A support and density theorem for Markovian rough paths

0.25 MB·

by  Ilya Chevyrev

A SUPPORT AND DENSITY THEOREM FOR MARKOVIAN
ROUGH PATHS
7
1 ILYACHEVYREVANDMARCELOGRODNIK
0
2
Abstract. We establish two results concerning a class of geometric rough
n paths X which arise as Markov processes associated to uniformly subelliptic
a Dirichletforms. Thefirstisasupport theorem forXinα-Hölderroughpath
J
topology for all α ∈ (0,1/2), which answers in the positive a conjecture of
1 Friz-Victoir [11]. The second is a Hörmander-type theorem for the existence
1 ofadensityofaroughdifferentialequationdrivenbyX,theproofofwhichis
basedonanalysisof(non-symmetric)Dirichletformsonmanifolds.
]
R
P
.
h 1. Introduction
t
a Consider a Markov process X associated with a symmetric Dirichlet form on
m
L2(Rd,λ)
[
d
1 (1.1) E(f,g)= ai,j(∂ f)(∂ g)dλ,
i j
v ZRdi,j=1
2 X
0 whereλistheLebesguemeasure,andaisauniformlyellipticfunctiontakingvalues
0 in symmetric d×d matrices (we make our exact set-up precise in Section 1.1).
3
We are interested in differential equations of the form
0
. (1.2) dY =V(Y )dX , Y =y ∈Re
1 t t t 0 0
0 driven by X along vector fields V = (V ,...,V ) on Re. When a is taken suffi-
1 d
7 ciently smooth, the process X can be realised as a semi-martingale for which the
1
classical framework of Itô gives meaning to the equation (1.2). However for irreg-
:
v ular functions a, this is no longer the case, and (1.2) falls outside the scope of Itô
i
X calculus.
One of the applications of Lyons’ theory of rough paths [16] has been to give
r
a meaning to differential equations driven by processes outside the range of semi-
martingales. One viewpoint of rough paths theory is that it factors the problem of
solvingequationsofthetype(1.2)intofirstenhancingXtoaroughpathbyappro-
priately defining its iterated integrals (which is typically done through stochastic
means), after which the theory allows (1.2) to be solved deterministically.
Probabilistic methods to enhance the Markov process X to a rough path and
the study of its fundamental properties appear in [15, 2, 13, 14], where primarily
the forward-backward martingale decomposition is used to show existence of the
stochasticarea. Asomewhatdifferentapproach,whichwefollowhere,istakenin[9]
where the authors define X directly as a diffusion on the free nilpotent Lie group
2010 Mathematics Subject Classification. Primary60H10;Secondary 60G17.
Key words and phrases. Markovian rough paths, support in Hölder topology, Hörmander’s
theorem.
1
2 ILYACHEVYREVANDMARCELOGRODNIK
GN(Rd)(inparticularthe iteratedintegralsaregivendirectly inthe construction).
One can show that in the situtation mentioned at the start, the two methods give
rise to equivalent definitions of rough paths, and the latter construction in fact
yields further flexibility in that the evolution of X, as a diffusion in GN(Rd), can
depend in a non-trivialonits higher levels (its iteratedintegrals). SuchMarkovian
rough paths have also recently been investigated in [6, 7] in connection with the
accumulated local-p variation functional and the moment problem for expected
signatures.
The goal of this paper is to contribute two results to the study of Markovian
rough paths in the sense of [9]. Our first contribution (Theorem 2.4) answers in
the positive a conjecture about the support of X in α-Hölder roughpath topology.
Such a support theorem appeared in [9] for α ∈ (0,1/6), and was improved to
α∈(0,1/4) in [11] where it was conjectured to hold for α∈(0,1/2) in analogy to
enhancedBrownianmotion. ComparingoursituationtothecaseofGaussianrough
paths, where such support theorems are known with sharp Hölder exponents (see
e.g.,[11]Section15.8,and[10]forrecentimprovements),thedifficultyofcourselies
in the lack of a Gaussianstructure, in particular the absence of a Cameron-Martin
space.
Oursolutiontothis probleminfactusesfew propertiesofXbeyondheatkernel
estimates;indeedwefirstderivegeneralconditionsunderwhichastochasticprocess
(takingvalues inaPolishspace)admits explicitlowerbounds onthe probabilityof
keeping a small α-Hölder norm, and then verify these conditions for the translated
(in general non-Markov) rough path T (X) for any h ∈ W1,2([0,T],Rd) (we also
h
note that, just like for enhanced Brownian motion, all relevant constants depend
on h only through ||h|| ). As usual, in combination with the continuity of the
W1,2
Itô-Lyons map from rough paths theory, an immediate consequence of improving
the Hölder exponent in the support theorem for X is a stronger Stroock-Varadhan
supporttheorem(inα-Höldertopology)forthesolutionY totheroughdifferential
equation (RDE) (1.2) along with the lower regularity assumptions on the driving
vector fields V (Lip2 instead of Lip4).
Our second contribution (Theorem 3.2 and its Corollary 3.4) may be seen as
a non-Gaussian Hörmander-type theorem, and provides sufficient conditions on
the driving vector fields V = (V ,...,V ) under which that the solution to the
1 d
RDE (1.2) admits a density with respect to the Lebesgue measure on Re. Once
again, while this result is reminiscent of density theorems for RDEs driven by
Gaussian rough paths (e.g., [3, 4, 5]), the primary difference in our setting is that
methodsfromMalliavincalculusarenolongeravailableduetothe lackofaGauss-
ian structure.
We replace the use of Malliavin calculus by direct analysis of (non-symmetric)
Dirichletformsonmanifold. Indeed,weidentify conditionsunderwhichthe couple
(X,Y)admitsadensityonitsnaturalstate-space,andconcludebyprojectingtoY.
Wenotehoweverthatourcurrentresultgivesnoquantitativeinformationaboutthe
density (beyond its existence) of even the couple (X,Y), and we strongly suspect
that the method can be improved to yield further information (particularly Lp
bounds and regularity results in the spirit of the De Giorgi–Nash-Mosertheorem).
1.1. Notation. Throughout the paper, we adopt the convention that the domain
of a path x : [0,T] 7→ E, for T > 0 and a set E, is extended to all of [0,∞) by
setting x =x for all t>T.
t T
A SUPPORT AND DENSITY THEOREM FOR MARKOVIAN ROUGH PATHS 3
WeletG=GN(Rd)denotethestep-N freenilpotentLiegroupoverRd forsome
N ≥ 2, and let U ,...,U be a set of generators for its Lie algebra g = gN(Rd),
1 d
which we identify with the space of left-invariant vector fields on G. We equip
Rd with the inner product for which U ,...,U form an orthonormal basis upon
1 d
canonically identifying Rd with a subspace of g.
For Λ > 0, let Ξ(Λ) = ΞN,d(Λ) denote the set of measurable functions a on G
taking values in symmetric d×d matrices which are sub-elliptic in the following
sense:
Λ−1|ξ|2 ≤hξ,a(x)ξi≤Λ|ξ|2, ∀ξ ∈Rd, ∀x∈G.
We let λ denote the Haar measure on G and define the associated Dirichlet form
E =Ea on L2(G,λ) for all f,g ∈C∞(G) by
c
(1.3) E(f,g)= ai,j(U f)(U g)dλ.
i j
ZG i,j
X
We let X = Xa,x denote the Markov diffusion on G associated to E with starting
point X = x ∈ G. We recall that the sample paths of X are a.s. geometric α-
0
Hölder roughpaths for all α∈(0,1/2),and when a(x) depends only on the level-1
projection π (x) ∈ Rd of x ∈ G, X serves as the natural rough path lift of the
1
Markovdiffusionassociatedtothe Dirichletform(1.1)onL2(Rd)discussedearlier.
For further details, we refer to [11].
Remark 1.1. Throughout the paper we assume the symmetric Dirichlet form (1.3)
is defined on the Hilbert space L2(G,λ) so that X is symmetric with respect to
λ. As pointed out in [6], it is natural to also consider E defined over L2(G,µ) for
a measure µ(dx) = v(x)λ(dx), v ≥ 0. While for simplicity we only work with E
defined on L2(G,λ), we note that appropriate assumptions of v and a Girsanov
transform(see, e.g.,[8]) canbe usedto relate the results ofthis paper to this more
general setting.
Acknowledgements. I.C.is funded by a Junior ResearchFellowshipofSt John’s
College, Oxford. M.O. is funded by a PhD scholarship from the Department of
Mathematics, Imperial College London. The authors would like to thank Thomas
Cass, Peter Friz, and Terry Lyons for helpful conversationson this topic.
2. Support theorem
2.1. Restricted Hölder norms. We first record some deterministic results on
Hölder norms which will be used in the sequel. Throughout this section, let (E,d)
be a metric space, α∈(0,1],T >0,and x∈C([0,T],E)a continuouspath. Let (cid:3)
denote any of the relations <,≤,=,≥,>,and consider the quantity
d(x ,x )
||x|| = sup u v ,
α-Höl,(cid:3)ε;[s,t] u,v∈[s,t],|u−v|(cid:3)ε |u−v|α
where we set ||x|| =0 if the set {(u,v)∈[s,t]2 ||u−v|(cid:3)ε} is empty.
α-Höl,(cid:3)ε;[s,t]
For ε,γ >0 and s∈[0,T], define also the times (τε,γ,s) =(τ ) by τ =s
n n≥0 n n≥0 0
and for n≥1
τ =inf{t>τ |||x|| ≥γ}.
n n−1 α-Höl,≥ε;[τn−1,t]
We call any such τ a Hölder stopping time of x.
i
4 ILYACHEVYREVANDMARCELOGRODNIK
Lemma 2.1. Let ε,γ >0 and s=0, and suppose that for some c>0
(2.1) ∀n≥0, sup d(x ,x )<c.
τn t
t∈[τn,τn+ε]
Then ||x|| <γ˜ :=(3cε−α)∨(4γ+cε−α).
α-Höl,=ε;[0,T]
Proof. For n ≥ 1 and t ∈ [τ −ε,τ ], we have one of the following three mutually
n n
exclusivecases: (a)τ =τ +ε,(b)τ ∈(τ +ε,τ +2ε]andt∈[τ ,τ +
n n−1 n n−1 n−1 n−1 n−1
ε], or (c) t > τ +ε. In case (a), (2.1) implies that d(x ,x )< 2c. In case (b),
n−1 t τn
d(x ,x )≤γ(2ε)α and (2.1) implies that d(x ,x )<c, so that
τn τn−1 t τn−1
d(x ,x )<c+γ(2ε)α ≤(2c)∨(4γεα).
t τn
In case (c), we have d(x ,x )<γεα and d(x ,x )≤γ(2ε)α, so that
t−ε t t−ε τn
d(x ,x )<γεα+γ(2ε)α ≤3γεα.
t τn
Hence, in all three cases
(2.2) d(x ,x )<(2c)∨(4γεα).
t τn
Consider now
τ =inf{t>0|||x|| =γ˜}.
α-Höl;=ε;[0,T]
Note that ||x|| ≥γ˜ ⇔τ < ∞. Arguing by contradiction, suppose that
α-Höl;=ε;[0,T]
τ < ∞, which means that d(x ,x ) = γ˜εα. Consider the largest n for which
τ−ε τ
τ ≤τ. Observe that τ ∈[τ −ε,τ], since otherwise d(x ,x )<γεα, which is a
n n τ−ε τ
contradiction since γ˜ > γ. It follows from (2.1) that d(x ,x ) ≤ c, and therefore
τn τ
by (2.2) and the triangle inequality
d(x ,x )<c+(2c)∨(4γεα)=γ˜εα,
τ−ε τ
which is again a contradiction. (cid:3)
Lemma 2.2. Suppose that ||x|| ≤γ for every n>N ∈Z. Then
α-Höl;=2−nε;[0,T]
γ
||x|| ≤ .
α-Höl;<2−Nε;[0,T] 1−2−α
Proof. Consider (t−s)/ε ∈ (0,2−N) with the binary representation (t−s)/ε =
∞ c 2−n with c ∈{0,1}, m>N, and c =1. It follows that
n=m n n m
∞
P
d(x ,x )≤γ εαc 2−nα.
s t n
n=m
X
Since 2−m ≤(t−s)/ε, we have εα2−nα ≤2α(m−n)(t−s)α. Hence
∞ γ(t−s)α
d(x ,x )≤γ 2α(m−n)(t−s)α = .
s t 1−2−α
n=m
X
(cid:3)
Lemma 2.3. Suppose there exist x∈E and r >0 such that for all integers k ≥0,
x ∈B(x,r) and ||x|| ≤γ. Then
kε α-Höl;≤ε;[kε,(k+1)ε]
||x|| ≤2γ+2rε−α.
α-Höl;[0,T]
A SUPPORT AND DENSITY THEOREM FOR MARKOVIAN ROUGH PATHS 5
Proof. Consider 0≤s<t≤[0,T], and denote s∈[kε,(k+1)ε), t∈[nε,(n+1)ε).
Ifk =nthereisnothingtoprove,sosupposek <n. If|t−s|≤ε,sothatn=k+1,
then
d(x ,x )≤d(x ,x )+d(x ,x )≤γ21−α|t−s|α.
s t s nε nε t
Finally, if |t−s|>ε then since x ,x ∈B(x,r), it follows that
kε nε
|t−s|−αd(x ,x )≤|t−s|−α(d(x ,x )+d(x ,x )+d(x ,x ))
s t kε s kε nε nε t
≤|t−s|−α(2εαγ+2r)
≤2γ+2rε−α.
(cid:3)
2.2. Positive probability of small Hölder norm. Suppose now(E,d) is aPol-
ish space. In this section, we give conditions under which an E-valued process has
anexplicitpositiveprobabilityofkeepingasmallHöldernorm. Wefixα∈(0,1/2),
aterminaltimeT >0,andanE-valuedstochasticprocessXadaptedtoafiltration
(F ) .
t t∈[0,T]
Consider the following conditions:
(1) ThereexistsC >0suchthatforeveryc,ε>0,andeveryHölderstopping
1
time τ of X, a.s.
−c2
P sup d(X ,X )>c|F ≤C exp .
τ t τ 1
C ε
"t∈[τ,τ+ε] # (cid:18) 1 (cid:19)
(2) There exist c ,C > 0 and x ∈ E such that for every s ∈ [0,T] and
2 2
ε∈(0,T −s], a.s.
P X ∈B(x,C ε1/2)|F ≥c 1{X ∈B(x,C ε1/2)}.
s+ε 2 s 2 s 2
h i
Roughly speaking, the first condition states that the probability of large fluctu-
ationsofXoversmalltime intervalsshouldhavethesameGaussiantailsasthatof
a Brownianmotion, while the second condition bounds from below the probability
that X is in a ball of radius ∼ε1/2 given that X was in the same ball.
s+ε s
Theorem 2.4. Assume conditions (1) and (2). Then there exist C2.4,c2.4 > 0,
depending only on C ,c ,C ,α and T, such that for every γ >0, a.s.
1 2 2
P ||X||α-Höl;[0,T] <γ |F0 ≥C2−.14exp γ2−/C(12−.24α) 1{X0 ∈B(x,c2.4γ1/(1−2α))}.
h i (cid:18) (cid:19)
Lemma 2.5. Assume condition (1). Then there exists C2.5 > 0, depending only
on C and α, such that for all 0≤s<t≤T and ε∈(0,t−s], a.s.
1
−γ2(1−2−α)2
P ||X||α-Höl;≤ε;[s,t] ≥γ |Fs ≤C2.5(t−s)ε−1(γ−2ε1−2α+1)exp 9C ε1−2α .
h i (cid:18) 1 (cid:19)
Proof. Let τ = τε,γ,s be defined as usual with τ = s. Note that (1) implies that
n n 0
for all c,γ >0, t>s and ε∈(0,t−s],
−c2
P ∃n≥0,τ ≤t, sup d(X ,X )>c|F ≤⌈(t−s)/ε⌉C exp ,
n τn u s 1 C ε
" u∈[τn,τn+ε] # (cid:18) 1 (cid:19)
6 ILYACHEVYREVANDMARCELOGRODNIK
so that by Lemma 2.1
−c2
P ||X|| ≥(3cε−α)∨(4γ+cε−α)|F ≤⌈(t−s)/ε⌉C exp .
α-Höl;=ε;[s,t] s 1 C ε
h i (cid:18) 1 (cid:19)
In particular, choosing c=2γεα yields that for all γ >0, t>s, and ε∈(0,t−s],
−(2γ)2
P ||X|| ≥6γ |F ≤⌈(t−s)/ε⌉C exp .
α-Höl;=ε;[s,t] s 1 C ε1−2α
h i (cid:18) 1 (cid:19)
Hence
∞ −2n(1−2α)γ2
P ∃n≥0,||X|| ≥γ |F ≤2C (t−s)ε−1 2nexp .
α-Höl;=2−nε;[s,t] s 1 9C ε1−2α
h i nX=0 (cid:18) 1 (cid:19)
The conclusion now follows from Lemma 2.2 and the observation that for every
θ >0 there exists C such that for all K >0
4
∞
2nexp −K2θn ≤C (K−1+1)e−K
4
n=0
X (cid:0) (cid:1)
(whichcanbe seen,forexample,by the integraltestandthe asymptoticbehaviour
of the incomplete gamma function Γ(p,K)). (cid:3)
Proof of Theorem 2.4. For γ,ε>0 and s∈[0,T], consider the event
A ={||X|| <γ,X ∈B(x,C ε1/2)}.
s α-Höl;≤ε;[s,s+ε] s+ε 2
Applying condition(2)andLemma2.5witht=s+ε,wesee thatforalls∈[0,T],
and ε,γ >0
−γ2(1−2−α)2
P[As |Fs]≥c21{Xs ∈B(x,C2ε1/2)}−C2.5(γ−2ε1−2α+1)exp 9C ε1−2α .
(cid:18) 1 (cid:19)
Observe also that Lemma 2.3 (with r =C ε1/2) implies that for all ε,γ >0
2
⌈T/ε⌉−1
P ||X|| <2γ+2C ε1/2−α |F ≥P A |F .
α-Höl;[0,T] 2 0  kε 0
h i k\=0
 
It remains to control the final last probability on the RHS. We set ε=c γ2/(1−2α)
1
(so that ε1/2−α ∼ γ), where c > 0 is sufficiently small (and depends only on
1
C1,c2,C2,C2.5 and α) such that
−(1−2−α)2
κ:=c2−C2.5(c11−2α+1)exp 36C c1−2α >0.
(cid:18) 1 1 (cid:19)
so in particular for all s∈[0,T] and γ >0,
P[A |F ]≥κ1{X ∈B(x,C ε1/2)}.
s s s 2
Inductively applying conditional expectations, it follows that for all n≥0
n
P A |F ≥κn+11{X ∈B(x,C ε1/2)}.
kε 0 0 2
" #
k=0
\
Taking n=⌈T/ε⌉−1 yields the desired result. (cid:3)
A SUPPORT AND DENSITY THEOREM FOR MARKOVIAN ROUGH PATHS 7
2.3. Support theorem for Markovian rough paths. We now turn to the sup-
porttheoremforMarkovianroughpathsinα-Höldertopology. RecallNotation1.1
and equip G=GN(Rd) with the Carnot-Carathéodorymetric d.
RecalltheSobolevpathspaceW1,2 =W1,2([0,T],Rd)andthetranslationopera-
torT (x)definedforx∈Cp-var([0,T],G),1≤p<N+1,andh∈C1-var([0,T],Rd)
h
(see [11] Section 9.4).
Let us fix α∈(0,1/2). We will show that for all h∈W1,2 and γ >0,
Pa,x dα-Höl;[0,T](X,SN(h))<γ >0,
where dα-Höl;[0,T] denotes th(cid:2)e (homogeneous) α-Hölde(cid:3)r metric. Indeed, this is an
immediate consequenceofthe continuityof Th(·)indα-Höl;[0,T] andofthe following
result.
Theorem 2.6. Let h∈W1,2. There exists a constant C2.6, depending only on Λ,
||h|| , α, and T, such that for all a∈Ξ(Λ), x∈G and γ >0
W1,2
Pa,x ||T (X)|| <γ ≥C−1 exp −C2.6 .
h α-Höl;[0,T] 2.6 γ2/(1−2α)
h i (cid:18) (cid:19)
For the proof, recall that the Fernique estimate ([11] Corollary 16.12) implies
that for every stopping time τ and p>2, a.s.
−c2
(2.3) P ||X|| >c|F ≤C exp ,
p-var;[τ,τ+ε] τ F C ε
h i (cid:18) F (cid:19)
where C depends only on Λ and p. Moreover,recall that
F
||h|| ≤C ε1/2||h||
1-var;[s,s+ε] W W1,2;[s,s+ε]
for a universal constant C . We now prove two lemmas which demonstrate that
W
the (in general non-Markov)process T (X) satisfies conditions (1) and (2).
h
Lemma 2.7. There exists a constant C, depending only on Λ, such that for all
c,ε>0 satisfying
c2
(2.4) ε≤ ,
4C2 ||h||2
W W1,2
it holds that for every stopping time τ, a.s.
−c2
P sup d(T (X) ,T (X) )>c|F ≤Cexp .
h τ h t τ
4Cε
"t∈[τ,τ+ε] # (cid:18) (cid:19)
Proof. Fix any 2<p<N +1. Note that (see, e.g., [11] Exercise 9.37)
sup d(T (X) ,T (X) )≤||T (X)||
h s h t h p-var;[s,s+ε]
t∈[s,s+ε]
≤C ||X|| +||h|| .
1 p-var;[s,s+ε] 1-var;[s,s+ε]
(cid:16) (cid:17)
Hence whenever c,ε > 0 satisfy (2.4), we have ||h|| ≤ c/2, and the
1-var;[s,s+ε]
conclusion follows from the Fernique estimate (2.3). (cid:3)
Lemma 2.8. For all C ≥C (Λ,||h|| )>0, there exists c=c(C,Λ,||h|| )>
0 W1,2 W1,2
0 such that for all x∈G, s∈[0,T], and ε∈(0,T −s], a.s.
P T (X) ∈B(x,Cε1/2)|F ≥c1{T (X) ∈B(x,Cε1/2)}.
h s+ε s h s
h i
8 ILYACHEVYREVANDMARCELOGRODNIK
Proof. We use the shorthand notation Y = T (X). For every x,y ∈ G, consider a
h
geodesic γy,x : [0,1] 7→ G with γy,x = y and γy,x = x parametrised at unit speed.
0 1
Let z(y,x):=γy,x denote its midpoint. For any x∈G, observe that
1/2
d(Y ,x)≤d(Y ,z(Y ,x))+d(z(Y ,x),x)
s+ε s+ε s s
≤d(Y ,X )+d(X ,Y−1z(Y ,x))+d(z(Y ,x),x).
s,s+ε s,s+ε s,s+ε s s s
If Y ∈ B(x,r), then evidently d(z(Y ,x),x) ≤ r/2. Moreover, by lower bounds
s s
on the heat kernel and the volume of balls ([11] Chapter 16), there exists C > 0,
1
depending only on Λ, such that for all x∈G, r,ε>0 and s∈[0,T]
−C r2
P d(X ,Y−1z(Y ,x))<r/4|F ≥C−1exp 1 1{Y ∈B(x,r)}.
s,s+ε s s s 1 ε s
(cid:18) (cid:19)
Fina(cid:2)lly, by standard rough paths estim(cid:3)ates (using that T (X) is equal to X
h s,t s,t
plus a combination cross-integralsof X and h over [s,t]) we have
1/i
i
d(X ,Y )≤C max ||h||k ||X||i−k
s,s+ε s,s+ε 2 1-var;[s,s+ε] p-var;[s,s+ε]
i∈{1...,N} !
k=1
X
1/i
i
≤C max εk/2||h||k ||X||i−k .
3 W1,2;[s,s+ε] p-var;[s,s+ε]
i∈{1...,N} !
k=1
X
Hence, if ||X|| ≤Rε1/2, then for some C >0 depending only on G
p-var;[s,s+ε] 4
d(X ,Y )≤C ε1/2(||h||1/N +||h|| )(1+R(N−1)/N).
s,s+ε s,s+ε 4 W1,2;[s,s+ε] W1,2;[s,s+ε]
We now let r =Cε1/2. It follows that if C and R satisfy
(2.5) C ≥4C (||h||1/N +||h|| )(1+R(N−1)/N),
4 W1,2;[s,s+ε] W1,2;[s,s+ε]
then by the Fernique estimate (2.3), for any 2<p<N +1,
P d(X ,Y )>Cε1/2/4|F ≤P ||X|| >Rε1/2,|F
s,s+ε s,s+ε s p-var;[s,s+ε] s
h i h −R2 i
≤C exp .
F
C
(cid:18) F (cid:19)
It follows that if C and R furthermore satisfy
−R2
(2.6) c:=C−1exp −C C2 −C exp >0
1 1 F C
(cid:18) F (cid:19)
(cid:0) (cid:1)
then we obtain
P d(Y ,x)<Cε1/2 |F ≥c1{Y ∈B(x,Cε1/2)}.
s+ε s s
h i
WenowobservethatduetothefactorR(N−1)/N in(2.5)above,thereexistsC >0,
0
depending only on ||h|| and Λ, such that for every C ≥C , we can find R>0
W1,2 0
for which (2.5) and (2.6) are satisfied. (cid:3)
Proof of Theorem 2.6. By Theorem 2.4, it suffices to check that T (X) satisfies
h
conditions (1) and(2) with constants C ,c ,C only depending on Λ and||h|| .
1 2 2 W1,2
However this follows directly from Lemmas 2.7 and 2.8. (cid:3)
A SUPPORT AND DENSITY THEOREM FOR MARKOVIAN ROUGH PATHS 9
3. Density theorem
3.1. Semi-Dirichlet associated with Hörmander vector fields. In this sub-
section, let O be a smooth manifold and W =(W ,...,W ) a collection of smooth
1 d
vector fields on O. For z ∈ O, let Lie W denote the subspace of T O spanned
z z
by the vector fields (W ,...,W ) and all their commutators at z. We say that W
1 d
satisfies Hörmander’s condition on O if Lie W = T O for every z ∈ O, in which
z z
case we call W a collection of Hörmander vector fields.
Fix a collectionW =(W ,...,W ) of Hörmandervector fields on O and U ⊂O
1 d
an open subset with compact closure. Consider a bounded measurable function a
onU takingvaluesin(notnecessarilysymmetric)d×dmatricessuchthatforsome
Λ≥1
(3.1) Λ−1|ξ|2 ≤hξ,a(z)ξi, ∀ξ ∈Rd, ∀z ∈U.
Let µ be a smooth measure on O and define the bilinear map
E :C∞(U)×C∞(U)7→R
c c
d
E :(f,g)7→− ai,j(z)(W f)(z)(W∗g)(z)µ(dz),
i j
i,j=1ZU
X
where W∗ = −W −div W is the formal adjoint of W with respect to µ. The
j j µ j j
following is the main result of this subsection. For background concerning (non-
symmetric,semi-)Dirichletforms,we referto [18]. The Lp norm||·|| for p∈[1,∞]
p
is assumed to be on Lp(U,µ).
Proposition 3.1. The bilinear form E is closable in L2(U,µ), lower bounded, and
satisfies the sector condition.
Let P denote the associated (strongly continuous) semi-group on L2(U,µ). Sup-
t
pose further that P is sub-Markov (so that the closed extension of E is a lower-
t
bounded semi-Dirichlet forms) and maps C (U) into itself. Then there exists ν >2
b
and b>0 such that for every x∈U and t>0 there exists p (x,·) ∈L2(U,µ) with
t
||p (x,·)|| ≤bt−ν/2 such that for all f ∈L2(U,µ)
t 2
P f(x)= p (x,y)f(y)µ(dy).
t t
ZU
TheproofofProposition3.1,whichwedefertoAppendixA,isbasedonthesub-
Riemannian Sobolev inequality combined with a classical argument of Nash [17].
3.2. Density for RDEs. We now specialise to the setting of Markovian rough
paths. Recall Notation 1.1 and consider the RDE
(3.2) dY =V(Y )dX , Y =y ∈Re,
t t t 0 0
for smooth vector fields V =(V ,...,V ) on Re. We suppose also that V are Lip2
1 d
so that (3.2) admits a unique solution. We fix also the starting point X =x ∈G
0 0
of X.
ConsiderthemanifoldG×Re. WecanonicallyidentifythetangentspaceT (G×
(x,y)
Re) with T G⊕T Re and define smooth vector fields on G×Re by W =U +V .
x y i i i
Let z = (x ,y ) ∈ G× Re and denote by O = O the orbit of z under the
0 0 0 z0 0
collection W = (W ,...,W ). By the orbit theorem (e.g., [1] Chapter 5), O is an
1 d
immersed submanifold of G×Re and Lie W ⊆T O for all z ∈O.
z z
10 ILYACHEVYREVANDMARCELOGRODNIK
Denote the coupleZ =(X ,Y )whichisaMarkovprocessonG×Re. Onecan
t t t
readily show that a.s. Zz0 ∈ O for all t > 0 (e.g., by approximating each sample
t
path of X in p-variation for some p>2 by piecewise linear paths).
Theorem 3.2. Suppose W satisfies Hörmander’s condition on O, i.e., Lie W =
z
T O for all z ∈ O. Then for all t > 0, Zz0 admits a density with respect to any
z t
smooth measure µ on O.
Remark 3.3. In the special case that a(x) depends only on the first level π (x)
1
for all x ∈ G2(Rd), the identical statement in Theorem 3.2 holds for the process
Z = (π (X ),Y ) upon setting N = 1 and G = G1(Rd) ∼= Rd, and going through
t 1 t t
the same discussion above. The reason for this is that Lemma 3.8 below can be
readily adjusted to give analogous infinitesimal behaviour of the process Z (now
t
takingvaluesinO⊆Rd×Re),afterwhichtheproofofthetheoremcarriesthrough
without change.
For a statement of the density of Y itself, let O′ ⊆ Re denote the orbit of
t
y ∈ Re under V (e.g., O′ = Re whenever V satisfies Hörmander’s condition on
0
Re). By the description of the tangent space T O in the orbit theorem, it holds
z
that the projection p : O 7→O′,(x,y) 7→y, is a (surjective) submersion (in fact a
2
smoothfibrebundle)fromOtoO′. Sincepre-imagesofnull-setsundersubmersions
arenull-sets, we see that Y admits a density in O′ wheneverZz0 admits a density
t t
in O.
Furthermore,theconditioninthetheoremmaybeeasilyrestatedintermsofjust
the driving vector fields V = (V ,...,V ). Indeed, by the Frobenius theorem and
1 d
the free step-N nilpotency of G, it holds that W satisfies Hörmander’s condition
on O if and only if
(3.3) the dimension of span{V (y):|I|>N}⊆T Re is constant in y ∈O′.
[I] y
In the above, we denote by V the vector field [[...[V ,V ],...],V ] for a multi-
[I] i1 i2 ik
index I =(i ,...,i )∈{1,...,d}k of length |I|=k.
1 k
We thus obtain the following corollary of Theorem 3.2.
Corollary3.4. Supposecondition (3.3)holds. Thenforallt>0,theRDEsolution
Y admits a density with respect to any smooth measure on O′.
t
Remark 3.5. FollowingRemark 3.3, in the case that a(x) depends only on the first
level π (x), we are able to take N =1 in (3.3) when applying Corollary 3.4.
1
Remark 3.6. Note that while (3.3) (for any N ≥ 0) implies that V satisfies Hör-
mander’sconditionon O′,the reverseimplicationis clearlynottrue. Inparticular,
wedonotknowifit sufficientfor V to onlysatisfy Hörmander’sconditiononO′ in
order for Y to admit a density on O′. The difficulty of course is that unless (3.3)
t
is satisfied, the couple (X ,Y ) will in general not admit a density in O, whereby
t t
our method of proof breaks down.
For the proof of Theorem 3.2, we first recall for the reader’s convenience the
infinitesimalbehaviourofthe coordinateprojectionsofXa. Asbefore,letλdenote
the Haar measure on G.