a*_{[αk]}=1 for all*a*_{[αk]}. Introducing the positive parameters*a*_{[αk]}, which can be conveniently chosen, allows us to have a better estimate for the decay rate λ (see note 5.1). To the best of our knowledge, the idea of introducing such parameters first appeared in [19] and was then further developed in [20]. However [19,20] work in weighted spaces, the weight being the invariant measure of the semigroup. Here, there is no obvious invariant measure to exploit, hence we have to work in a pointwise setting, similar to [16].

The result of proposition 3.1 part (a) hinges only on proving (3.1). The following elementary lemma gives a sufficient condition to verify (3.1). Before stating the next lemma we observe that, with our assumptions on the coefficients of the SDE (1.1), classic arguments show that the operator $\mathcal{L}$ and the semigroup commute on a set of sufficiently smooth functions (say, for example, on the set ${C}_{V}^{\mathrm{\infty}}$, defined just before definition 2.1).

### Proposition 3.4.

*Assume the same setting of proposition 3.1 part* (*a*). *If there exists a real number* λ>0 *such that*

*then*(

*3.1*)

*holds*.

### Proof.

This is again standard so we only sketch it.

### 4. Main results: long-time behaviour of derivatives of the semigroup

If *X* is a first-order differential operator on ${\mathbb{R}}^{N}$, $\mathcal{L}$ is the operator (1.4) and ${f}_{t}(x):=({\mathcal{P}}_{t}{f}_{0})(x),$ then

*f*

_{0}is smooth. The identity (4.1) is obtained by using (2.3) and the fact that

*X*and all the

*V*

_{j}’s are first-order differential operators (see [21], Lemma 2.2). Recall that if a multi-index

*α*is of length

*k*we will denote it by

*α*

_{k}. In the view of (3.9), we use (4.1) to calculate the following:

Note that if *V*_{[αk]} is a field of length *k*, from (2.11) one can see that the commutator between *V*_{[αk]} and $\mathcal{L}$ will contain the second-order operators summed up with the first-order operators of length at most *k*+2. For this reason, when we calculate the commutators $[{V}_{{\alpha}_{m}},\mathcal{L}]$ and $[{V}_{{\alpha}_{m-1}},\mathcal{L}]$, we can make use of the UFG condition and express such commutators in terms of fields of length at most *m*. This fact will be repeatedly used in the proofs of §5.

We now split the above expression as follows:

### Theorem 4.1

*Let m be a positive integer and* ${\mathcal{P}}_{t}{f}_{0}=:{f}_{t}$ *be the semigroup associated with the SDE* (*1.1*), *i.e. the semigroup* (*1.2*). *With the notation introduced so far, assume the following two conditions are satisfied by the vector fields* {*V*_{0}*,V*_{1}*,…,V*_{d}} *appearing in* (*1.1*):

—

*there exists a collection of strictly positive constants*${\{{a}_{[\alpha ]}\}}_{\alpha \in {\mathcal{A}}_{m}}$*such that the corresponding bilinear form*(*2.6*)*satisfies*$$\mathcal{S}(\hspace{0.17em}{f}_{t})=-2\sum _{j=1}^{d}\parallel {V}_{j}\mathbf{\text{V}}{f}_{t}{\parallel}_{{\mathcal{A}}_{m}}^{2}+4\sum _{j=1}^{d}{\u27e8{V}_{j}{\mathit{\Lambda}}_{j}\mathbf{\text{V}}{f}_{t},\mathbf{\text{V}}{f}_{t}\u27e9}_{{\mathcal{A}}_{m}}\le \gamma \parallel \mathbf{\text{V}}{f}_{t}{\parallel}_{{\mathcal{A}}_{m}}^{2},$$4.6*for some constant γ*>0 (*possibly dependent on the collection*${\{{a}_{[\alpha ]}\}}_{\alpha \in {\mathcal{A}}_{m}}$*);*—

*there exists μ*>*γ such that*$$\mathcal{F}(\hspace{0.17em}{f}_{t})=2{\u27e8\mathit{\Lambda}\mathbf{\text{V}}{f}_{t},\mathbf{\text{V}}{f}_{t}\u27e9}_{{\mathcal{A}}_{m}}\le -\mu \parallel \mathbf{\text{V}}{f}_{t}{\parallel}_{{\mathcal{A}}_{m}}^{2},$$4.7*where*${\u27e8\cdot ,\cdot \u27e9}_{{\mathcal{A}}_{m}}$*is the bilinear form defined by the same constants for which*(*4.6*)*holds.*

*then* (*3.9*) *holds with* λ=*μ*−*γ. Therefore*, (*3.2*) *holds for any smooth initial datum.*

### Proof of theorem 4.1.

Trivially, from (4.2), (4.4)–(4.7) and recalling the notation (2.7):

If we divide both sides of (4.7) by $\parallel \mathbf{\text{V}}{f}_{t}{\parallel}_{{\mathcal{A}}_{m}}^{2}$, then it becomes clear that imposing condition (4.7) is equivalent to requiring that (there exists a bilinear form on ${\mathfrak{R}}^{|{\mathcal{A}}_{m}|}\simeq {\mathbb{R}}^{N|{\mathcal{A}}_{m}|}$ such that) the ‘angle’ between the vectors *Λ***V***f*_{t} and **V***f*_{t} is obtuse.

We now establish conditions under which (4.6) and (4.7) hold.

### Theorem 4.2

*Let* {*V*_{i}:*i*=0,…,*d*} *be the vector fields appearing in* (*1.1*). *Then the following holds*:

(

*i*)*If the vector fields*{*V*_{i}:*i*=0,…,*d*}*satisfy the UFG condition for some*$m\in \mathbb{N}$*, then there exists a choice of the constants*${\{{a}_{[\alpha ]}\}}_{\alpha \in {\mathcal{A}}_{m}}$*such that*(*4.6*)*is satisfied*.(

*ii*)*Suppose the assumption of the above point*(*i*)*is satisfied and assume that there exists a real number*λ_{0}>0*such that, for every*$\alpha \in {\mathcal{A}}_{m}$*and every smooth enough function f,*$$({V}_{[\alpha ]}\hspace{0.17em}f)([{V}_{[\alpha ]},{V}_{0}]f)\le -{\mathrm{\lambda}}_{0}|{V}_{[\alpha ]}f{|}^{2}.$$4.8*If*λ_{0}*is big enough then*(*4.7*)*holds with μ*=λ_{0}.*Hence there exists*λ>0*such that*(*3.2*)*holds for any smooth initial datum. Therefore, by proposition 3.1 part*(*c*),*if the initial datum f*_{0}*is continuous and bounded, then for any t*_{0}∈(0,1)*and any K*>0*there exists a constant c*_{t0,K}>0*such that*$$\underset{x\in \mathbb{B}(0,K)}{sup}|{V}_{[\alpha ]}\hspace{0.17em}{f}_{t}(x){|}^{2}\le {c}_{{t}_{0},K}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{\lambda}(t-{t}_{0})}\parallel {f}_{0}(x){\parallel}_{\mathrm{\infty}}\phantom{\rule{1em}{0ex}}\mathit{for\; all}\alpha \in {\mathcal{A}}_{m}\hspace{0.17em}\mathit{and\; all}\hspace{0.17em}t{t}_{0}.$$*If the coefficients of the vector fields*{*V*_{0},…,*V*_{d}}*are bounded, then*$$|{V}_{[\alpha ]}\hspace{0.17em}{f}_{t}(x){|}^{2}\le {c}_{{t}_{0}}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{\lambda}(t-{t}_{0})}\parallel {f}_{0}(x){\parallel}_{\mathrm{\infty}}\phantom{\rule{1em}{0ex}}\mathit{for\; all}x\in {\mathbb{R}}^{N},\alpha \in {\mathcal{A}}_{m}\hspace{0.17em}\mathit{and\; all}\hspace{0.17em}t{t}_{0}.$$4.9

### Note 4.3

Let us clarify the statement of theorem 4.2. According to part (i) of theorem 4.2, if the UFG condition holds then one can fix a bilinear form ${\u27e8\cdot ,\cdot \u27e9}_{{\mathcal{A}}_{m}}$ such that (4.6) holds. In the statement of part (ii) of the theorem, we intend (4.7) to be satisfied for the same bilinear form. An explicit estimate on how big λ_{0} is will be given in the proof, see (5.10). Obviously, the estimate (5.10) is quite general and can be made more precise when explicit knowledge of the functions *φ*’s appearing in the UFG condition is available. We also remark that (4.8) is a slight generalization of the so-called *dilation condition*, which has been considered in the literature for elliptic and hypoelliptic semigroups (see [16], Section 2 and references therein). More generally, (4.8) replaces in a quantitative way the exact dilation structure of stratified Lie groups. Still regarding (4.8), note that one cannot expect that the UFG condition alone could yield exponential decay of the derivatives of the semigroup (as we have already pointed out in the Introduction, if $\mathcal{L}$ is uniformly elliptic, then it satisfies the UFG condition, but not every elliptic dynamics has derivatives that decay exponentially fast), therefore some quantitative condition on the vector fields has to be imposed.

Let us present some examples of UFG generators that satisfy the assumptions of theorem 4.2, in particular condition (4.8).

### Example 4.4 (UFG–Grušin Plane).

Let *d*=1 and *N*=2, i.e. consider the operator $\mathcal{L}={V}_{0}+{V}_{1}^{2}$ on ${\mathbb{R}}^{2}$, with

*V*

_{0},

*V*

_{1}} satisfy the UFG condition with

*m*=1, as [

*V*

_{1},

*V*

_{0}]=−

*kV*

_{1}. It is easy to see that (3.9) holds with λ=2

*k*( for every

*k*>0). Indeed, by direct calculation

### Example 4.5

The operators {*V*_{0},*V*_{1},*V*_{2}} defined in example 2.5 satisfy the assumptions of theorem 4.2. In particular in this case, one can obtain the following result, the proof of which can be found in §5.

### Lemma 4.6

*Let d*=2 *and consider the operator* $\mathcal{L}$ *of the form* (*1.4*) *acting on* ${\mathbb{R}}^{3}$, *where the fields* *V*_{0},*V*_{1},*V*_{2} *are those defined in example 2.5. With the notation introduced in example 2.5, we have that for every* *k*>0, (*3.9*) *holds with* λ=*k*, *i.e*.

We include the proof of the above lemma in §5 for two reasons: (i) to show on a simple example how the proof of theorem 4.2 works in practice, without all the cumbersome notation that one needs to prove the result in general; (ii) to show that, thanks to the freedom to choose the constants appearing in the definition of *Γ* (see note 5.1), the general lower bound for λ given in (5.10) can be improved when we explicitly know the functions *φ*’s deriving from the UFG condition.

### Note 4.7

The quadratic form *Γ*( *f*_{t}) includes the derivatives of the semigroup but not the semigroup *f*_{t} itself. Therefore, the results of this paper only give information on the behaviour of the derivatives; in §4a, we use such results to obtain some (partial) information on the asymptotic behaviour of the semigroup ${f}_{t}={\mathcal{P}}_{t}f$. An analogous observation holds for the derivatives in the direction *V*_{0}. Note that our result does not imply anything regarding the behaviour in the direction *V*_{0}, as *V*_{0} is not contained in the definition of *Γ*. This is again a structural fact. Indeed, under just the UFG condition, one is not even guaranteed differentiability in the direction *V*_{0}, let alone decay, see [6], Section 2.9. However, it was proved that under the so-called * V_{0}-condition* (see definition 4.8), the semigroup ${\mathcal{P}}_{t}$ is also differentiable in the direction

*V*

_{0}. In this case, our results also cover such a direction.

### Definition 4.8 (*V*_{0}-condition).

With the notation introduced so far, we say that the *V*_{0}-condition is satisfied if there exist functions ${\phi}_{\beta}\in {C}_{V}^{\mathrm{\infty}}$ such that

### Corollary 4.9

*Suppose that the assumptions of theorem 4.1 are satisfied. If the* *V*_{0} *condition holds, then there exist positive constants* *c*,λ>0 *such that*

*say for any smooth*

*f*

_{0}.

Finally, we observe, although without proof, that the same strategy used in this paper can be adapted to obtain estimates on the derivatives of any order along the semigroup. This can be done inductively (on the order of the derivative) using, at step *n* of the induction, the quadratic form

*Γ*

^{(n)}contains all the derivatives of order at most

*n*, in all the directions contained in ${\mathfrak{R}}_{m}$. A similar inductive procedure has been used, for hypoelliptic semigroups of hypocoercive type, in [21,22].

#### (a) Decay of the semigroup

Let *x* and *y* be two points in ${\mathbb{R}}^{N}$ and, for some given $\alpha \in {\mathcal{A}}_{m}$, suppose there exists an integral curve of *V*_{[α]} joining *x* and *y*. That is, suppose there exists $\eta (\tau ):[0,1]\to {\mathbb{R}}^{N}$ such that

*V*

_{[α]}has to be intended as a vector field rather than as a differential operator. More generally, we say that

*y*is reachable from

*x*, and write

*x*∼

*y*, if there exists an integer

*M*>0 and

*M*points in ${\mathbb{R}}^{N}$,

*z*

_{1},…,

*z*

_{M}, such that

*z*

_{1}=

*x*,

*z*

_{M}=

*y*and for every

*i*=1,…,

*M*−1, there exists an ${\alpha}^{(i)}\in {\mathcal{A}}_{m}$ such that the integral curve of

*V*

_{[α(i)]}is well defined and joins

*z*

_{i}with

*z*

_{i+1}. The relation ∼ is an equivalence relation. We denote by ${\mathcal{U}}_{x}$ the set of points reachable from

*x*(clearly, if

*y*∼

*x*then ${\mathcal{U}}_{x}={\mathcal{U}}_{y}$).

#### Corollary 4.10

*Let* ${\mathcal{P}}_{t}$ *be the semigroup* (*1.2*) *and assume for simplicity that the fields V*_{[α]} *have bounded coefficients. Suppose that the assumptions of theorem 4.2 hold. Then for any f*(*x*) *continuous and bounded, for any* $x\in {\mathbb{R}}^{N}$ *and for any* $y\in {\mathcal{U}}_{x}$ *there exists* λ>0 *such that*

*where c*>0

*is a constant independent of t*.

#### Proof.

We just need to prove the result for *M*=1 (that is, when *y* can be reached from *x* moving along the integral curve of one of the *V*_{[α]}’s). For any fixed *t*>0, by definition of directional derivative we have

### 5. Proofs of main results

Throughout this section, if *φ*(*x*) is a function, we denote

*α*of length at most

*m*, we have

### Proof of part (i) of theorem 4.2.

The case *m*=1 is straightforward and can be dealt with directly, so throughout the proof we take *m*>1. Looking at (4.3), note that if ∥*α*_{k}∥=*m* then ∥*α*_{k}**j*∥=*m*+1 when *j*∈{1,…,*d*}, so we can apply the UFG condition to the operator *V*_{[αk*j]}. So from (4.3) we obtain

Let us now set, for any multi-index $\gamma \in {\mathcal{A}}_{m}$,

*Terms with*△: For each ${\alpha}_{m}\in {\overline{\mathcal{A}}}_{m}$,$$\begin{array}{rl}& 4{a}_{[{\alpha}_{m}]}\sum _{j=1}^{d}\sum _{\beta \in {\mathcal{A}}_{m}}{\phi}_{{\alpha}_{m}\ast j,\beta}({V}_{j}{V}_{[\beta ]}\hspace{0.17em}{f}_{t})({V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t})\\ & \phantom{\rule{1em}{0ex}}\le 2\sum _{j=1}^{d}\sum _{\beta \in {\mathcal{A}}_{m}}[{\mathbf{\text{1}}}_{\{{\mathcal{J}}_{{\alpha}_{m}}\ne 0\}}|{V}_{j}{V}_{[\beta ]}\hspace{0.17em}{f}_{t}{|}^{2}+{a}_{[{\alpha}_{m}]}^{2}{\mathcal{J}}_{{\alpha}_{m}}^{2}|{V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t}{|}^{2}]\\ & \phantom{\rule{1em}{0ex}}=2{a}_{[{\alpha}_{m}]}^{2}{\mathcal{J}}_{{\alpha}_{m}}^{2}d|{\mathcal{A}}_{m}||{V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t}{|}^{2}+2{\mathbf{\text{1}}}_{\{{\mathcal{J}}_{{\alpha}_{m}}\ne 0\}}\sum _{j=1}^{d}\sum _{\beta \in {\mathcal{A}}_{m}}|{V}_{j}{V}_{\beta}{f}_{t}{|}^{2}\end{array}$$*Terms***: For each ${\alpha}_{m}\in {\overline{\mathcal{A}}}_{m}$, we have$$\begin{array}{rl}& 4{a}_{[{\alpha}_{m}]}\sum _{j=1}^{d}\sum _{\beta \in {\mathcal{A}}_{m}}{\phi}_{{\alpha}_{m}\ast j,\beta}^{j}({V}_{[\beta ]}\hspace{0.17em}{f}_{t})({V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t})\\ & \phantom{\rule{1em}{0ex}}\le 4{a}_{[{\alpha}_{m}]}\sum _{j=1}^{d}{\phi}_{{\alpha}_{m}\ast j,{\alpha}_{m}}^{j}|{V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t}{|}^{2}+4d{\mathcal{H}}_{{\alpha}_{m}}{a}_{[{\alpha}_{m}]}\sum _{\beta \in {\mathcal{A}}_{m}\mathrm{\setminus}\{{\alpha}_{m}\}}({V}_{[\beta ]}\hspace{0.17em}{f}_{t})({V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t})\\ & \phantom{\rule{1em}{0ex}}\le 4{a}_{[{\alpha}_{m}]}\sum _{j=1}^{d}{\phi}_{{\alpha}_{m}\ast j,{\alpha}_{m}}^{j}|{V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t}{|}^{2}\\ & \phantom{\rule{2em}{0ex}}+2d{\mathbf{\text{1}}}_{\{{\mathcal{H}}_{{\alpha}_{m}}\ne 0\}}\sum _{\beta \in {\mathcal{A}}_{m}\mathrm{\setminus}\{{\alpha}_{m}\}}|{V}_{[\beta ]}\hspace{0.17em}{f}_{t}{|}^{2}+2d{a}_{[{\alpha}_{m}]}^{2}{\mathcal{H}}_{{\alpha}_{m}}^{2}(|{\mathcal{A}}_{m}|-1)|{V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t}{|}^{2}\end{array}$$*Terms ⋆*: for every*k*=1,…,*m*−1,$$4{a}_{[{\alpha}_{k}]}\sum _{j=1}^{d}({V}_{j}{V}_{[{\alpha}_{k}\ast j]}\hspace{0.17em}{f}_{t})({V}_{[{\alpha}_{k}]}\hspace{0.17em}{f}_{t})\le 2d|{V}_{[{\alpha}_{k}]}\hspace{0.17em}{f}_{t}{|}^{2}+\sum _{j=1}^{d}2{a}_{[{\alpha}_{k}]}^{2}|{V}_{j}{V}_{[{\alpha}_{k}\ast j]}\hspace{0.17em}{f}_{t}{|}^{2}$$5.3

Putting the above estimates together, after setting

*k*=1,…,

*m*−1

*a*

_{[α1]}have been fixed, all the other coefficients can be chosen through the above recursive relation. This choice allows to fix all the constants in the expression for the quadratic form

*Γ*. Assuming that any choice satisfying (5.6) has been made, one then has

### Proof of part (ii) of theorem 4.2.

For simplicity, suppose *m*>3. The case *m*≤3 can be studied analogously (and it is in fact less involved). We notice again that if ∥*α*_{k}∥=*m*−1 (*m*, respectively) then ∥*α*_{k}**j***j*∥=*m*+1 (*m*+2, respectively). Therefore, we can again apply the UFG condition to the vector fields *V*_{[αk*j*j]} (appearing in (4.3) and (4.4)) when ∥*α*_{k}∥=*m*−1 or *m*, obtaining

*Terms with **: For each ${\alpha}_{m}\in {\overline{\mathcal{A}}}_{m}$,$$\begin{array}{rl}& 2{a}_{[{\alpha}_{m}]}\sum _{j=1}^{d}\sum _{\beta \in {\mathcal{A}}_{m}}{\phi}_{{\alpha}_{m}\ast j\ast j,\beta}({V}_{[\beta ]}\hspace{0.17em}{f}_{t})({V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t})\\ & \phantom{\rule{1em}{0ex}}\le 2{a}_{[{\alpha}_{m}]}\sum _{j=1}^{d}{\phi}_{{\alpha}_{m}\ast j\ast j,{\alpha}_{m}}|{V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t}{|}^{2}+2d{\mathcal{I}}_{{\alpha}_{m}}{a}_{[{\alpha}_{m}]}\sum _{\beta \in {\mathcal{A}}_{m}\mathrm{\setminus}\{{\alpha}_{m}\}}({V}_{[\beta ]}\hspace{0.17em}{f}_{t})({V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t})\\ & \phantom{\rule{1em}{0ex}}=2{a}_{[{\alpha}_{m}]}\sum _{j=1}^{d}{\phi}_{{\alpha}_{m}\ast j\ast j,{\alpha}_{m}}|{V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t}{|}^{2}\\ & \phantom{\rule{2em}{0ex}}+d\sum _{\beta \in {\mathcal{A}}_{m}\mathrm{\setminus}\{{\alpha}_{m}\}}[{a}_{[{\alpha}_{m}]}^{2}{\mathcal{I}}_{{\alpha}_{m}}^{2}|{V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t}{|}^{2}+|{V}_{[\beta ]}\hspace{0.17em}{f}_{t}{|}^{2}{\mathbf{\text{1}}}_{\{{\mathcal{I}}_{{\alpha}_{m}}\ne 0\}}]\\ & \phantom{\rule{1em}{0ex}}=2{a}_{[{\alpha}_{m}]}\sum _{j=1}^{d}{\phi}_{{\alpha}_{m}\ast j\ast j,{\alpha}_{m}}|{V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t}{|}^{2}\\ & \phantom{\rule{2em}{0ex}}+d{a}_{[{\alpha}_{m}]}^{2}{\mathcal{I}}_{{\alpha}_{m}}^{2}|{V}_{[{\alpha}_{m}]}\hspace{0.17em}{f}_{t}{|}^{2}(|{\mathcal{A}}_{m}|-1)+d\hspace{0.17em}{\mathbf{\text{1}}}_{\{{\mathcal{I}}_{{\alpha}_{m}}\ne 0\}}\sum _{\beta \in {\mathcal{A}}_{m}\mathrm{\setminus}\{{\alpha}_{m}\}}|{V}_{[\beta ]}\hspace{0.17em}{f}_{t}{|}^{2}\end{array}$$*Terms*$\u25fb$: For each ${\alpha}_{m-1}\in {\overline{\mathcal{A}}}_{m-1}$,$$\begin{array}{rl}& 2{a}_{[{\alpha}_{m-1}]}\sum _{j=1}^{d}\sum _{\beta \in {\mathcal{A}}_{m}}{\phi}_{{\alpha}_{m-1}\ast j\ast j,\beta}({V}_{[\beta ]}\hspace{0.17em}{f}_{t})({V}_{[{\alpha}_{m-1}]}\hspace{0.17em}{f}_{t})\\ & \phantom{\rule{1em}{0ex}}\le 2{a}_{[{\alpha}_{m-1}]}\sum _{j=1}^{d}{\phi}_{{\alpha}_{m-1}\ast j\ast j,{\alpha}_{m-1}}|{V}_{[{\alpha}_{m-1}]}\hspace{0.17em}{f}_{t}{|}^{2}\\ & \phantom{\rule{2em}{0ex}}+d{a}_{[{\alpha}_{m-1}]}^{2}{\mathcal{I}}_{{\alpha}_{m-1}}^{2}(|{\mathcal{A}}_{m}|-1)|{V}_{[{\alpha}_{m-1}]}\hspace{0.17em}{f}_{t}{|}^{2}+d{\mathbf{\text{1}}}_{\{{\mathcal{I}}_{{\alpha}_{m-1}}\ne 0\}}\sum _{\beta \in {\mathcal{A}}_{m}\mathrm{\setminus}\{{\alpha}_{m-1}\}}|{V}_{[\beta ]}\hspace{0.17em}{f}_{t}{|}^{2}.\end{array}$$*Terms*$\diamond $: for every*k*=1,…,*m*−2,$$2{a}_{[{\alpha}_{k}]}\sum _{j=1}^{d}({V}_{[\alpha \ast j\ast j]}\hspace{0.17em}{f}_{t})({V}_{[{\alpha}_{k}]}\hspace{0.17em}{f}_{t})\le d|{V}_{[{\alpha}_{k}]}\hspace{0.17em}{f}_{t}{|}^{2}+\sum _{j=1}^{d}{a}_{[{\alpha}_{k}]}^{2}|{V}_{[{\alpha}_{k}\ast j\ast j]}\hspace{0.17em}{f}_{t}{|}^{2}.$$5.7

Overall, one obtains

*k*=1,…,

*m*−2

*a*

_{[αk]}and ℓ

_{αk}, there always exists at least one λ

_{0}>0 satisfying the above. In particular, one can choose any λ

_{0}such that

_{0}satisfies (5.10), and hence (5.9), from (5.8) one has

### Proof of lemma 4.6.

Consider the quadratic form

*ϵ*=

*a*

_{1}for the first inequality and

*ϵ*=

*a*

_{2}in the second), we can estimate the terms on the last line as

*a*

_{1}and

*a*

_{2}such that

*a*

_{12}such that −4

*a*

_{12}

*k*≤−

*a*

_{12}

*k*, which is true, e.g. for any

*a*

_{12}>1. Then, looking at the terms $\mathrm{\u2661}$, we choose

*a*

_{12}much bigger than

*a*

_{1}and

*a*

_{2}, more precisely we choose

*a*

_{12}such that

*k*>0, one can find

*a*

_{1}>0 and

*a*

_{2}>0 such that (5.13) is satisfied, this concludes the proof. ▪

### Note 5.1

If the constants *a*_{1},*a*_{2},*a*_{12} had not been introduced, i.e. if *a*_{1}=*a*_{2}=*a*_{12}= 1, then we would have only been able to prove the result for $k>\frac{1}{2}$ (by making better use of the Young inequality in (5.11) and (5.12)).

### Author contributions

Both authors carried out the work described in this paper and gave final approval for this manuscript.

### Competing interests

We declare we have no competing interests.

### Funding statement

No grants or funding to acknowledge.

## Appendix A

We define here the notion of classical solution *u* of the PDE (1.5). The notion is quite natural: we will require *u* to be continuously differentiable (twice) in the direction of every vector field *V*_{i}, *i*=1,…,*d*. As a consequence of the need that *u* satisfies (1.5), we will also require *u* to be continuously differentiable in the direction ${\mathcal{V}}_{0}={\mathrm{\partial}}_{t}-{V}_{0}$, when viewed as a function (*t*,*x*)↦*u*(*t*,*x*) over the product space $(0,\mathrm{\infty})\times {\mathbb{R}}^{N}$. However, we will not require *u* to be differentiable in either the time direction ∂_{t} or the direction *V*_{0}.

The analysis of *u* hinges on being well approximated by solutions of the PDE (1.5) with smooth initial condition (and therefore smooth for all *t*≥0). The approximation is done in such way that, in the limit, only the differentiability in the directions *V*_{i} *i*=1,…,*d* and ${\mathcal{V}}_{0}={\mathrm{\partial}}_{t}-{V}_{0}$ is preserved, but not that in the time direction ∂_{t} or in the direction *V*_{0}. This is to be expected as the smoothing effect only takes place in the directions *V*_{i}, *i*=1,…,*d*. An extreme case where the UFG condition holds is when all the *V*_{i}, *i*=1,…,*d* are equal to zero. Take, for example, the transport equation

*N*=1,

*V*

_{0}=∂/∂

*x*. In this case, the solution is explicitly given by

*u*(

*t*,

*x*)=

*f*(

*t*+

*x*), $x\in \mathbb{R}$ and

*t*≥0. Obviously should

*f*not be differentiable (choose for example

*f*(

*x*)=|

*x*|, $x\in \mathbb{R}$), we will not expect differentiability in either the time direction ∂

_{t}, or the space direction ∂/∂

*x*. However,

*u*will be differentiable in the direction ${\mathcal{V}}_{0}={\mathrm{\partial}}_{t}-{V}_{0}$. In fact,

*u*is constant in the direction ${\mathcal{V}}_{0}={\mathrm{\partial}}_{t}-{V}_{0}$, as ${\mathcal{V}}_{0}u=0$. In this extreme case, no additional smoothness is gained because of the absence of any second-order differential operator in the PDE (1.5).

At the other end of the spectrum, we have the case when the vector fields *V*_{i}, *i*=0,…,*d*, satisfy the Hörmander condition. In this case, the smoothing effect occurs in every direction. In particular, *u* becomes differentiable in the *V*_{0} direction, and since *u* is differentiable in the direction ${\mathcal{V}}_{0}={\mathrm{\partial}}_{t}-{V}_{0}$, *u* will also be differentiable in the time direction. In this case, the notion of a classical solution defined below coincides with the standard notion of a classical solution.

Finally, we remark that the Hörmander condition is not necessary to ensure that *u* becomes differentiable in the *V*_{0} direction (and therefore also in the time direction). If the vector fields *V*_{i} *i*=0,…,*d* satisfy the UFG condition and *V*_{0} belongs to $\mathcal{A}$ ^{3} , then it is still the case that *u* becomes differentiable in the *V*_{0} direction and in the time direction.

To introduce rigourously the classical solution of the PDE (1.5), we need several spaces of functions, which we come to introduce. For an open ball $\mathbb{B}\subset {\mathbb{R}}^{N}$ and for a function *φ* in ${C}_{V}^{\mathrm{\infty}}(\mathbb{B})$ (that is, for any smooth bounded real-valued function *φ* with bounded derivatives on $\mathbb{B}$ of any order in the directions *V*_{[α]}, $\alpha \in {\mathcal{A}}_{m}$), we set

^{4}More generally, for

*k*>1, we can define by induction

*d*-dimensional ball of centre 0 and radius

*r*. For $v\in {\mathcal{D}}_{V}^{k,\mathrm{\infty}}({\mathbb{R}}^{N})$,

*V*

_{[α1]}…

*V*

_{[αk]}

*v*is understood as the derivative of

*v*in the directions

*V*

_{[α1]}…

*V*

_{[αk]}, with ${\alpha}_{1},\dots ,{\alpha}_{k}\in {\mathcal{A}}_{m}$. Similarly, for $\phi \in {\mathcal{C}}_{V}^{\mathrm{\infty}}(\mathbb{B})$ and

*k*≥0, we set

## Note A.1

Note that any function in ${\mathcal{D}}_{V}^{1,\mathrm{\infty}}({\mathbb{R}}^{N})$ is differentiable along the solutions of the ordinary differential equation ${\dot{\gamma}}_{t}=V({\gamma}_{t})$, *t*≥0, for $V\in {\mathcal{A}}_{m}$. In particular, any function in ${\mathcal{D}}_{V}^{1,\mathrm{\infty}}({\mathbb{R}}^{N})$ is continuously differentiable on ${\mathbb{R}}^{N}$ when the uniform Hörmander condition is satisfied.

To define the notion of a classical solution to (1.5), we will need to introduce the set of functions that are continuously differentiable in the direction ${\mathcal{V}}_{0}={\mathrm{\partial}}_{t}-{V}_{0}$. Again, we proceed via a closure argument. For any *r*≥1 and any time-space function $\phi \in {\mathcal{C}}_{V}^{\mathrm{\infty}}([1/r,r]\times \mathbb{B}(0,r))$ with bounded derivatives of any order, we set

*r*≥1. We are now in position to define a classical solution to the PDE (1.5)

## Definition A.2

We call a function $v=\{v(t,x),(t,x)\in [0,+\mathrm{\infty})\times {\mathbb{R}}^{N}\}$ a *classical* solution of the PDE (1.5) if the following conditions are satisfied:

(1)

*v*belongs to ${\mathcal{D}}_{{\mathcal{V}}_{0}}^{1,\mathrm{\infty}}((0,+\mathrm{\infty})\times {\mathbb{R}}^{N})$ and, for any*t*>0,*v*(*t*,⋅) is in ${\mathcal{D}}_{V}^{2,\mathrm{\infty}}({\mathbb{R}}^{N})$; moreover, for any ${\alpha}_{1},{\alpha}_{2}\in \mathcal{A}$, the function $(t,x)\in (0,+\mathrm{\infty})\times {\mathbb{R}}^{N}\mapsto ({V}_{[{\alpha}_{1}]}v(t,x),{V}_{[{\alpha}_{1}]}{V}_{[{\alpha}_{2}]}v(t,x))$ is continuous.(2) For any $(t,x)\in (0,+\mathrm{\infty})\times {\mathbb{R}}^{N}$, it holds

$${\mathcal{V}}_{0}v(t,x)=\sum _{i=1}^{N}{V}_{i}^{2}v(t,x).$$(3) The boundary condition $\underset{(t,y)\to (0,x)}{lim}v(t,y)=f(x)$ holds as well for any $x\in {\mathbb{R}}^{N}$.

## Note A.3

Again, we emphasize that we do not assume that a classical solution of the PDE (1.5) must be differentiable in the time direction or in the direction *V*_{0}. However, this is the case if vector fields satisfy the uniform Hörmander condition. In this case, the above definition coincides with the standard definition of a classical solution.

The following proposition is a particular case of proposition 2.8 in [23]:

## Proposition A.4

*Under the UFG condition, if f is a continuous function of polynomial growth, the function* $(t,x)\mapsto ({\mathcal{P}}_{t}f)(x)$ *is a classical solution to the PDE* (*1.5*) *in the sense of definition A.2. Moreover, any other classical solution v of the linear PDE* (*1.5*) *that has polynomial growth matches the solution* $(t,x)\mapsto ({\mathcal{P}}_{t}f)(x)$.

## Lemma A.5

*With the notation introduced so far, if* (*3.1*) *holds for any* ${g}_{0}\in {\mathcal{C}}_{V}^{\mathrm{\infty}}$ *then* (*3.3*) *holds for any* ${g}_{0}\in {\mathcal{D}}_{V}^{1,\mathrm{\infty}}\cap Pol$.

## Proof.

This is a standard density argument, so we just sketch it. Let ${g}_{0}^{n}$ be a sequence in ${\mathcal{C}}_{V}^{\mathrm{\infty}}$, such that ${g}_{0}^{n}\stackrel{\parallel \cdot {\parallel}_{\mathrm{\infty}}^{V,1}}{\u27f6}{g}_{0}\in {\mathcal{D}}_{V}^{1,\mathrm{\infty}}$. Then ${g}_{0}^{n}$ and ${V}_{[\alpha ]}{g}_{0}^{n}$ converge uniformly on compacts to *g*_{0} and *V*_{[α]}*g*_{0}, respectively, for any $\alpha \in {\mathcal{A}}_{m}$. Applying (3.1) to the sequence ${g}_{0}^{n}$ gives

*V*

_{[α]}

*g*

_{t}(

*x*). As for the right-hand side, as ${V}_{[\alpha ]}{g}_{0}^{n}$ converges uniformly on compacts to

*V*

_{[α]}

*g*

_{0}, then $(\mathit{\Gamma}{g}_{0}^{n})(x)$ converges uniformly on compacts, and therefore pointwise, to (

*Γg*

_{0})(

*x*). By definition of ${\mathcal{P}}_{t}$, we have

*p*

_{t}(

*x*,

*dy*) are the transition probabilities of the process

*X*

_{t}in (1.1). Because

*g*

_{0}∈

*Pol*, we can always choose the approximating sequence so that $\mathit{\Gamma}{g}_{0}^{n}$ grows polynomially (with the degree of the polynomial independent of

*n*). Therefore, by the dominated convergence theorem, $({\mathcal{P}}_{t}\mathit{\Gamma}{g}_{0}^{n})(x)$ converges pointwise to $({\mathcal{P}}_{t}\mathit{\Gamma}{g}_{0})(x)$ as $n\to \mathrm{\infty}$. Taking the (pointwise) limit as $n\to \mathrm{\infty}$ on both sides of (A 14) gives then

*h*(

*x*)=1+|

*x*|

^{q}and therefore, by proposition A.4, $({\mathcal{P}}_{t}h)(x)$ is polynomially bounded. Taking the supremum over compact sets on both sides of the above gives the desired result. ▪

## Footnotes

1 We hope that this does not create confusion when $x\in {\mathbb{R}}^{N}$, in which case |*x*| is the euclidean norm of *x*.

2 Actually, differentiability holds in all the directions spanned by the vector fields ${V}_{[\alpha ]},\hspace{0.17em}\alpha \in \mathcal{A}$. Note that differentiability in the direction *V*_{0} does not in general hold under the UFG condition. This is one of the major differences with the uniform Hörmander condition, see [6], Section 2.9.

3 For example, if *V*_{0} is a linear combination of the vector fields *V*_{i},[*V*_{i},*V*_{j}], *i*,*j*=1,…,*d*.

4 Note that this closure is well defined, see [23], Section 2.3.

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