COUPLING IN THE HEISENBERG GROUP AND ITS APPLICATIONS TO GRADIENT ESTIMATES

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1 COUPLING IN THE HEISENBERG GROUP AND ITS APPLICATIONS TO GRADIENT ESTIMATES SAYAN BANERJEE, MARIA GORDINA, AND PHANUEL MARIANO Absrac. We consruc a non-markovian coupling for hypoellipic diffusions which are Brownian moions in he hree-dimensional Heisenberg group. We hen derive properies of his coupling such as esimaes on he coupling rae, and upper and lower bounds on he oal variaion disance beween he laws of he Brownian moions. Finally we use hese properies o prove gradien esimaes for harmonic funcions for he hypoellipic Laplacian which is he generaor of Brownian moion in he Heisenberg group. Conens. Inroducion. Preliminaries 4.. Sub-Riemannian basics 4.. The Heisenberg group 6 3. Coupling resuls 7 4. Gradien esimaes 9 5. Concluding remarks 3 References 3. Inroducion Recall ha a coupling of wo probabiliy measures µ and µ, defined on respecive measure spaces Ω, A and Ω, A, is a measure µ on he produc space Ω Ω, A A wih marginals µ and µ. In his aricle, we will be ineresed in coupling of he laws of wo Markov processes X : and Y : in a geomeric seing of a sub-riemannian manifold such as he Heisenberg group H 3. Namely, we discuss couplings of wo Markov processes having he same generaor bu saring from differen poins joining ogeher coupling a some random ime, and how hese can be used o obain oal variaion bounds and prove gradien esimaes for harmonic funcions on H 3. Couplings have been an exremely useful ool in probabiliy heory and has resuled in esablishing deep connecions beween probabiliy, analysis and geomery. 99 Mahemaics Subjec Classificaion. Primary 58G3; Secondary 35H, 6J6, 35R3. Key words and phrases. coupling, hypoellipic diffusion, Heisenberg group. Research was suppored in par by EPSRC Research Gran EP/K3939. Research was suppored in par by he Simons Fellowship. Research was suppored in par by NSF Gran DMS-7496.

2 BANERJEE, GORDINA, AND MARIANO We sar by providing some background on couplings and hen on gradien esimaes in our seing. The coupling is said o be successful if he wo processes couple wihin finie ime almos surely, ha is, he coupling ime for X and Y defined as τx, Y = inf{ : X s = Y s for all s }. is almos surely finie. A major applicaion of couplings arises in esimaing he oal variaion disance beween he laws of wo Markov processes a ime which in general is very hard o compue explicily. Such an esimae can be obained from he Aldous inequaliy. µ {τx, Y > } LX LY T V, where µ is he coupling of he Markov processes X and Y, LX and LY denoe he laws disribuions of X and Y respecively, and ν T V = sup{ νa : A measurable} denoes he oal variaion norm of he measure ν. This, in urn, can be used o provide sharp raes of convergence of Markov processes o heir respecive saionary disribuions, when hey exis see [8] for some such applicaions in sudying mixing imes of Markov chains. This raises a naural quesion: how can we couple wo Markov processes so ha he probabiliy of failing o couple by ime coupling rae is minimized in an appropriae sense for some, preferably all,? Griffeah [6] was he firs o prove ha maximal couplings, ha is, he couplings for which he Aldous inequaliy becomes an equaliy for each in he ime se of he Markov process, exis for discree ime Markov chains. This was laer grealy simplified by Piman [33] and generalized o non-markovian processes by Goldsein [4] and coninuous ime càdlàg processes by Sverchkov and Smirnov [35]. These consrucions, hough exremely elegan, have a major drawback: hey are ypically very implici. Thus, i is very hard, if no impossible, o perform deailed calculaions and obain precise esimaes using hese couplings. Par of he impliciness comes from he fac ha hese couplings are non-markovian. A Markovian coupling of wo Markov processes X and Y is a coupling where, for any, he join process {X s, Y s : s } condiioned on he filraion σ{x s, Y s : s } is again a coupling of he laws of X and Y, bu now saring from X, Y. These are he mos widely used couplings in deriving esimaes and performing deailed calculaions as heir consrucions are ypically explici. However, hese couplings usually do no aain he opimal raes. In fac, i has been shown in [3] ha he exisence of a maximal coupling ha is also Markovian imposes enormous consrains on he generaor of he Markov process and is sae space. Furher, [] describes an example using Kolmogorov diffusions defined as a wo dimensional diffusion given by a sandard Brownian moion along wih is running ime inegral, where for any Markovian coupling, he probabiliy of failing o couple by ime does no even aain he same order of decay wih as he oal variaion disance. More precisely, hey showed ha if he driving Brownian moions sar from he same poin, hen he oal variaion disance beween he corresponding Kolmogorov diffusions decays like 3/ whereas for any Markovian coupling, he coupling rae is a bes of order /.

3 HEISENBERG COUPLING 3 This brings us o he main subjec of his aricle: when can we produce non- Markovian couplings ha are explici enough o give us good bounds on he oal variaion disance beween he laws of X and Y when Markovian couplings fail o do so? And wha informaion can such couplings provide abou he geomery of he sae space of hese Markov processes? In his aricle, we look a he Heisenberg group which is he simples example of a sub-riemannian manifold and Brownian moion on i. The laer is he Markov process whose generaor is he sub-laplacian on he Heisenberg group as described in Secion. We consruc an explici successful non-markovian coupling of wo copies of his process saring from differen poins in H 3 and use i o derive sharp bounds on he oal variaion disance beween heir laws a ime. We also use his coupling o produce gradien esimaes for harmonic funcions on he Heisenberg group more deails below, hus providing a non-rivial link beween probabiliy and geomeric analysis in he sub-riemannian seing. We noe here ha successful Markovian couplings of Brownian moions on he Heisenberg group have been consruced in [3] and raes of hese couplings have been sudied in [4]. However, he raes for he coupling we consruc are much beer. In fac, we show in Remark 3. ha i is impossible o derive he raes we ge from Markovian couplings. Moreover, he coupling we consider is efficien, ha is, he coupling rae and he oal variaion disance decay like he same power of as poined ou in Remark 3.7. Now we would like o describe gradien esimaes in geomeric seings and how couplings have been used o prove hem previously. Le us sar wih a classical gradien esimae for harmonic funcions in R d. Suppose u is a real-valued funcion u on R d which is harmonic in a ball B δ x, hen here exiss a posiive consan C d which depends only on he dimension d and no on u such ha sup ux C d sup ux. x B δ x δ x B δ x In 975, Cheng and Yau see [, 34, 37] generalized he classical gradien esimae o complee Riemannian manifolds M of dimension d wih Ricci curvaure bounded below by d K for some K. They proved ha any posiive harmonic funcion on a Riemannian ball B δ x saisfies ux sup C d x B δ/ x ux δ + K. Moreover, in addiion o such esimaes, here is a vas lieraure on funcional inequaliies such as hea kernel gradien esimaes, Poincaré inequaliies, hea kernel esimaes, ellipic and parabolic Harnack inequaliies ec on Riemannian manifolds or more generally on measure meric spaces. Quie ofen hese resuls require assumpions such as volume doubling and curvaure bounds. In 99, M. Cranson in [] used he mehod of coupling wo diffusion processes o obain a similar gradien esimae for soluions o he equaion. u + Zu = on a Riemannian manifold M, g whose Ricci curvaure is bounded below and Z is a bounded vecor field. This coupling is known as he Kendall-Cranson coupling

4 4 BANERJEE, GORDINA, AND MARIANO as i was based on he echniques in []. In paricular, M. Cranson proved he following gradien esimae. Theorem. Cranson. Suppose M, g is a complee d-dimensional Riemannian manifold wih disance ρ M and assume Ric M Kg. Le Z be a C vecor field on M such ha Zx m for all x M. There is a consan c = c K, d, m such ha whenever δ > and. is saisfied in some Riemannian ball B δ x, we have ux c δ + sup ux, x B x, δ. x Bx,3δ/ If. is saisfied on M and u is bounded and posiive, hen ux K d + m u. Cranson s approach generalized he coupling of Brownian moions on manifolds of Kendall [] o couple processes wih he generaor L = + Z. The mehods in ha paper required ools from Riemannian geomery such as he Laplacian comparison heorem and he index heorem o obain esimaes on he processes ρ M X, Y and ρ M X, X where ρ M is he Riemannian disance. M. Cranson also proved similar resuls on R d in []. In his paper we consider he simples sub-riemannian manifold, he Heisenberg group H 3 as a saring poin of using couplings for proving gradien esimaes in such a seing. As he generaor of H 3 -valued Brownian moion is a hypoellipic operaor, funcional inequaliies for he corresponding harmonic funcions or hypoellipic hea kernels are much more challenging o prove. There was recen progress in using generalized curvaure-dimension inequaliies for such resuls e.g. [, 4, 5], as well as resuls in he spiri of opimal ranspor e.g. [6]. The main poin of he curren paper is no wheher a coupling can be consruced, as hese have been known since [6], bu raher finding a necessarily non-markovian coupling ha gives sharp oal variaion bounds and explici gradien esimaes. The properies of he coupling we consruc in he curren paper are crucial in his, and i is ineresing o conras his wih opimaliy or he lack of i for he Kendall- Cranson coupling in he Riemannian manifolds as described in [5, 7]. The paper is organized as follows. Secion gives basics on sub-riemannian manifolds and he Heisenberg group H 3 including Brownian moion on H 3. In Secion 3 we consruc he non-markovian coupling of Brownian moions in H 3, and describe is properies. Finally, in Secion 4 we prove he gradien esimaes for harmonic funcions for he hypoellipic Laplacian which is he generaor of Brownian moion in he Heisenberg group.. Preliminaries.. Sub-Riemannian basics. A sub-riemannian manifold M can be hough of as a Riemannian manifold where we have a consrained movemen. Namely, such a manifold has he srucure M, H,,, where allowed direcions are only he ones in he horizonal disribuion, which is a suiable subbundle H of he angen bundle T M. For more deail on sub-riemannian manifolds we refer o [3]. Namely, for a smooh conneced d-dimensional manifold M wih he angen bundle T M, le H T M be an m-dimensional smooh sub-bundle such ha he

5 HEISENBERG COUPLING 5 secions of H saisfy Hörmander s condiion he bracke generaing condiion formulaed in Assumpion. We assume ha on each fiber of H here is an inner produc, which varies smoohly beween fibers. In his case, he riple M, H,, is called a sub-riemannian manifold of rank m, H is called he horizonal disribuion, and, is called he sub-riemannian meric. The vecors resp. vecor fields X H are called horizonal vecors resp. horizonal vecor fields, and curves γ in M whose angen vecors are horizonal, are called horizonal curves. Assumpion. Hörmander s condiion We will say ha H saisfies Hörmander s bracke generaing condiion if horizonal vecor fields wih heir Lie brackes span he angen space T p M a every poin p M. Hörmander s condiion guaranees analyic and opological properies such as hypoellipiciy of he corresponding sub-laplacian and opological properies of he sub-riemannian manifold M. We explain briefly boh aspecs below. Firs we define he Carno-Carahéodory meric d CC on M by. d CC x, y = { } inf γ H d where γ = x, γ = y, γ is a horizonal curve, where as usual inf :=. Here he norm is induced by he inner produc on H, namely, v H := v, v p for v H p, p M. The Chow-Rashevski heorem says ha Hörmander s condiion is sufficien o ensure ha any wo poins in M can be conneced by a finie lengh horizonal curve. Moreover, he opology generaed by he he Carno-Carahéodory meric coincides wih he original opology of he manifold M. As we are ineresed in a Brownian moion on a sub-riemannian manifold M, H,,, a naural quesion is wha is generaor is. While here is no canonical operaor such as he Laplace-Belrami operaor on a Riemannian manifold, here is a noion of a sub-laplacian on sub-riemannian manifolds. A second order differenial operaor defined on C M is called a sub-laplacian H if for every p M here is a neighborhood U of p and a collecion of smooh vecor fields {X, X,..., X m } defined on U such ha {X,..., X m } are orhonormal wih respec o he sub-riemannian meric and m H = Xk + X. k= By he classical heorem of L. Hörmander in [8, Theorem.] Hörmander s condiion Assumpion guaranees ha any sub-laplacian is hypoellipic. For more properies of sub-laplacians which are generaors of a Brownian moion on a sub- Riemannian manifold we refer o [5]. Finally, he horizonal gradien H is a horizonal vecor field such ha for any smooh f : M R we have ha for all X H, H f, X = X f. We define he lengh of he gradien as in [6]. For a funcion f on M, le. H f x := lim sup f x f x r d CC x, x, and se H f := sup x H 3 H f x. <d CC x, x r

6 6 BANERJEE, GORDINA, AND MARIANO.. The Heisenberg group. The Heisenberg group H 3 is he simples non-rivial example of a sub-riemannian manifold. Namely, le H 3 = R 3 wih he muliplicaion defined by x, y, z x, y, z := x + x, y + y, z + z + x y x y, wih he group ideniy e =,, and he inverse given by x, y, z = x, y, z. We define X, Y, and Z as he unique lef-invarian vecor fields wih X e = x, Y e = y, and Z e = z, so ha X = x y z, Y = y + x z, Z = z. The horizonal disribuion is defined by H = span{x, Y} fiberwise. Observe ha [X, Y] = Z, so Hörmander s condiion is easily saisfied. Moreover, as any ieraed Lie bracke of lengh greaer han wo vanishes, H 3 is a nilpoen group of sep. The Lebesgue measure on R 3 is a Haar measure on H 3. We endow H 3 wih he sub- Riemannian meric, so ha {X, Y} is an orhonormal frame for he horizonal disribuion. As poined ou in [5, Example 6.], he sum of squares operaor.3 H = X + Y is a naural sub-laplacian for he Heisenberg group wih his sub-riemannian srucure. In general i is very cumbersome o compue he Carno-Carahéodory disance d CC explicily. In he case of he Heisenberg group an explici formula for he disance is known. Le r x = d CC x, e be he disance beween x = x, y, z H 3 and he ideniy e =,,. In [9] he disance is given by he formula r x = ν θ c x + y + z, where θ c is he unique soluion of µ θ x + y = z in he inerval [, π and µz = co z and where z sin z νz = z sin z + µz = z z + sin, ν =. z sin z cos z Since he disance is lef-invarian, we have d CC x, x = d CC x x, e which gives us an explici expression for d CC on he Heisenberg group. Alhough ν is no coninuous i was shown in [8] ha d CC is coninuous. We will no use his explici expression for d CC. Insead, since ν and bounded below and above by posiive consans in he inerval [, π, i is clear ha he Carno-Carahéodory disance is equivalen o he pseudo-meric.4 ρ x, y = x x + y ỹ + z z + xỹ y x. Finally, we can describe Brownian moion whose generaor is H / explicily as follows. Le B, B be real-valued independen Brownian moions saring from.

7 HEISENBERG COUPLING 7 Define Brownian moion on he Heisenberg group X : [, Ω H o be he soluion of he following Sraonovich sochasic differenial equaion SDE dx = X X db + Y X db, X = b, b, a. Leing X = X, X, X 3 we see ha he SDE reduces o dx = X db + X db, so ha one needs o solve he following sysem of equaions dx = db dx = db, dx 3 = X db + X db. Since he covariaion of wo independen Brownian moions is zero we ge ha.5 X = b + B, X = b + B, X 3 = a + B s + b db s 3. Coupling resuls B s + b db s. Le B, B be independen real-valued Brownian moions, saring from b and b respecively. We call he process 3. X = B, B, a + B sdb s B sdb s Brownian moion on he Heisenberg group, wih driving Brownian moion B = B, B, saring from b, b, a. Le X and X be coupled copies of his process saring from b, b, a and b, b, ã respecively. Denoe he coupling ime { τ = inf : X s = X } s for all s. We will consruc a non-markovian coupling X, X of wo Brownian moions on he Heisenberg group. This, via he Aldous inequaliy, will yield an upper bound on he oal variaion disance beween he laws of X and X. Before we sae and prove he main heorem, we describe he ools required in is proof. For T >, le B br, B br be a coupling of sandard Brownian bridges defined on he inerval [, T ]. If G T is a Gaussian variable wih mean zero and variance T independen of B br, B br, a sandard covariance compuaion shows ha he assignmen B = B br + T GT 3. B = B br + T GT

8 8 BANERJEE, GORDINA, AND MARIANO gives a non-markovian coupling of wo sandard Brownian moions on [, T ] saisfying B T = B T. This coupling is similar in spiri o he one developed in []. The usefulness of his coupling sraegy arises when we wan o couple wo copies of he process B, F [B] :, where B is a Brownian moion, [B] denoes he whole Brownian pah up unil ime hough of as an elemen of C [, ], and F is a possibly random funcional on C [, ]. We firs reflecion couple he Brownian moions unil hey mee. Then, by dividing he fuure ime ino inervals [T n, T n+ ] usually of growing lengh and consrucing a suiable non-markovian coupling of he Brownian bridges on each such inerval, we can obain a coupling of he Brownian pahs by he above recipe in such a way ha he corresponding pah funcionals agree a one of he deerminisic imes T n. As by consrucion, he coupled Brownian moions agree a he imes T n, we achieve a successful coupling of he join process B, F. Furher, he rae of coupling aained by his non-markovian sraegy is usually significanly beer han Markovian sraegies, and is ofen near opimal see []. We will be ineresed in he paricular choice of he random funcional, namely, F [w] = wsdb s, where B is a sandard Brownian moion and w C [, ]. Our coupling sraegy for he Brownian bridges on [, T ] will be based on he Karhunen-Loève expansion which goes back o [, 3] and for examples of such expansions see [36, p.]. For he Brownian bridge we have 3.3 B br = sin kπ T T Z k = T Z k g T,k kπ k= for [, T ], where Z k are i.i.d. sandard Gaussian random variables. Thus, in order o couple wo Brownian bridges on [, T ], we will couple he random variables {Z k } k. We now sae and prove he following lemmas. Lemma 3.. There exiss a non-markovian coupling of he diffusions { B, B, a + { B, B, ã + k= } B sdb s :, } B sd B s :, B = B = b, B = B = b, and a > ã, for which he coupling ime τ saisfies a ã P τ > C for some consan C > ha does no depend on he saring poins and a ã. Proof. We will wrie I = a + B sdb s and Ĩ = ã + B sd B s. From Brownian scaling, i is clear ha for any r R, he following disribuional

9 HEISENBERG COUPLING 9 equaliy holds B 3.4, r B, a + B sdb s r r d = B /r, B /r, a /r r + B sdb s, where B, B are independen Brownian moions wih B = b /r, B = b /r. Thus we can assume a ã =. For he general case, we can obain he corresponding coupling by applying he same coupling sraegy o he scaled process using 3.4 wih r = a ã. Le us divide he non-negaive real line ino inervals [ n, n+ ], n. We will synchronously couple B and B a all imes. Thus, we sample he same Brownian pah for B and B. Condiional on his Brownian pah {B : } we describe he coupling sraegy for B and B inducively on successive inervals. Suppose we have consruced he coupling on [, n ] in such a way ha he coupled Brownian moions B and B { saisfy B n = B n = b and I n > Ĩn. Condiional on B, B } : n and he whole Brownian pah B, we will consruc he coupling of B b and B b for [ n, n+ ]. To his end, we will couple wo Brownian bridges B br and B br on [ n, n+ ], hen sample an independen Gaussian random variable G n wih mean zero, variance n and finally use he recipe 3. o ge he coupling of B and B on [ n, n+ ]. Le Z n, Z n,... and Zn, n Z,... denoe he Gaussian coefficiens in he Karhunen-Loève expansion 3.3 corresponding o B br and B br respecively. Sample i.i.d Gaussians Z k and se Z n n k = Z k = Z k for k. Now we consruc he coupling of Z n n and Z {. Le W n be a sandard Brownian moion saring from zero, independen of B, B } : n, {Z k } k and B. In wha follows we will repeaedly use he following random funcional 3.5 λ n = π n Define he random ime σ n by πs n + sin n db s, n n+. { } σ n inf : W = n = In Ĩn λ n n+, if λ n n+,, oherwise. As λ n n+ is a Gaussian random variable wih mean zero and variance 4 π n+ n πs sin n + n ds = n+ π,

10 BANERJEE, GORDINA, AND MARIANO he ime σ n is finie for almos every realizaion of he Brownian pah B. Now, define W n as follows { W n W n if σ n = W n W n σ n if > σ n. { Condiional on B, B } : n, {Z k } k and B, σ n is a sopping ime { for W n. Thus W n defined above is also a Brownian moion independen of B, B } : n, {Z k } k and B. Finally, we se Z n = n/ W n n and coupling we ge Z n = n/ W n n. Under his 3.6 I Ĩ = I n Ĩ n + W n n σ n λ n, for [ n, n+ ]. In paricular, I n+ Ĩ n+ and equals o zero if and only if σ n n. If I n Ĩ n =, we synchronously couple B, B afer ime n. By inducion, he coupling is defined for all ime. Now, we claim ha he coupling consruced above gives he required bound on he coupling rae. Using Lévy s characerizaion of Brownian moion and he fac ha he { W n} n are independen of he Brownian pah B, we obain a Brownian moion B independen of B such ha for all, where λ k k+ W k k + + k = B T, k= T = k= λ k k+ k < s k+ ds. Noe ha for any n, he coupling happens afer ime n+ if and only if σ k > k for all k n, ha is, B > ã a = for all T n+. Therefore, if for y R, τy denoed he hiing ime of level y for he Brownian moion B, hen we have P τ > n+ = P τ > T n+. By a sandard hiing ime esimae for Brownian moion, we see ha here is a consan C > ha does no depend on b, b, a, ã such ha 3.7 P τ > n+ [ ] CE. T n+ Thus, we need o obain an esimae for he righ hand side in 3.7. Noe ha n T n+ has he same disribuion as Ψ n := 4 π n k= k U k, where he U k are i.i.d. sandard Gaussian random variables.

11 HEISENBERG COUPLING For n, Ψ / n Ψ / π /. U + U As U + U has densiy re[ r / dr wih respec o he Lebesgue measure for r, we conclude ha E π ] U + U / <. Thus, for n [ ] E n T n+ [ = E Ψ / n ] [ E Ψ / ] [ E π U + U / ] <. This, along wih 3.7, implies ha here is a posiive consan C no depending on b, b, a, ã such ha for n, P τ > n+ C n. I is easy o check ha he above inequaliy implies he lemma. Remark 3.. Under he hypohesis of Lemma 3., i is no possible o obain he given rae of decay of he probabiliy of failing o couple by ime coupling rae wih any Markovian coupling. The proof of his proceeds similar o ha of [, Lemma 3.]. We skech i here. Under any Markovian coupling µ, a simple Fubini argumen shows ha here exiss a deerminisic ime > such ha µ B B >. Le τ B represen he firs ime when he Brownian moions B and B mee afer ime which should happen a or before he coupling ime of X and X. Le F denoe he filraion generaed by B and B up o ime and le E µ denoe expecaion under he coupling law µ. Then, from he fac ha he maximal coupling rae of Brownian moion equivalenly he oal variaion disance beween B and B decays like /, we deduce ha for sufficienly large µτ > = E µ E µ [τ > F ] E µ E µ [ τ B > F ] C µ / C µ /, where C µ denoes a posiive consan ha depends on he coupling µ. Thus, any Markovian coupling has coupling rae a leas /, bu he non-markovian coupling described in Lemma 3. gives a rae of. The nex lemma gives an esimae of he ail of he law of he sochasic inegral B sdb s run unil he firs ime B his zero. Lemma 3.3. Le B, B be independen Brownian moions wih B = b >. For z R, le τ z denoe he hiing ime of level z by B. Then τ P B sdb s > y b for y b. y

12 BANERJEE, GORDINA, AND MARIANO Proof. For any level z b, we can wrie τ P B sdb s > y = τ P B sdb s > y, τ z < τ + P [ ] τ τ E z B sds P τ z < τ + y P τ z < τ + z y E [τ τ z ], τ B sdb s > y, τ z τ where he second sep follows from Chebyshev s inequaliy. From sandard esimaes for Brownian moion, P τ z < τ = b/z and E [τ τ z ] = bz b bz. Using hese in he above, we ge P τ B sdb s > y bz + bz3 y. As his bound holds for arbirary z b, he resul follows by choosing z = y. Consider wo coupled Brownian moions X, X on he Heisenberg group saring from b, b, a and b, b, ã respecively. A key objec in our coupling consrucion for Brownian moions on he Heisenberg group H 3 will be he invarian difference of sochasic areas given by 3.8 A = a ã + B sdb s B sdb s B sd B s B sd B s + B B B B. Noe ha he Lévy sochasic area is invarian under roaions of coordinaes. If he Brownian moions B and B are synchronously coupled a all imes, hen as he covariaion beween B and B and beween B and B is zero, 3.9 A A = where B sdb s + B sdb s, 3. A = a ã + b b b b, for. The nex lemma esablishes a conrol on he invarian difference evaluaed a he ime when he Brownian moions B and B firs mee, provided hey are reflecion coupled up o ha ime. Lemma 3.4. Le B be a real-valued Brownian moion saring from b, and le B, B be reflecion coupled one-dimensional Brownian moions saring from b and b respecively. Consider he invarian difference of sochasic areas given by 3.8 wih B = B {. Define T = inf : B = B }. Then here exiss

13 HEISENBERG COUPLING 3 a posiive { consan C ha does no depend on b, b, b, a, ã such ha for any b max b } a, ã + b b b b, [ ] A T E C b b + a ã + b b b b. Proof. In he proof, C, C will denoe generic posiive consans ha do no depend on b, b, b, a, ã, whose values migh change from line o line. For any >, [ ] A T E 3. [ ] A T E ; k < A T k + P A T > k P k < A T k + P A T > k= k= k P A T k + P A T >. k= As B and B are reflecion coupled, we can rewrie 3.9 as A A = B s B s db s where B B is a Brownian moion saring from b b and independen of B. By Lemma 3.3, for max b { b }, A, 3. P A T > P A T A > A P A T A > b b C. { b Furher, for max b }, A, 3.3 k P A T k k= = k P A T k } k: max{ k b b, A + } k: >max{ k b b, A k P A T k.

14 4 BANERJEE, GORDINA, AND MARIANO To esimae he firs erm on he { righ hand side of 3.3, le k be he smalles b ineger k such ha k max b }, A. Then, 3.4 k P A T k } k: max{ k b b, A k = k+ = 4 k 4 { b max b }, A k=k b b 8 + A b b a ã + b b b b 8 +, where we used he facs ha b b b b for b b and A = a ã + b b b b o ge he las inequaliy. To esimae he second erm on he righ hand side of 3.3, we use Lemma 3.3 o ge 3.5 } k: >max{ k b b, A k P A T k C k/ b b } k: >max{ k b b, A C b b b b k/ C. Using 3.4 and 3.5 in 3.3, 3.6 k P A T k b b a ã + b b b b C +. k= Using 3. and 3.6 in 3., we complee he proof of he lemma. Now, we sae and prove our main heorem on coupling of Brownian moions on he Heisenberg group H 3. Theorem 3.5. There exiss a non-markovian coupling X, X of wo Brownian moions on he Heisenberg group saring from b, b, a and b, b, ã respecively, and a consan C > which does no depend on he saring poins such ha he coupling ime τ saisfies b b a ã + b b b b P τ > C + { b for max b, a ã + b b b b }. Here b = b, b and b = k= b, b.

15 HEISENBERG COUPLING 5 Proof. We will explicily consruc he non-markovian coupling. In he proof, C will denoe a generic posiive consan ha does no depend on he saring poins. Since he Lévy sochasic area is invarian under roaions of coordinaes, i suffices o consider he case when b = b. Recall he invarian difference of sochasic areas A defined by 3.8. We will synchronously couple he Brownian moions B and B a all imes. Recall ha under his seup, he invarian difference akes he form 3.9. The coupling comprises he following wo seps. Sep. We { use a reflecion coupling for B and B unil he firs ime hey mee. Le T = inf : B = B }. Sep. Afer ime T we apply he coupling sraegy described in Lemma 3. o he diffusions { B, B, AT + { B, B, T B sdb s T } B sd B s : T. : T }, By sandard esimaes for he Brownian hiing ime we have C b b 3.7 P T > for b b. By Lemma 3. and Lemma 3.4, for max { b b, A }, [ ] A T 3.8 P τ T > CE b b a ã + b b b b C +. Equaions 3.7 and 3.8 ogeher yield he required ail bound on he coupling ime probabiliy saed in he heorem. An ineresing observaion o noe from Theorem 3.5 is ha, if he Brownian moions sar from he same poin, hen he coupling rae is significanly faser. The above coupling can be used o ge sharp esimaes on he oal variaion disance beween he laws of wo Brownian moions on he Heisenberg group saring from disinc poins. Theorem 3.6. If d T V denoes he oal variaion disance beween probabiliy measures, and L X, L X denoe he laws of Brownian moions on he Heisenberg group saring from b, b, a and b, b, ã respecively, hen here exiss posiive

16 6 BANERJEE, GORDINA, AND MARIANO consans C, C no depending on he saring poins such ha b d T V L b a ã + b b b b X, L X C + b d T V L b X, L X C b b a ã + b = b { b for max b, a ã + b b b b }. Proof. The upper bound on he oal variaion disance follows from Theorem 3.5 and he Aldous inequaliy.. To prove he lower bound, we firs address he case b b. I is sraighforward o see from he definiion of he oal variaion disance ha d T V L X, L X d T V L B, L B. Thus, when b b, he lower bound in he heorem follows from he sandard esimae on he oal variaion disance beween he laws of Brownian moions using he reflecion principle d T V L B, L B b b = P N, b b. πe where N, denoes a sandard Gaussian variable. Now, we deal wih he case b = b. As he generaor of Brownian moion on he Heisenberg group is hypoellipic, he law of Brownian moion saring from u, v, w has a densiy wih respec o he Lebesgue measure on R 3 which coincides wih he Haar measure on H 3. We denoe by p u,v,w,, his densiy he hea kernel a ime. The hea kernel p u,v,w x, y, z is a symmeric funcion of u, v, w, x, y, z H 3 H 3 and is invarian under lef muliplicaion, ha is, p u,v,w x, y, z = p e u, v, w x, y, z = p e x, y, z u, v, w. Using he fac ha u, v, w = u, v, w we see ha 3.9 p u,v,w x, y, z = p e x u, y v, z w uy + vx, where e =,,. Then d T V L X, L X = p b,b,a x, y, z p b,b,ã x, y, z dxdydz R 3 = p e x b, y b, z a b y + b x R 3 p e x b, y b, z ã b y + b x dxdydz = p e x, y, z a p e x, y, z ã dxdydz R 3 f z a f z ã dz, R

17 HEISENBERG COUPLING 7 where f denoes he densiy wih respec o he Lebesgue measure of he Lévy sochasic area a ime when he driving Brownian moion sars a he origin. The hird equaliy above follows by a simple change of variable formula and he las sep follows from wo applicaions of he inequaliy R fxdx R fx dx for real-valued measurable f. From Brownian scaling, i is easy o see ha f z = z f, z R. Subsiuing his in he above and using he change of variable formula again, we ge d T V L X, L X f z a f z ã dz R = f z a ã f z R dz f z a ã f z dz. z The explici form of f is well-known see, for example, [38] or [3, p. 3] f z = cosh πz, z R. Wihou loss of generaliy, we assume a > ã. By he mean value heorem and he assumpion made in he heorem ha a ã, f z a ã f z a ã We can explicily compue a ã inf ζ [z a ã f ζ = π eπζ e πζ e πζ + e πζ.,z] f ζ inf ζ [z,z] f ζ. This is an even funcion which is sricly decreasing for ζ /. Thus, for z, Thus, d T V L X, L X inf ζ [z,z] f ζ f 3z/. z a ã f which complees he proof of he heorem. Several remarks are in order. z a ã z f z dz f 3z/ dz a ã = C, Remark 3.7. Theorem 3.6 shows ha he non-markovian coupling sraegy we consruced is, in fac, an efficien coupling sraegy in he sense ha he coupling rae decays according o he same power of as he oal variaion disance beween

18 8 BANERJEE, GORDINA, AND MARIANO he laws of he Brownian moions X and X. We refer o [, Definiion ] for he precise noion of efficiency. Remark 3.8. Alhough we have saed our resuls wihou any quaniaive bounds on he consans appearing in he coupling ime and oal variaion esimaes, i is possible o rack concree numerical bounds from he proofs presened above. We need he following elemenary fac. For any x and y 3. x + y x + y. Indeed, x + y x + y x + y, since y. This immediaely gives us he following resul. Proposiion 3.9. Assume ha a ã + b b b b <. Then here exiss a consan C > such ha P τ > C d CC b, b, a, b, b, ã { b for max b, a ã + b b b b, }. Proof. Since >, hen, so by Theorem 3.5 b b a ã + b b b b P τ > C + C b b + a ã + b b b b C b b + a ã + b b b b where we used 3. in he las inequaliy. Now we consider ρ b, b, a, b,, b, ã b = b + a ã + b b b b as defined by.4. Recall from Secion ha his pseudo-meric is equivalen o he Carno-Carahéodory disance d CC b, b, a, b, b, ã. This gives us he desired inequaliy. Liouville ype heorems have been known for he Heisenberg group and oher ypes of Carno groups e.g. [7, Theorem 5.8.]. Using he coupling we consruced, we derive a funcional inequaliy a form of which appeared as [, Equaion 4] which consequenly gives us he Liouville propery raher easily. In he following, for any bounded measurable funcion u : H 3 R and any x H 3, we define P ux = Eu X x, where X x is a Brownian moion on he Heisenberg group saring from x. By we denoe he sup norm.,

19 HEISENBERG COUPLING 9 Corollary 3.. For any bounded u C H 3 here exiss a posiive consan C, which does no depend on u, such ha for any 3. H P u C u. Consequenly, if H u =, hen u is a consan. Proof. Fix. Take wo disinc poins b, b, a and b, b, ã in H 3, d CC sufficienly close o b, b, a wih respec o he disance d CC in such a way ha { b max b }, a ã + b b b b. Then, using he coupling X, X consruced in Theorem 3.5 and by Proposiion 3.9, we ge P u b, b, a P u b, b E, ã = u X u X : τ > u P τ > C u d CC b, b, a, b, b, ã. Dividing by d CC b, b, a, b, b, ã on boh sides above and aking a supremum over all poins b, b, ã b, b, a, we ge 3.. Finally if H u =, hen P u = u for all. Taking in 3., we ge H u and hence u C H 3 is consan by [7, Proposiion.5.6]. 4. Gradien esimaes The goal of his secion is o prove gradien esimaes using he coupling consrucion inroduced earlier. Le x = b, b, a and x = b, b, ã. We le X, X be he non-markovian coupling of wo Brownian moions X and X on he Heisenberg group saring from x and x respecively as described in Theorem 3.5. For a se Q, define he exi ime of a process X from his se by τ Q X = inf { > : X / Q}. The oscillaion of a funcion over a se Q is defined by osc u sup u inf u. Q Q Q Before we can formulae and prove he main resuls of his secion, Theorems 4.3 and 4.4, we need wo preliminary resuls. Lemma 4. gives second momen esimaes for sup τ B s b db s, sup τ B b and sup τ B b under he coupling consruced above, when he coupled Brownian moions sar from he same poin b, b. I would be naural o wan o apply here Burkholder- Davis-Gundy BDG inequaliies such as [9, p. 63] which give sharp esimaes of momens of sup T M for any coninuous local maringale M in erms of he momens of is quadraic variaion M T when T is a sopping ime. Bu he coupling ime τ is no a sopping ime wih respec o he filraion generaed by B, B, and herefore we can no apply hese inequaliies o ge he momen esimaes.

20 BANERJEE, GORDINA, AND MARIANO Lemma 4.. Consider he coupling of he diffusions { B, B, a + { B, B, ã + } B sdb s : } B sd B s :, described in Lemma 3., wih B = B = b, B = B = b and a > ã, wih coupling ime τ. Then here exiss a posiive consan C no depending on b, b, a, ã such ha we have he following i E sup τ B s b db s CEτ, ii E sup τ B b 4 CEτ, iii E sup τ B b 4 CEτ. Proof. In his proof, C will denoe a generic posiive consan whose value does no depend on b, b, a, ã. Our basic sraegy will be o find appropriae enlargemens of he naural filraion generaed by B, B under which τ becomes a sopping ime, and hen use he Burkholder-Davis-Gundy inequaliy. I suffices o prove he saemen for b = b =. Moreover, using scaling of Brownian moion, i is sraighforward o check ha i is sufficien o prove he saemen wih a ã = and τ replaced by τ M for arbirary M >. We wrie B = Y + Y, where Y = n/ Z n g n, n + + n 4. Y = n= n= n/ n + + n n Z n + k= Z n k g n,k n + + n wih g n,k = g n,k as defined in he Karhunen-Loève expansion 3.3 and Z n = n/ G n for a a Gaussian variable wih mean zero and variance n as we used in 3.. Consider he filraion F = σ {B s : s } {W n s : n, s } {Z n k : n, k }. We assume wihou loss of generaliy ha {F } is augmened, in he sense ha all he null ses of F and heir subses lie in F. We claim ha τ is a sopping ime under he above filraion. To see his, recall ha by he definiion of coupling ime, he coupled processes mus evolve ogeher afer he coupling ime and hus, by he coupling consrucion given in Lemma 3. in paricular, see 3.6, 4. P[τ { n+ : n }] =. Thus, o show ha τ is a sopping ime wih respec o F, i suffices o show ha {τ > n+ } is measurable wih respec o F n+ for each n. This is because, for [ n+, n+ n, {τ > } = {τ > n+ }

21 HEISENBERG COUPLING almos surely wih respec o he coupling measure P, by 4.. Noe ha for any n, n {τ > n+ } = {σ m > m }. Recall ha σ m = inf { : W m = I m Ĩm / and on he even {τ > m+ }, m= m+ m g m, s m + db s B s B s = Y s Ỹs = Y s, for all s m+. As {Y : m+ } depends measurably on {Z k : k m} and hence on {W k s : k, s < }, he above represenaion for σ m implies ha he even {σ m > m } is measurable wih respec o F m+. Thus, for each n, {τ > n+ } is measurable wih respec o F n+ and hence, τ is indeed a sopping ime wih respec o {F }. Also, noe ha B sdb s remains a coninuous maringale under his enlarged filraion. Thus, by he Burkholder-Davis-Gundy inequaliy, we ge τ M E sup B sdb s CE Bsds τ M CE sup B τ M τ M Now, by he Cauchy-Schwarz inequaliy 4 / Eτ E sup B τ M E sup B M /. τ M τ M Thus, o complee he proof i and iii, i suffices o show ha 4 E sup B CEτ M. τ M To show his, define he Brownian moion W = W n n + + n n= and he following augmened filraion F = σ {B s, W s : s } {Z n k : n, k }. Exacly as before, we can check ha τ is a sopping ime wih respec o his new filraion and W is a Brownian moion hence a coninuous maringale under i. From he represenaion 4., noe ha sup Y = τ M π } sup W n+ W n sup W. n: n+ τ M π τ M

22 BANERJEE, GORDINA, AND MARIANO Thus, by he he Burkholder-Davis-Gundy inequaliy 4 E sup Y τ M π 4 E sup W CEτ M. τ M To esimae sup τ M Y, noe ha Y and τ are independen. condiioning argumen, i suffices o show ha for fixed T >, E sup Y CT. T To see his, observe ha Y = B Y for each and hus sup T Y sup T B + sup Y. T Again by he he Burkholder-Davis-Gundy inequaliy 4 E sup B CT. T Thus, by a By exacly he same argumen as he one used o esimae he supremum of Y, bu now applied o a fixed ime T, we ge 4 E sup Y CT. T The wo esimaes above yield 4.4, and hence complee he proof of i and iii. Similarly, ii follows from he fac ha B is a Brownian moion under he filraion {F } and he Burkholder-Davis-Gundy inequaliy. The nex lemma esimaes Eτ. Lemma 4.. Under he coupling of Lemma 3., here exiss a posiive consan C no depending on b, b, a, ã such ha Eτ C a ã. Proof. Wihou loss of generaliy, we assume a ã. We can wrie Eτ = a ã + Pτ > d a ã Pτ > d. From Lemma 3., we ge a consan C ha does no depend on b, b, a, ã such ha for > a ã, Pτ > a ã C. Using his we ge Eτ a ã + C a ã which proves he lemma. d + C a ã,

23 HEISENBERG COUPLING 3 Le D H 3 be a domain. Laer in Theorem 4.4 we give gradien esimaes for harmonic funcions in D, bu we sar by a resul on he coupling ime τ. Define he Heisenberg ball of radius r > wih respec o he disance ρ Bx, r = {y H 3 : ρx, y < r}. Recall ha ρ is he pseudo-meric equivalen o d CC defined by.4. For x D, le δ x = ρ x, D c. Consider he coupling of wo Brownian moions on he Heisenberg group X and X saring from poins x, x D respecively as described by Theorem 3.5. We choose hese poins in such a way ha ρx, x is small enough compared o δ x. The following heorem esimaes he probabiliy as a funcion of δ x and ρx, x ha one of he processes exis he ball Bx, δ x before coupling happens. This urns ou o be pivoal in proving he gradien esimae. Theorem 4.3. Le x = b, b, a D, x = b, b, ã D such ha ρx, x < δ x /3, b b and a ã + b b b b /. Then, under he same coupling of Theorem 3.5, here exiss a consan C > ha does no depend on x, x such ha P τ > τ Bx,δx X τ Bx,δx X C + + δ x δx δ x 3 δx 4 ρx, x. Proof. In his proof, C will denoe a generic posiive consan whose value migh change from line o line ha does no depend on x, x. Le ˆb i = bi+ b i for i =, and â = a+ã. We define he Heisenberg cube by { Q = y, y, y 3 R 3 : max y i ˆb δ x â i i=, 8, y3 + ˆb y ˆb δ } y x. 6 Wrie ˆx = ˆb, ˆb, â. I is sraighforward o check ha ρx, ˆx ρx, x/ < δ x /3. Moreover, for y Q ρˆx, y = y ˆb + y ˆb + â y 3 + ˆb y ˆb / y y ˆb + y ˆb + â y 3 + ˆb y ˆb / y δx /. Thus, by he riangle inequaliy, for any y Q ρx, y ρx, ˆx + ρˆx, y < δ x and hence, Q Bx, δ x. Noe ha we can wrie Q = Q Q where { Q = y, y, y 3 R 3 : max y i ˆb } δ x i, i=, 8 { Q = y, y, y 3 R 3 : â y 3 + ˆb y ˆb δ } y x. 6 As he Lévy sochasic area is invarian under roaions of coordinaes, i suffices o assume ha b = b. We define Noe ha U = a â + B sdb s B sdb s + B ˆb B ˆb. du = B ˆb db B ˆb db.

24 4 BANERJEE, GORDINA, AND MARIANO Wriing σ u = inf{ : U > u}, we observe ha τ Q X = σ δ x /6 and hence, τ Q X = τ Q X τ Q X = τ Q X σ δ x /6. We can wrie P τ > τ Bx,δx X τ Bx,δx X Pτ > τ Q X τ Q X Pτ > τ Q X + Pτ > τ Q X. Now we esimae Pτ > τ Q X, he second erm in he inequaliy above can be esimaed similarly. Firs we define { Q = y, y, y 3 R 3 : max y i ˆb } δ x i. i=, 6 We have 4.5 Pτ > τ Q X = Pτ > τ Q X σ δ x /6 PT > τ Q X + Pτ > τ Q X σ δ x /6, T τ Q X PT > τ Q X + Pσ δ x /3 T τ Q X + Pτ > τ Q X σ δ x /6, T τ Q X σ δ x /3. I follows from a compuaion involving sandard Brownian esimaes see, for example, he proof of [, Theorem ] ha 4.6 b b PT > τ Q X C. δ x To esimae he second erm in 4.5, noe ha Pσ δ x /3 T τ Q X = P sup U > δ x T τ Q X 3 Now, as T τ Q X is a sopping ime wih respec o he naural filraion generaed by B, B, by he he Burkholder-Davis-Gundy inequaliy E sup U U T τ Q X T τ Q X CE Bs ˆb ds CE T τ Q X δ xds Cδ xet τ Q X. We can again appeal o sandard Brownian esimaes e.g. see he proof of [, Theorem ] o see ha 4.7 E T τ Q X Cδ x b ˆb..

25 HEISENBERG COUPLING 5 Using his esimae gives us E sup U E T τ Q X sup U U T τ Q X + U Cδ 3 x b ˆb + a â + b ˆb b ˆb C δ3 x b b + a ã + b b b b. By assumpion a ã + b b b b <, and herefore E sup U C + δ x 3 b b + a ã + b b b b T τ Q X C + δ x 3 ρx, x, where he las inequaliy follows from 3.. Thus, by he Chebyshev inequaliy P sup U > δ x T τ Q X 3 C + δ x 3 δx 4 ρx, x, which, in urn, gives us 4.8 Pσ δ x /3 T τ Q X C + δ x 3 ρx, x. δ 4 x To esimae he las erm in 4.5, we wrie 4.9 Pτ > τ Q X σ δ x /6, T τ Q X σ δ x /3 Pτ T > + Pτ > τ Q X σ δ x /6, T τ Q X σ δ x /3, τ T. By Lemma 3., we ge Pτ T > CE AT, where A is he invarian difference of sochasic areas defined in 3.8. Applying Lemma 3.4 wih = and appealing o our assumpion ha b b and a ã + b b b b /, we have which gives E AT C b b + a ã + b b b b Cρx, x. 4. Pτ T > Cρx, x.

26 6 BANERJEE, GORDINA, AND MARIANO Finally, we need o esimae Pτ > τ Q X σ δ x /6, T τ Q X σ δ x /3, τ T. Noe ha 4. Pτ >τ Q X σ δ x /6, T τ Q X σ δ x /3, τ T P P sup B B T δ x /6 T T +τ T sup T T +τ T + P B B T δ x /6 sup U UT δx/3, T T +τ T sup B B T < δ x /6, T τ Q X T T +τ T By he srong Markov propery applied a T, along wih pars ii and iii of Lemma 4. and he Chebyshev inequaliy, we ge P sup B i B i T δ x /6 T T +τ T + C Eτ T δ 4 x for i =,. From he explici consrucion of he coupling sraegy given in Theorem 3.5 and Lemma 4. and Lemma 3.4, we obain Eτ T E AT Cρx, x.. and hus, 4. P sup B i B i T δ x /6 T T +τ T C ρx, x δx 4. for i =,. To handle he las erm in 4., define Noe ha U = U B ˆb B ˆb. du = B ˆb db. and U T = UT as B T = ˆb. Furher, observe ha sup U UT T T +τ T sup U U T + sup B ˆb B ˆb. T T +τ T T T +τ T

27 HEISENBERG COUPLING 7 Using his, we can bound he las erm in 4. as 4.3 P sup U UT δx/3, T T +τ T sup T T +τ T P + P B B T < δ x /6, T τ Q X sup T T +τ T U U T δ x/64 sup B ˆb B ˆb δx/64, T T +τ T sup B B T < δ x /6, T τ Q X T T +τ T By condiioning a ime T and par i of Lemma 4., followed by applicaions of Lemma 4. and Lemma 3.4, we obain E sup U U T T T +τ T 4E sup B s ˆb db s T T +τ T T CEτ T E AT Cρx, x. Consequenly, by he Chebyshev inequaliy 4.4 Moreover, 4.5 P P sup U U T δx/64 T T +τ T sup B ˆb B ˆb δx/64, T T +τ T sup T T +τ T P + P C B B T < δ x /6, T τ Q X sup T T +τ T B ˆb δ x /8 sup B ˆb δ x /8, T T +τ T ρx, x δx 4. sup B B T < δ x /6, T τ Q X T T +τ T..

28 8 BANERJEE, GORDINA, AND MARIANO We use he fac B T = ˆb and proceed exacly along he lines of he proof of 4. o obain 4.6 P sup B ˆb ρx, x δ x /8 C T T +τ T δx 4. The second probabiliy appearing on he righ hand side of 4.5 can be bounded as follows 4.7 P P sup B ˆb δ x /8, T T +τ T sup T T +τ T B B T < δ x /6, T τ Q X sup B ˆb δ x /8, T τ Q X T τ Q X+τ T τ Q X sup T τ Q X T τ Q X+τ T τ Q X B B T τ Q X < δ x /6 P B T τ Q X ˆb > δ x /6. We will use he fac ha b = ˆb. By an applicaion of he Chebyshev inequaliy followed by he Burkholder-Davis-Gundy inequaliy, and using 4.7, we ge P B T τ Q X ˆb > δ x /6 C E B T τ Q X ˆb δx E sup T τ Q X B b C δx C ET τ Q X b ˆb δx C. δ x Using his in 4.7, 4.8 P sup B ˆb δ x /8, T T +τ T sup B B T < δ x /6, T τ Q X T T +τ T Using 4.6 and 4.8 in 4.5, we obain 4.9 P sup B ˆb B ˆb δx/64, T T +τ T b ˆb C. δ x sup B B T < δ x /6, T τ Q X T T +τ T C + δ x δx 4 ρx, x.

29 HEISENBERG COUPLING 9 Finally, using 4.4 and 4.9 in 4.3, 4. P sup U UT δx/3, T T +τ T sup B B T < δ x /6, T τ Q X T T +τ T C + δ x δx 4 ρx, x. Using he esimaes from 4. and 4. in 4., we ge 4. Pτ > τ Q X σ δ x /6, T τ Q X σ δ x /3, τ T C + δ x δx 4 ρx, x. Using 4. and 4. in 4.9, we ge 4. Pτ > τ Q X σ δ x /6, T τ Q X σ δ x /3 C Using he esimaes 4.6, 4.8 and 4. in 4.5, we obain Pτ > τ Q X C + + δ x δx δ x δx 4 ρx, x. + + δ x δx 4 ρx, x. The same esimae for Pτ > τ Q X is obained by inerchanging he roles of x and x. This complees he proof of he heorem. The above heorem yields he gradien esimae formulaed in Theorem 4.4. Before we can formulae our resul, we explain he argumen in he proof of [6, Proposiion 4.] ha leads o 4.4. Recall ha H denoes he sub-laplacian which is he generaor of he Brownian moion on H 3, and for any funcion f on H 3, H f denoes he associaed lengh of he horizonal gradien of f defined by.. As before H denoes he norm induced by he sub-riemannian meric on horizonal vecors. We can use he fac ha {X, Y} is an orhonormal frame for he horizonal disribuion, herefore for any Lipschiz coninuous funcion u defined on a domain D in H 3, H u H = X u + Yu holds in D where X u and Yu are inerpreed in he disribuional sense. Now we can use [7, Theorem.7] for he vecor fields {X, Y} in H 3 idenified wih R 3. We need o check some assumpions in his heorem. Firs, if u is Lipschiz coninuous on D, i is clear ha uz u z H u x sup <, z, z D,z z d CC z, z for all x D, and hence H u is locally inegrable. In addiion, as u is Lipschiz coninuous, H u is an upper gradien of u by [6, Lemma.], so [7, Theorem.7] is applicable and we have ha 4.4 H u H H u, a.e. wih respec o he Lebesgue measure.

30 3 BANERJEE, GORDINA, AND MARIANO Le C D be he space of funcions ha are coninuous on he closure of he domain D. We also le C D be he space of funcions ha are wice coninuously differeniable in D. Theorem 4.4. Suppose u C D C D such ha H u = on D H 3. Fix any consan α, ]. There exiss a consan C > ha does no depend on u such ha for every x D 4.5 H ux H H u x C + + δ x δx δ x 3 δx 4 osc u. Bx,αδ x Proof. I clearly suffices o consider he case α =. Since u is coninuous on D, osc Bx,δx u <. Le x = b, b, a D, x = b, b, ã D such ha ρx, x < δ x /3, b b and a ã + b b b b /. Consider he coupling from Theorem 3.5 of wo Brownian moions, X and X, on he Heisenberg group saring from he poins x and x respecively. By Theorem 4.3 and he equivalence of he Carno-Carahéodory meric d CC and he pseudo-meric ρ, we have P τ > τ Bx,δx X τ Bx,δx X C + + δ x δx δ x 3 δx 4 d CC x, x. Using he coupling from Theorem 3.5 and Iô s formula we have ha [ ] u x u x = E u X τbx,δx X u X τbx,δx [ X X ] u E X τbx,δx X u X τbx,δx osc u P τ > τ Bx,δx X τ Bx,δx X Bx,δ x C osc u + + Bx,δ x δ x δx δ x 3 δx 4 d CC x, x. Since u C D C D herefore 4.4 holds for every x D. Dividing ou by d CC x, x and using 4.4 we have ha for every x D, as needed. u x u x H ux H H u x = lim sup r <d CC x, x r d CC x, x C + + δ x δx δ x 3 δx 4 osc u, Bx,δ x Corollary 4.5. Le u C D C D be a non-negaive soluion o H u = on D H 3. There exiss a consan C > ha does no depend on u, δ x, x, D such ha H u x H H u x C + + δ x δx δ x 3 δx 4 ux for every x D. Proof. By [7, Corollary 5.7.3] we have he following Harnack inequaliy 4.6 sup u C inf u Bx,α δ x Bx,α δ x

31 HEISENBERG COUPLING 3 for x D H 3, where α, ], C > are consans no depending on u, δ x, x, D. Then Equaions 4.5 and 4.6 give he desired resul. We can use Corollary 4.5 and he sraified srucure of H 3 o prove he Cheng- Yau gradien esimae. In paricular, his recovers he fac ha non-negaive harmonic funcions on he Heisenberg group mus be consan. We hank F. Baudoin for poining ou he connecion beween he gradien esimae in Corollary 4.5 and he Cheng-Yau inequaliy. Corollary 4.6. If u is any posiive harmonic funcion in a ball B x, r H 3, hen here exiss a universal consan C > no dependen on u and x such ha sup H log ux H C Bx,r, r. Moreover, if u is any posiive harmonic funcion on H 3, hen u mus be a consan. Proof. Suppose u > is harmonic in B,. By Corollary Hux H ux C = C sup x B, δ x 3 δ x δ x δx 4, x B,, where C is he same consan as in Corollary 4.5. This implies ha 4.8 sup H log u H C. B,, Now suppose ha u > is harmonic in B x, r for r >. By lef invariance and he dilaion properies of H 3 we see ha 4.8 implies sup H log u H C Bx,r, r. If u is harmonic on all of H 3, aking r gives us ha u mus be consan. 5. Concluding remarks Our work gives he firs use of explici non-markovian coupling echniques o ge geomeric informaion in he sub-riemannian seing. We would like o poin ou some poenially significan connecions wih a differen approach o such a seing. K. Kuwada in [6] proved an imporan resul on he dualiy of L q -gradien esimaes for he hea kernel of diffusions and heir L p -Wassersein disances under he assumpions of volume doubling and a local Poincaré inequaliy, for any p [, ], p + q =. Using his dualiy, he used he L -gradien esimae of he hea kernel for Brownian moion on he Heisenberg group obained in [9] and [] o derive L -Wassersein bounds. More precisely, he proved ha if d W x, y; denoes he L -Wassersein disance beween he laws of Brownian moion on H 3 saring from x and y a ime >, hen 5. d W x, y; Kd CC x, y for some consan K ha does no depend on x, y,. The consan K is no known, he bes esimae obained so far is K see [3]. Alhough we work wih he oal variaion disance insead of he Wassersein disance, Theorem 3.6 gives a beer esimae of he disance beween he laws of he wo Brownian moions on