Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Pointwise gradient bounds for degenerate semigroups (of UFG type)

D. Crisan

D. Crisan

Department of Mathematics, Imperial College London, Huxley Building, 180 Queen’s Gate, London SW7 2AZ, UK

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Published:https://doi.org/10.1098/rspa.2016.0442

    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 RN

    Xt=X0+0tV0(Xs)ds+2i=1d0tVi(Xs)dWi(s),1.1
    where V0,…,Vd are smooth vector fields on RN, ° denotes Stratonovich integration and, for each i, Wi(t) is an N-dimensional standard Brownian motion. The Markov semigroup {Pt}t0 associated with the SDE (1.1) is defined on the set Cb of continuous and bounded functions, as
    (Ptf)(x):=E[f(Xt|X0=x)].1.2
    We recall that, given a vector field V on RN, we can think of V both as a vector-valued function on RN and as a first-order differential operator on RN:
    V=(V1(x),V2(x),,VN(x))orV=j=1NVj(x)j,xRN,j=xj,1.3
    and we shall do so throughout the paper. With this notation, the Kolmogorov operator associated with the semigroup Pt is the second-order differential operator given on smooth functions by
    L=V0+i=1dVi2.1.4
    The study of Markov semigroups associated with SDEs of the form (1.1) has a long history and the literature on the matter is vast. Most of such literature deals with the case in which the operator L is elliptic or hypoelliptic; more specifically, a large body of work has been dedicated to the study of the diffusion semigroup (1.2) in the case in which the vector fields V0,…,Vd 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 Pt have a smooth density; furthermore, Ptf is differentiable in every direction and u(t,x):=(Ptf)(x) is a classical solution of the Cauchy problem
    tu(t,x)=Lu(t,x)
    and
    u(0,x)=f(x).

    In this paper, we relax the hypoellipticity assumption and work in the setting in which the vector fields V0,…,Vd satisfy a weaker condition, the so-called UFG condition. The acronym UFG stands for uniformly finitely generated. Informally, denoting by Cb(RN) the set of smooth bounded functions with bounded derivatives, this condition states that the Cb(RN)-module W generated by the vector fields {Vi,i=1,…,d} within the Lie algebra generated by {Vi,i=0,1,…,d} is finite dimensional. In particular, we emphasize that the UFG condition does not require that the vector space {W(x)|WW} is homeomorphic to RN for any xRN; indeed, the dimension of the space {W(x)|WW} is not even required to be constant over RN. 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 [25], Kusuoka and Stroock have analysed the smoothness properties of diffusion semigroups {Pt}t0 associated with the stochastic dynamics (1.1) when the vector fields {Vi,i=0,1,…,d} satisfy the UFG condition. In particular they showed that, under the UFG condition, the semigroup Pt is no longer differentiable in the direction V0; however, it is still differentiable in the direction V:=tV0 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 fCb, the function u(t,x):=(Ptf)(x) is a classical solution of the Cauchy problem

    Vu(t,x)=i=1dVi2u(t,x)andu(0,x)=f(x).}1.5
    More precisely, u is twice continuously differentiable in the directions of the vector fields Vi, i=1,…,d and once continuously differentiable in the direction V0=tV0, when viewed as a function (t,x)↦u(t,x) over the product space (0,)×Rd (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 (Ptf)(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 [810]. 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 [24] 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 Pt. 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)=(Ptf)(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 {Vi,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 t0∈(0,1) and any vector field, V, belonging to the Cb(RN)-module W, there exist constants ct0,λ>0 such that

    |VPtf(x)|2ct0eλtfor allxRNand allt>t0.

    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 xPtf=eatPt(xf) (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 Lp 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 [1113], 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 dN and let A be the set of all n-tuples, of any size n≥1, of integers of the following form:

    A:={α=(α1,,αn),nN:αj{0,1,,d} for all j1}{(0)}.
    For the sake of clarity, we stress that all n-tuples of any length n≥1 are allowed in A, except the trivial one, α=(0) (however, α=(j) belongs to A if j∈{1,…,d}). We endow A with the product
    αβ:=(α1,,αh,β1,,β),
    for any α=(α1,…,αh) and β=(β1,…,β) in A. If α is an element of A, we define the length of α, denoted by ∥α∥, the integer
    α:=h+card{i:αi=0},if α=(α1,,αh).
    For any mN,m1, we then introduce the sets
    Am={αA:αm}
    and if 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 RN, we refer to the functions {V j(x)}1≤jN as to the components or coefficients of the vector field. We say that a vector field on RN is smooth or that it is C if all the components V j(x), j=1,…,N, are C functions. Given two differential operators V and W, the commutator between V and W is defined as

    [V,W]:=VWWV.
    Let now {Vi:i=0,…,d} be a collection of vector fields on RN and let us define the following ‘hierarchy’ of operators:
    V[i]:=Vii=0,1,,d
    and
    V[αi]:=[V[α],V[i]],αA,i=0,1,,d.
    Note that if ∥α∥=h, then ∥α*i∥=h+1 if i∈{1,…,d} and ∥α*i∥=h+2 if i=0. If αA 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 Rm to be the space containing all the operators of the above hierarchy, up to and including the operators of length m (but excluding V0):
    Rm:={V[α],αAm}.2.1
    Let
    span{Rm}:={vector fields V on RN:V=βAmφα,βV[β](x)},
    where the functions φα,β in the above belong to the set CV(RN) of bounded smooth functions, φα,β=φα,β(x):RNR, such that
    supxRN|V[γ(1)]V[γ(n)]φα,β|<2.2
    for all n and all γ(1),…,γ(n),α and β in Am. With this notation in place, we can now introduce the definition that will be central in this paper.

    Definition 2.1 (UFG Condition).

    Let {Vi:i=0,…,d} be a collection of smooth vector fields on RN and assume that the coefficients of such vector fields have bounded partial derivatives (of any order). We say that the fields {Vi:i=0,…,d} satisfy the UFG condition if there exists mN such that for any αA of the form

    α=αi,αAm,i{0,,d},
    there exist bounded smooth functions φα,βCV(RN) such that
    V[α](x)=βAmφα,βV[β](x).

    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 V0.

    Example 2.2

    Consider the following first-order differential operators on R2

    V0=sinxyandV1=sinxx.
    Then {V0,V1} 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
    V0=sinxxandV1=sinxy
    then {V0,V1} still satisfy the UFG condition, this time with m=1 (indeed, [V0,V1]=cosxV1).

    Note 2.3

    Under assumption (2.2) on the functions φ, if the UFG condition holds for some mN, then it also holds for any nm,nN. In other words, if the UFG condition holds for some m in N, then for any V[γ] with ∥γ∥>m one has

    V[γ]=βAmφγ,βV[β](x)
    for some bounded functions φγ,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 {Pt}t0 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 Pt when the vector fields {V0,V1,…,Vd} satisfy the UFG condition. We recall that a semigroup Pt of bounded operators is Markov if

    Pt1=1andPtf0 when f0,
    where, in the above, 1 denotes the function identically equal to one. Denoting by the supremum norm, the above implies that if f< then Ptff, i.e. the semigroup is a contraction in the supremum norm.

    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 {V0,V1,…,Vd} appearing in the generator (1.4), the semigroup Pt still enjoys good smoothing properties: if f(x) is continuous, then (Ptf)(x) is differentiable in all the directions spanned by the vector fields contained in Rm (we recall that the set Rm is defined in (2.1)).2

    Moreover, while the function u(t,x):=(Ptf)(x) may not be differentiable in the direction V0 or in the time variable, it is still differentiable in the direction V:=tV0, and it is the unique classical solution of the Cauchy problem

    Vu(t,x)=j=1dVj2u(t,x)andu(0,x)=f(x),}2.3
    provided the initial datum 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 {V0,V1,…,Vd} satisfy the UFG condition for some m>0. We can then construct the vector field V, containing all the vector fields (operators) V[α], αAm:

    V:=(V1,,Vd,,V[α],).2.4
    The vector V has |Am| entries; using the notation (1.3), each entry (i.e. each vector V[α], αAm) can be expressed as follows:
    V[α]=(V[α]1,,V[α]N).
    Therefore, V can be rewritten as
    V=(V11,,V1N,,Vd1,,VdN,,V[α]1,,V[α]N,).
    It is clear from the above that we can think of V as a function from RN to RN|Am|. 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 |Am| entries, where each entry is an operator in span{Rm}, will be denoted by R|Am|. Clearly, VR|Am|.

    We emphasize that if XR|Am|, 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 Vj is any first-order differential operator, we also write

    VjV=(VjV1,,VjVd,,VjV[α],).
    Given a collection of strictly positive numbers {a[α]}αAm and any f(x):RNR (smooth enough so that the expression below makes sense), we can define the following quadratic form:
    (Γf)(x):=αAma[α]|(V[α]f)(x)|2,xRN.2.5
    If a multi-index α is of length k, we will denote it by αk (when we want to emphasize its length) and V[αk] will be the corresponding first-order operator of length k (obviously, for a given kN, 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
    (Γf)(x)=k=1m{αk,αk=k}a[αk]|V[αk]ft(x)|2.
    Also, if we define the following bilinear form on R|Am|:
    Xf,YfAm:=k=1mαk:αk=ka[αk](X[αk]f)(Y[αk]f),X,YR|Am|,2.6
    where f is any smooth enough function, then the quadratic form Γ can be rewritten as
    Γ(f)=VfAm2,2.7
    where Am is the (semi) norm induced by the bilinear form ,Am. We stress that the definition of the bilinear form ,Am depends on the choice of the constants {a[α]}αAm. For j≥0, we also define the linear mappings Λj,Λ:span{Rm}span{Rm} as follows:
    ΛjV[α]={V[αj]if αjmβAmφαj,βV[β]if αj>m2.8
    and
    Λ:=Λ0+j=1dΛjΛj.2.9
    With abuse of notation, we keep denoting by Λj also the linear mapping Λj:R|Am|R|Am| that acts on the component [α] of the vector V as follows:
    (ΛjV)[α]:=ΛjV[α].
    Analogous use of notation holds for 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 Λj, 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[α],αAm, (but not necessarily in other directions) and such that

    |(V[α]f)(x)|2κ(1+|x|q),2.10
    for some κ,q>0. When, given a function fPol, we want to stress the value of the constant κ such that the above holds, we write fPol(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 L is the operator (1.4), using the above we find that for any vector field V[α]:

      [V[α],L]=V[α0]+j=1dV[αjj]+2j=1dVjV[αj]=ΛV[α]+2j=1dVjΛjV[α].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 R3, see [18]). More precisely, set d=2 and N=3 and consider the operators

    X1:=xy2z,X2:=y+x2z,X3:=[X1,X2]:=z,
    and
    X0:=xX1+yX2+2zX3=xx+yy+2zz.
    The Lie algebra generated by {X0,X1,X2} is usually referred to as the Heisenberg Lie algebra. The vector fields {X0,X1,X2} satisfy the Hörmander condition hence the operator L=X0+X12+X22 is hypoelliptic on R3. If the above fields are slightly modified, we obtain new vector fields, {V0,V1,V2}, 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
    V0:=k(xx+yy+2zz),V1:=yz,V2:=y+xz,k>0.
    The operators {V0,V1,V2} satisfy the UFG condition with m=2, as
    [V1,V0]=kV1,[V2,V0]=kV2[V1,V2]=z=V12,[V12,V0]=2kV12and[V12,V1]=[V12,V2]=0.
    Therefore, in this example, we have A2:={1,2,(1,2),(2,1)} and span{R2}=span{V1,V2,V[12]=:V12}. Because V21:=V[2*1]=−V12, V21 does not need to be in the list of the base fields of R2 ( for the same reason, it can also be omitted in the definition of Γ below, as the constants a1,a2,a12 are anyway arbitrary). Using definition (2.5), the quadratic form Γ associated with the UFG-Heisenberg group is
    (Γf)(x)=a1|V1ft|2+a2|V2ft|2+a12|V12ft|2.
    The vector V is V=(V1,V2,V12) and the mappings Λ1 and Λ2 give
    Λ1V=(0,V12,0)andΛ2V=(V12,0,0),
    while for Λ0 we have
    Λ0V=(kV1,kV2,kV3).

    3. Preliminary results: a Bakry-type approach

    In this section, we consider Markov semigroups associated with operators L of the form (1.4), for a given set {V0,V1,…,Vd} of vector fields. We recall that the class of functions ‘Pol’ has been defined immediately after note 2.4.

    Proposition 3.1

    Let ft:=Ptf0 be the diffusion semigroup defined in (1.2).

    (a) Let m be any positive integer and assume the initial datum f0 is a bounded smooth (in every direction) function such that V[α]f0< for all αAm. Consider the quadratic form Γ defined in (2.5):

    (Γft)(x):=αAma[α]|V[α]ft(x)|2,
    for some strictly positive constants {a[α]}{αAm} (to be chosen). Suppose there exists λ>0 such that the following inequality holds:
    ddsPtsΓ(fs(x))λPtsΓ(fs(x))for any xRN.3.1
    Then
    Γ(ft)eλtΓ(f0),for all t0;3.2
    therefore,
    |V[α]ft(x)|21a[α]Γ(f0)eλtfor all αAm,t0.

    (b) Suppose, in addition, that the vector fields {V0,…,Vd} satisfy the UFG condition (for some m). In this case, if (3.1) holds when f0 is smooth (and V[α]f0<), then the following holds for any f0Pol(κ): for every open ball B(0,K) of radius K and for all αAm,

    supxB(0,K)|V[α]ft(x)|2κcKeλt,3.3
    where cK>0 is a constant dependent on K.

    (c) If the vector fields {V0,…,Vd} satisfy the UFG condition (for some m) and (3.1) is satisfied for any smooth initial datum, then then following holds when f0 is only continuous and bounded (but not necessarily smooth): for any t0∈(0,1) and any K>0 there exists a constant ct0,K>0 such that

    supxB(0,K)|V[α]ft(x)|2ct0,Keλ(tt0)f0(x)2for all αAmand allt>t0.3.4
    Moreover, if the coefficients of the vector fields {V0,…,Vd} are bounded, then the constant ct0,K does not depend on K and we have the uniform bound
    |V[α]ft(x)|2ct0eλ(tt0)f0(x)2for all αAmand allt>t0.3.5

    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 f0 is bounded and continuous and the UFG condition holds, then (Ptf0) is differentiable in the directions V[α],αAm [25]. Because we are assuming that the vector fields {V0,…,Vd} 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, B(0,K) and for any αAm,

    supxB(0,K)|V[α](Ptf0)(x)|ctα/2supxB(0,K)[(1+|x|α)]f0(x),3.6
    for some constant c>0 (which does not depend on x,t or f0). 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:
    |V[α](Ptf0)(x)|c~tα/2.3.7

    Proof of proposition 3.1.

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

      PtsΓ(fs)eλsPtΓ(f0),for all 0<st.3.8
      Therefore, using (3.8) for s=t and the contractivity of the semigroup Pt in the supremum norm gives the result. Note, in particular, that if f0 is smooth in every direction, then also (Ptf0)(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 t0∈(0,1) the function Ptf0 belongs to the set Pol(κ); in particular, by (3.6), the constant κ appearing in (2.10) is, for this function, κ=t0mcf0(x). Therefore, by part (b), for any fixed 0<t0<1 and for any tt0, we can write

      |V[α](Ptf0(x))|2=|V[α](Ptt0Pt0f0(x))|2eλ(tt0)ct0,Kf0(x).
      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 f0(x) is a smooth function with bounded derivatives. This is to make sense of the expression (Γf0)(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 f0(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 L and the semigroup commute on a set of sufficiently smooth functions (say, for example, on the set CV, 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

    (L+t)Γ(ft)λΓ(ft)t>03.9
    then (3.1) holds.

    Proof.

    This is again standard so we only sketch it.

    ddsPtsΓ(fs(x))=LPtsΓ(fs)+PtssΓ(fs).
    We can now use the fact that the semigroup commutes with its generator (on a set of sufficiently smooth functions) and the positivity preserving property of Markov semigroups, and therefore conclude the proof. ▪

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

    If X is a first-order differential operator on RN, L is the operator (1.4) and ft(x):=(Ptf0)(x), then

    (L+t)|Xft|2=2j=1d|VjXft|2+2([X,L]ft)(Xft),4.1
    whenever f0 is smooth. The identity (4.1) is obtained by using (2.3) and the fact that X and all the Vj’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 αk. In the view of (3.9), we use (4.1) to calculate the following:
    (L+t)Γ(ft)=(4.1)2k=1mαka[αk]j=1d|VjV[αk]ft|2+2k=1mαka[αk]([V[αk],L]ft)(V[αk]ft)=(2.11)2k=1mαka[αk]j=1d|VjV[αk]ft|2+4k=1mαka[αk]j=1d(VjV[αkj]ft)(V[αk]ft)+2k=1mαka[αk]([V[αk],V0]ft)(V[αk]ft)+2k=1mαka[αk]j=1d(V[αkjj]ft)(V[αk]ft).

    Note that if V[αk] is a field of length k, from (2.11) one can see that the commutator between V[αk] and 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 [Vαm,L] and [Vαm1,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:

    (L+t)Γ(ft)=S(ft)+F(ft)4.2
    where
    S(ft):=2k=1mαka[αk]j=1d|VjV[αk]ft|2+4k=1mαka[αk]j=1d(VjV[αkj]ft)(V[αk]ft)4.3
    =(2.8)2j=1dVjVftAm2+4j=1dVjΛjVft,VftAm4.4
    and
    F(ft):=+2k=1mαka[αk]([V[αk],V0]ft)(V[αk]ft)+2k=1mαka[αk]j=1d(V[αkjj]ft)(V[αk]ft)=(2.9)2ΛVft,VftAm.4.5
    Note that F(ft) contains only the first-order operators (vector fields), while S(ft) contains second order as well as first-order operators (see also the expression for S(ft) at the beginning of the proof of lemma 4.2, in particular the terms with (**)).

    Theorem 4.1

    Let m be a positive integer and Ptf0=:ft 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 {V0,V1,…,Vd} appearing in (1.1):

    • — there exists a collection of strictly positive constants {a[α]}αAm such that the corresponding bilinear form (2.6) satisfies

      S(ft)=2j=1dVj VftAm2+4j=1dVjΛj Vft, VftAmγ VftAm2,4.6
      for some constant γ>0 (possibly dependent on the collection {a[α]}αAm);

    • — there exists μ>γ such that

      F(ft)=2Λ Vft, VftAmμ VftAm2,4.7
      where ,Am is the bilinear form defined by the same constants for which (4.6) holds.

    then (3.9) holds with λ=μγ. 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):

    (L+t)Γ(ft)(γμ)Vft2=λΓ(ft).

    If we divide both sides of (4.7) by VftAm2, then it becomes clear that imposing condition (4.7) is equivalent to requiring that (there exists a bilinear form on R|Am|RN|Am| such that) the ‘angle’ between the vectors ΛVft and Vft is obtuse.

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

    Theorem 4.2

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

    • (i) If the vector fields {Vi:i=0,…,d} satisfy the UFG condition for some mN, then there exists a choice of the constants {a[α]}αAm 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 λ0>0 such that, for every αAm and every smooth enough function f,

      (V[α]f)([V[α],V0]f)λ0|V[α]f|2.4.8
      If λ0 is big enough then (4.7) holds with μ0. 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 f0 is continuous and bounded, then for any t0∈(0,1) and any K>0 there exists a constant ct0,K>0 such that
      supxB(0,K)|V[α]ft(x)|2ct0,Keλ(tt0)f0(x)for all αAmand allt>t0.
      If the coefficients of the vector fields {V0,…,Vd} are bounded, then
      |V[α]ft(x)|2ct0eλ(tt0)f0(x)for all xRN,αAmand allt>t0.4.9

    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 ,Am 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 λ0 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 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 L=V0+V12 on R2, with

    V0=kxx,V1=xy,k>0.
    The fields {V0,V1} satisfy the UFG condition with m=1, as [V1,V0]=−kV1. It is easy to see that (3.9) holds with λ=2k ( for every k>0). Indeed, by direct calculation
    (L+t)|V1ft|2=(4.1)2|V12ft|2+2([V1,L]ft)(V1ft)=2|V12ft|22k|V1ft|22k|V1ft|2.
    We name this example the UFG-Grušin plane as it results from a small modification of the so-called UFG-Grušin plane, given by the operators
    X0:=xx+2yy,X1:=xandX2:=xy.
    It is easy to verify that the operator X0+X12+X22 verifies the Hörmander condition.

    Example 4.5

    The operators {V0,V1,V2} 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 L of the form (1.4) acting on R3, where the fields V0,V1,V2 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.

    (L+t)Γ(ft)kΓ(ft).

    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( ft) includes the derivatives of the semigroup but not the semigroup ft 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 ft=Ptf. An analogous observation holds for the derivatives in the direction V0. Note that our result does not imply anything regarding the behaviour in the direction V0, as V0 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 V0, let alone decay, see [6], Section 2.9. However, it was proved that under the so-called V0-condition (see definition 4.8), the semigroup Pt is also differentiable in the direction V0. In this case, our results also cover such a direction.

    Definition 4.8 (V0-condition).

    With the notation introduced so far, we say that the V0-condition is satisfied if there exist functions φβCV such that

    V0=βA2φβV[β].

    Corollary 4.9

    Suppose that the assumptions of theorem 4.1 are satisfied. If the V0 condition holds, then there exist positive constants c,λ>0 such that

    |V0ft|2ceλt,
    say for any smooth f0.

    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)f)(x):=k=1nα(j)Ama[α(k),,α(1)]|V[α(k)]V[α(1)]ft(x)|2.
    That is, the quadratic form Γ(n) contains all the derivatives of order at most n, in all the directions contained in Rm. 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 RN and, for some given αAm, suppose there exists an integral curve of V[α] joining x and y. That is, suppose there exists η(τ):[0,1]RN such that

    ddτη(τ)=V[α](η(τ)),η(0)=x,η(1)=y.
    We stress that in the above 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 xy, if there exists an integer M>0 and M points in RN, z1,…,zM, such that z1=x, zM=y and for every i=1,…,M−1, there exists an α(i)Am such that the integral curve of V[α(i)] is well defined and joins zi with zi+1. The relation ∼ is an equivalence relation. We denote by Ux the set of points reachable from x (clearly, if yx then Ux=Uy).

    Corollary 4.10

    Let Pt 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 xRN and for any yUx there exists λ>0 such that

    |(Ptf)(x)(Ptf)(y)|ceλt,for all t>0,
    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

    ddτ(Ptf)(η(τ))=(V[α]Ptf)(η(τ))
    (see also note A.1 in appendix A). Integrating the above between 0 and 1 and using (4.9), we obtain the result. ▪

    5. Proofs of main results

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

    φ¯:=supx|φ(x)|.
    We also set
    A¯m:={αA:α=m}.
    We make the (obvious) remark, that if the UFG condition holds for some mN, then for all the multi-indices α of length at most m, we have
    φ¯αj,βj:=supx|Vj(φαj,β)|<,j=1,,d.5.1
    We also recall Young’s inequality
    |ab|a22ϵ+b2ϵ2,for all a,bR and ϵ>0,5.2
    which we will repeatedly use throughout the proofs of this section.

    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 ∥αk∥=m then ∥αk*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

    S(ft)=2k=1mαka[αk]j=1d|VjV[αk]ft|2+4k=1m1αka[αk]j=1d(VjV[αkj]ft)(V[αk]ft)+4αma[αm]j=1dβAmφαmj,βj(V[β]ft)(V[αm]ft)+4αma[αm]j=1dβAmφαmj,β(VjV[β]ft)(V[αm]ft)

    Let us now set, for any multi-index γAm,

    Jγ:=supj=1,,dβAmφ¯γj,βandHγ:=supj=1,,dβAmγφ¯γj,βj.
    With these definitions in mind, let us start estimating each of the above terms, beginning with the last.
    • Terms with △: For each αmA¯m,

      4a[αm]j=1dβAmφαmj,β(VjV[β]ft)(V[αm]ft)2j=1dβAm[1{Jαm0}|VjV[β]ft|2+a[αm]2Jαm2|V[αm]ft|2]=2a[αm]2Jαm2d|Am||V[αm]ft|2+21{Jαm0}j=1dβAm|VjVβft|2

    • Terms **: For each αmA¯m, we have

      4a[αm]j=1dβAmφαmj,βj(V[β]ft)(V[αm]ft)4a[αm]j=1dφαmj,αmj|V[αm]ft|2+4dHαma[αm]βAm{αm}(V[β]ft)(V[αm]ft)4a[αm]j=1dφαmj,αmj|V[αm]ft|2+2d1{Hαm0}βAm{αm}|V[β]ft|2+2da[αm]2Hαm2(|Am|1)|V[αm]ft|2

    • Terms ⋆: for every k=1,…,m−1,

      4a[αk]j=1d(VjV[αkj]ft)(V[αk]ft)2d|V[αk]ft|2+j=1d2a[αk]2|VjV[αkj]ft|25.3

    Putting the above estimates together, after setting

    J:=αmA¯m1{Jαm0},
    we obtain
    S(ft)k=1mαkA¯kcαk|V[αk]ft|25.4
    +k=1mj=1dαkA¯k|VjV[αk]ft|2[2a[αk]+2(J+1{k>1}1{αk=αk1j}a[αk1]2)]5.5
    where
    cαm:=2a[αm]2Jαm2d|Am|+4a[αm]j=1dφαmj,αmj+2dβmA¯m{αm}1{Hβm0}+2da[αm]2Hαm2(|Am|1)
    and, for k=1,…,m−1
    cαk:=2d+2dαmA¯m1{Hαm0}.
    With the purpose of making sure that the terms in (5.5) are negative, we can simply choose
    a[α1]>max{0,J}anda[αk]>J+a[αk1]2for all k=2,,m.5.6
    Therefore, once all the 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
    S(ft)k=1mαkA¯kcαk|V[αk]ft|2γVft2,
    having set
    γ:=maxαkAkk=1,,mcαka[αk].

    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 ∥αk∥=m−1 (m, respectively) then ∥αk*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 ∥αk∥=m−1 or m, obtaining

    2ΛVft,Vft(4.8)2k=1mαka[αk]λ0|V[αk]ft|2+2k=1m2αka[αk]j=1d(V[αkjj]ft)(V[αk]ft)+2αm1A¯m1a[αm1]j=1dβAmφαm1jj,β(V[β]ft)(V[αm1]ft)+2αmA¯ma[αm]j=1dβAmφαmjj,β(V[β]ft)(V[αm]ft)
    Like in the proof of part (i) of theorem 4.2, we set
    Iγ:=supj=1,,dβAmγφ¯γjj,β
    and estimate all the above terms, starting from the last.
    • Terms with *: For each αmA¯m,

      2a[αm]j=1dβAmφαmjj,β(V[β]ft)(V[αm]ft)2a[αm]j=1dφαmjj,αm|V[αm]ft|2+2dIαma[αm]βAm{αm}(V[β]ft)(V[αm]ft)=2a[αm]j=1dφαmjj,αm|V[αm]ft|2+dβAm{αm}[a[αm]2Iαm2|V[αm]ft|2+|V[β]ft|21{Iαm0}]=2a[αm]j=1dφαmjj,αm|V[αm]ft|2+da[αm]2Iαm2|V[αm]ft|2(|Am|1)+d1{Iαm0}βAm{αm}|V[β]ft|2

    • Terms : For each αm1A¯m1,

      2a[αm1]j=1dβAmφαm1jj,β(V[β]ft)(V[αm1]ft)2a[αm1]j=1dφαm1jj,αm1|V[αm1]ft|2+da[αm1]2Iαm12(|Am|1)|V[αm1]ft|2+d1{Iαm10}βAm{αm1}|V[β]ft|2.

    • Terms : for every k=1,…,m−2,

      2a[αk]j=1d(V[αjj]ft)(V[αk]ft)d|V[αk]ft|2+j=1da[αk]2|V[αkjj]ft|2.5.7

    Overall, one obtains

    2ΛVft,Vftk=1mαkA¯k|V[αk]ft|2(2a[αk]λ0+αk),5.8
    where
    αm:=da[αm]2Iαm2(|Am|1)+dβmA¯m{αm}1{Iαm0}+2a[αm]j=1dφαmjj,αm+dαm1A¯m11{Iαm10}+1{αm=αm2jj}a[αm2]2αm1:=2a[αm1]j=1dφαm1jj,αm1+da[αm1]2Iαm12(|Am|1)+1{αm1=αm3jj}a[αm3]2+dαmA¯m1{Iαm0}+dαm1A¯m11{Iαm10}
    and, for k=1,…,m−2
    αk:=d+1{k2>0}1{αk=αk2jj}a[αk2]2+dαmA¯m1{Iαm0}+dαm1A¯m11{Iαm10}.
    Looking at (5.8), we then impose
    2a[αk]λ0+αka[αk]λ0k=1,,m,5.9
    that is,
    a[αk]λ0αk,k=1,,m.
    It is clear that given any two sets of positive constants, a[αk] and ℓαk, there always exists at least one λ0>0 satisfying the above. In particular, one can choose any λ0 such that
    λ0>maxαkA¯kk=1,,mαka[αk].5.10
    If λ0 satisfies (5.10), and hence (5.9), from (5.8) one has
    2ΛVft,Vftλ0k=1mαkA¯ka[αk]|V[αk]ft|2=λ0Vft2.
    This concludes the proof. ▪

    Proof of lemma 4.6.

    Consider the quadratic form

    Γ(ft)=a1|V1ft|2+a2|V2ft|2+a12|V12ft|2.
    From propositions 3.1 and 3.4, it is clear that we only need to show the inequality (L+t)Γ(ft)kΓ(ft). Using (4.1), let us therefore calculate the following:
    (L+t)Γ(ft)=2a1|V12ft|2+2a1([V1,L]ft)(V1ft)2a1|V2V1ft|22a2|V22ft|2+2a2([V2,L]ft)(V2ft)2a2|V1V2ft|22a12|V1V12ft|+2a12([V12,L]ft)(V12ft)2a12|V2V12ft|2.
    The commutators appearing in the above can be calculated, and they are
    [V1,L]=kV1+2V2V12[V2,L]=kV2+2V1V21=kV22V1V12and[V12,L]=2kV12.
    Therefore,
    (L+t)Γ(ft)=2a1|V12ft|22a1|V2V1ft|22a1k|V1ft|22a2|V22ft|22a2|V1V2ft|22a2k|V2ft|22a12|V1V12ft|22a12|V2V12ft|24a12k|V12ft|2+4a1(V2V12ft)(V1ft)4a2(V1V12ft)(V2ft).
    If we use Young’s inequality (5.2) (with ϵ=a1 for the first inequality and ϵ=a2 in the second), we can estimate the terms on the last line as
    4a1(V2V12ft)(V1ft)2a12|V2V12ft|2+2|V1ft|25.11
    and
    4a2(V1V12ft)(V2ft)2a22|V1V12ft|2+2|V2ft|25.12
    Therefore,
    (L+t)Γ(ft)(2a1k+2)|V1ft|2+(2a2k+2)|V2ft|24a12k|V12ft|22a1|V12ft|22a1|V2V1ft|22a2|V22ft|22a2|V1V2ft|2(2a12+2a22)|V1V12ft|2+(2a12+2a12)|V2V12ft|2
    Looking at the terms ♠, we choose a1 and a2 such that
    2aik+2aikai2ki=1,25.13
    and a12 such that −4a12k≤−a12k, which is true, e.g. for any a12>1. Then, looking at the terms , we choose a12 much bigger than a1 and a2, more precisely we choose a12 such that
    2a12+2ai2<0,i=1,2.
    Because for any k>0, one can find a1>0 and a2>0 such that (5.13) is satisfied, this concludes the proof. ▪

    Note 5.1

    If the constants a1,a2,a12 had not been introduced, i.e. if a1=a2=a12= 1, then we would have only been able to prove the result for k>12 (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 Vi, 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 V0=tV0, when viewed as a function (t,x)↦u(t,x) over the product space (0,)×RN. However, we will not require u to be differentiable in either the time direction ∂t or the direction V0.

    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 Vi i=1,…,d and V0=tV0 is preserved, but not that in the time direction ∂t or in the direction V0. This is to be expected as the smoothing effect only takes place in the directions Vi, i=1,…,d. An extreme case where the UFG condition holds is when all the Vi, i=1,…,d are equal to zero. Take, for example, the transport equation

    tu(t,x)=V0u(t,x)
    and
    u(0,x)=f(x).
    For example, assume that N=1, V0=∂/∂x. In this case, the solution is explicitly given by u(t,x)=f(t+x), xR and t≥0. Obviously should f not be differentiable (choose for example f(x)=|x|, xR), we will not expect differentiability in either the time direction ∂t, or the space direction ∂/∂x. However, u will be differentiable in the direction V0=tV0. In fact, u is constant in the direction V0=tV0, as V0u=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 Vi, i=0,…,d, satisfy the Hörmander condition. In this case, the smoothing effect occurs in every direction. In particular, u becomes differentiable in the V0 direction, and since u is differentiable in the direction V0=tV0, 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 V0 direction (and therefore also in the time direction). If the vector fields Vi i=0,…,d satisfy the UFG condition and V0 belongs to A 3 , then it is still the case that u becomes differentiable in the V0 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 BRN and for a function φ in CV(B) (that is, for any smooth bounded real-valued function φ with bounded derivatives on B of any order in the directions V[α], αAm), we set

    φB,V,1=φB,+αAmV[α]φB,
    and then define DV1,(B) as the closure of CV(B) in Cb(B¯) w.r.t. B,V,1.4 More generally, for k>1, we can define by induction
    φB,V,k=φB,V,k1+α1,,αkAmV[α1]V[αk]φB,,φCV(B).
    We then define DVk,(B) as the closure of CV(B) in Cb(B¯) w.r.t. B,V,k. In particular, we can define DVk,(RN) as
    DVk,(Rd)=r1DVk,(B(0,r)),
    where B(0,r) stands for the d-dimensional ball of centre 0 and radius r. For vDVk,(RN), V[α1]V[αk]v is understood as the derivative of v in the directions V[α1]V[αk], with α1,,αkAm. Similarly, for φCV(B) and k≥0, we set
    φB,V,k+1/2=φB,V,k+i=1Nα1,,αkAmV[α1]V[αk]ViφB,.
    (Above, B,V,0=B,.) We then define DVk+1/2,(B) as the closure of CV(B) in Cb(B¯) w.r.t. B,V,k+1/2 and we set
    DVk+1/2,(RN)=r1DVk+1/2,(B(0,r)),k0.

    Note A.1

    Note that any function in DV1,(RN) is differentiable along the solutions of the ordinary differential equation γ˙t=V(γt), t≥0, for VAm. In particular, any function in DV1,(RN) is continuously differentiable on RN 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 V0=tV0. Again, we proceed via a closure argument. For any r≥1 and any time-space function φCV([1/r,r]×B(0,r)) with bounded derivatives of any order, we set

    φ[1/r,r]×B(0,r),V0,1=φ[1/r,r]×B(0,r),+V0φ[1/r,r]×B(0,r),.
    We then define DV01,([1/r,r]×B(0,r)) as the closure of C([1/r,r]×B(0,r)) w.r.t. [1/r,r]×B(0,r),V0,1 and then define DV01,((0,+)×RN) as the intersection of the spaces DV01,([1/r,r]×B(0,r)) over 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)[0,+)×RN} a classical solution of the PDE (1.5) if the following conditions are satisfied:

    • (1) v belongs to DV01,((0,+)×RN) and, for any t>0, v(t,⋅) is in DV2,(RN); moreover, for any α1,α2A, the function (t,x)(0,+)×RN(V[α1]v(t,x),V[α1]V[α2]v(t,x)) is continuous.

    • (2) For any (t,x)(0,+)×RN, it holds

      V0v(t,x)=i=1NVi2v(t,x).

    • (3) The boundary condition lim(t,y)(0,x)v(t,y)=f(x) holds as well for any xRN.

    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 V0. 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)(Ptf)(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)(Ptf)(x).

    Lemma A.5

    With the notation introduced so far, if (3.1) holds for any g0CV then (3.3) holds for any g0DV1,Pol.

    Proof.

    This is a standard density argument, so we just sketch it. Let g0n be a sequence in CV, such that g0nV,1g0DV1,. Then g0n and V[α]g0n converge uniformly on compacts to g0 and V[α]g0, respectively, for any αAm. Applying (3.1) to the sequence g0n gives

    |V[α]gtn(x)|2eλt(PtΓg0n)(x)for all αAm,t0,A 14
    where in the above gtn:=Ptg0n. From the integration by parts formulae in [7], ch. 3 the left-hand side of the above converges uniformly on compacts to V[α]gt(x). As for the right-hand side, as V[α]g0n converges uniformly on compacts to V[α]g0, then (Γg0n)(x) converges uniformly on compacts, and therefore pointwise, to (Γg0)(x). By definition of Pt, we have
    |[Pt(Γg0nΓg0)](x)|=[(Γg0n)(y)(Γg0)(y)]pt(x,dy),
    where pt(x,dy) are the transition probabilities of the process Xt in (1.1). Because g0Pol, we can always choose the approximating sequence so that Γg0n grows polynomially (with the degree of the polynomial independent of n). Therefore, by the dominated convergence theorem, (PtΓg0n)(x) converges pointwise to (PtΓg0)(x) as n. Taking the (pointwise) limit as n on both sides of (A 14) gives then
    |V[α]gt(x)|2eλt(PtΓg0)(x)=eλt(Γg0)(y)pt(x,dy)(2.10)|Am|κeλt(1+|y|q)pt(x,dy).
    Now (1+|y|q)pt(x,dy)=(Pth)(x), where h(x)=1+|x|q and therefore, by proposition A.4, (Pth)(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 xRN, in which case |x| is the euclidean norm of x.

    2 Actually, differentiability holds in all the directions spanned by the vector fields V[α],αA. Note that differentiability in the direction V0 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 V0 is a linear combination of the vector fields Vi,[Vi,Vj], i,j=1,…,d.

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

    Published by the Royal Society. All rights reserved.

    References