Hölder norms and the support theorem for diffusions

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1 Hölder norms and the support theorem for diffusions Gérard Ben Arous, Mihai Gradinaru Uniersité de Paris-Sud, Mathématiques, Bât. 45, 9145 Orsay Cedex, France and Michel Ledoux Laboratoire de Statistique et Probabilités, Uniersité Paul Sabatier, 316 Toulouse Cedex, France Summary. One shows that the Stroock-Varadhan [S-V] support theorem is alid in α-hölder norm. The central tool is an estimate of the probability that the Brownian motion has a large Hölder norm but a small uniform norm. Contents :. Introduction p Conditional tails for oscillations of the Brownian motion p... Hölder balls of different exponent positiely correlated p Conditional tails for oscillations of stochastic integrals p Support theorem in Hölder norm p. 17. Appendix p.. References p. Introduction What is the probability that the Brownian motion oscillates rapidly conditionally on the fact that it is small in uniform norm? More precisely, 1

2 what is the probability that the α-hölder norm of the Brownian motion is large conditionally on the fact that its uniform norm or more generally its β-hölder norm with β < α is small? This is the kind of question that naturally appears if one wants to extend the Stroock-Varadhan characterization of the support of the law of diffusion processes [S-V] to sharper topologies than the one induced by the uniform norm. We deal with this question in 1 and show that those tails are much smaller than the gaussian tails one would get without the conditioning. This gies a family of examples where the conjecture stated in [DG-E-...] that two conex symmetric bodies are positiely correlated for gaussian measures is true. Our proofs are based on the Ciesielski isomorphism [C] see [B-R] for other applications of this theorem and on the correlation inequality. We gie in appendix a proof which aoids these tools. This enables us to control in 3 the probability that a Brownian stochastic integral oscillates rapidly conditionally on the fact that the Brownian motion is small in uniform norm. This is the tool to extend the Stroock-Varadhan support theorem to α-hölder norms. 1. Conditional tails for oscillations of the Brownian motion If x is a continuous real function on [, 1], anishing at zero, we define the sequence ξ m x m 1 by the formula: ξ m x = ξ n +kx = n k 1 k k 1 x x x, n+1 n n for n and k = 1,..., n. Denote 1.1 x = sup x t, t 1 1. x α = sup s t 1 x t x s, α ], 1], t s α 1.3 x α = sup m α 1 ξm x, α [, 1]. m 1 It is now classical that, for α ], 1[, the norms α and α equialent see [C]: are

3 1.4 α 1 x α x α 1 k α x α, α ], 1[, where and k α = α 1 1 α 1, 1.5 x x. Let w be a liniar Brownian motion started from zero. We want to estimate the probability that w. α is large conditionally on the fact that w. β is small. We shall first tackle the same problem with the norms. 1.6 Theorem. Let r, R be a couple of real positie numbers, = 1 R b b a and denote r a 1.7 Λ α,β r, R = ϕ + 1 a R 1 a where ϕt = e t, a = 1 π α, b = 1 β. Then, ϕt t 1 a dt, 1.8 P w. α > R w. β < r 1 ϕt dt Λ α,βr, R ; 1.9 P w. α > R w. β r Λ α,β p β r, R, where p β = 1 β, if β > and p = 4; 1.1 P w. α > R w. β r Λ α,β p β r, 1 k 1 α R. To proe the theorem we need the following: 1.11 Lemma. Let us denote n = 3 [ 1 ] R b a. Then r

4 1.1 Proof. n n +1 By the classical bound: Rn a ϕt dt Λ α,β r, R. t ϕs ds ϕt t and the fact that ψ is decreasing, we get: n n +1 Rn a ϕt dt ψt, t >, n n +1 ψrn a = ψrn + 1 a + ϕ n n a R 1 a From this the conclusion follows. ψrn a ψ + n +1 Rn +1 a ψt t 1 a 1 dt. ψrt a dt = q.e.d. We make another essential obseration. If C, C are two symmetric conex sets in IR d, a general conjecture stated in [DG-E-...] predicts that they are positiely correlated for the canonical Gaussian measure γ d, that is, 1.13 γ d C C γ d C γ d C. This is true for d = see [P], and for arbitrary d proided C is a symmetric strip see [Sc] or [Si]. The general case is stil open. Proof of the Theorem 1.6. Proof of 1.8. We note that g n = ξ n w is a sequence of independent identically distributed standard Gaussian random ariables. Then, P w. α > R w. β < r = P sup n 1 n a g n > R sup m b g m < r = m 1 P n 1 g n > Rn a m 1 g m < rm b P m 1 g m < rm b 4

5 n 1 n 1 P Rn a < g n < rn b m 1,m n g m < rm b = m 1 P g m < rm b P Rn a < g n < rn b P g n < rn b n n +1 rn b 1 Rn a < rn b = n 1 ϕt dt Rn a rn b ϕt dt 1 ϕt dt s Rn a e π ds rn b rn b n n +1 e s π ds Rn a ϕt dt. 1 Rn a <rn b = Clearly rn b rn + 1 b so the last inequality is true. Then 1.8 is a consequence of the Lemma Proof of 1.9. We can write again P w. α > R w. β r = P sup n a g n > R w. β r = n 1 P n 1 g n > Rn a w. β r n 1 P g n > Rn a w. β r. But for w. β r, by 1.4 or 1.5 we get or g n 1 β rn b, if β > g n 4rn 1, if β =. So, the preceding sum is taken oer all integer n 1 such that 1 β rn b Rn a, if β >, or 4rn 1 Rn a, if β =, that is, n β 1 R 1 b a r or n α R r 1 α. On the other hand, g n = ξ n w is a linear form on the Wiener space. By the correlation inequality with one of sets a symmetric strip and by a simple finite dimensional approximation see also [S-Z], we obtain P g n > Rn a, w. β r P g n > Rn a P w. β r 5

6 or Therefore P g n > Rn a w. β r P g n > Rn a. P w. α > R w. β r where n 1 = [ p 1 b a β n n 1 +1 P g n > Rn a = n n ] R b a. By the Lemma 1.11 we get 1.9. r Rn a ϕt dt, Proof of 1.1. It is a consequence of 1.4 or 1.5 and 1.9. q.e.d. We shall estimate Λ α,β r, R: 1.14 Lemma. With the notations of the Theorem 1.6, there exists a polynomial function Ψ a, increasing on ], [, such that 1.15 Λ α,β r, R ϕ a R a a Ψ a. Proof. We shall simply gie an upper bound for ϕt t 1 a dt. Noting that ϕ t = t ϕt and integrating by parts, we get If a 1 3, ϕt t 1 a dt = ϕ t t 1 a 3 dt = ϕ 1 a a 3 ϕt t 1 a 4 dt. ϕt t 1 a dt ϕ 1 a 3, which gies 1.14 with Ψ a x 1. If a < 1 3, similarly, So, if 1 5 a < 1 3, ϕt t 1 a 4 dt = ϕ 1 1 a 5 + a 5 ϕt t 1 a 6 dt. ϕt t 1 a dt ϕ 1 1 a 3 + a 3 ϕ 1 a 5, 6

7 which is exactly 1.14 with p = 1 in the following expression: Ψ a x = 1 + a 3 x a 3 a 5... a p 1 x p. Repeating the same reasoning the result is easily obtained for any p and any 1 a such that a < 1. Ψ p+3 p+1 a has positie coefficients, it is therefore increasing on ], [. q.e.d. Combining this result with the Theorem 1.6 we obtain the following: 1.16 Corollary. Let R, r be such that ε >. Then, 1.17 P w. α > R w. β < r cε ϕ ε 1.18 P w. α > R w. β < r ϕq β q β ε 1 + Ψ a 1 ε 1 + q 1 a 1 β Ψ a ε R β 1.19 P w. α > R w. β < r ϕc α,β c α,β ε k 1 α 1 1 a c 1 R β. Ψ a. ε r α r α R β ; r α a α,β ; Here q β = p a b a β, c α,β = q β 1 kα 1 b 1 b a and cε = ε ϕt dt. if ε then cε and Ψ a 1 1. ε Note that We proe now a stronger result: 1. Theorem. Let α, β be two real numbers such that β < α < 1. There exists a positie number u α,β = 1 α, such that, for eery u [, u 1 β α,β[, there exists M α, β, u and positie constants k i α, β, u, i = 1,, such that, for eery M M, 7

8 1.1 sup P w. α > Mδ u w. β < δ k 1 M β exp k M 1 β <δ 1 Proof. First of all we take in the Corollary 1.16, R = Mδ u and r = δ. So, for eery δ ], 1], P w. α > Mδ u w. β < δ c α,β M 1 b b a δ u1 b 1 a b a exp c α,β M b b a δ ub a b a. b a It is clear that, when M u1 b 1 a b a b the right hand side c b a a ub α,β of the last inequality is an increasing function of δ, for δ ], 1]. So, sup P w. α > Mδ u w. β < δ c α,β M 1 b b a exp c α,β M b b a, <δ 1. namely the conclusion. q.e.d.. Hölder balls of different exponent positiely correlated We show here that the conjecture on the correlation inequality is true for Hölder balls. We denote B α ρ = { w. α ρ } and B αρ = { w. α ρ }..1 Theorem. If R is sufficient large and if r is fixed, then B α R and B β r are positiely correlated. Proof. We proed in Corollary 1.16, for example when r = 1, that, for large R,. P B α R c B β 1 c α,β exp c α,β R 1 β, for eery β < α < 1. We can compare this estimate with the classical gaussian estimate, for large R,.3 P w. α > R exp c α R see [BA-Le] or [B-BA-K] for other consequences of this inequality. 8

9 By large deiations principle we obtain in fact, P B α R c e cα R, proided R is sufficiently large. Therefore, by., for large R,.4 P B α R B β 1 P B α R. So, in this particular case, the general conjecture is alid: the two symmetric conex sets B α R and B β 1 are positiely correlated, for large R. q.e.d. Remark. We can also show that, for any R, r >, the pairs of balls B αr, B βr and B αr, B β r are positiely correlated. Indeed, by 1.3, B αr = m 1 gm Rm 1 α = m 1 S m, so, it is an intersection of independent symmetric strips. Then, with the same argument as in the proof of 1.9, we get, for any conex symmetric C, P C B αr = P C m 1 S m P C m S m P S 1... P C m 1 P S m = P C P m 1 S m = P C P B αr. Here we used the independence of S m. The conclusion is obtained taking C = B βr or C = B β r. 3. Conditional tails for oscillations of stochastic integrals We shall estimate the Hölder norm of some stochastic integrals. Let X j t, x, j = 1,..., m, X t, x be smooth ector fields on IR d+1 and denote B 1,..., B m a m-dimensional Brownian motion. Let P x be the law of the diffusion x t, the solution of the Stratonoich equation m 3.1 dx t = X j t, x t db j t + X t, x t dt, x = x. j=1 9

10 Let us introduce the following class of stochastic processes: 3. Definition. For α, β [, 1 [ and u [, 1], we shall denote by the set of stochastic processes Y, such that M α, β u 3.3 lim sup M <δ 1 P Y. α > Mδ u B. β < δ =. Here and elsewhere B. α = max 1 i m B. i α. We collect our results in the following: 3.4 Lemma. Let f : IR d IR be a smooth function and, for i, j {1,..., r}, denote 3.5 η ij t = 1 Bs i dbs j Bs j db i s, ξ ij t = Bs i dbs j. Then, i B. i M α, β u, for β < α < 1 and u [ [, 1 α 1 β. ii η. ij M α u,, for α [, 1 [ and u [, 1]. iii ξ. ij M α u,, for α [, 1 [ and u [, 1]. i fx s dξs ij Mu α,, for α [, 1 [ and u [, 1]. fx s dbs i Mu α,, for α [, 1 [ and u [, 1 α[. Proof. Clearly, i is proed in the Theorem 1.. ii We proceed as in [S-V]. There exists a one dimensional Brownian motion w, such that, when i j, η ij t = wat, at = 1 B 4 s i + Bs j ds, where w is independent of the process B i t + B j t and so, independent of B.. There exists a positie constant c, such that a, a 1 are bounded 1

11 by c B.. Then we can write P η. ij α > Mδ u B. < δ = P B. < δ 1 P wa α > Mδ u, B. < δ. If z is α-hölder, z is β-hölder then z z is αβ-hölder and z z αβ z α, z z α β. Here and elsewhere α,t denotes the Hölder norm on [, T ]. So, wa α w α, a a α 1. Therefore P wa α > Mδ u, B. < δ P w α, c B. c B. α > Mδ u, B. < δ. A scaling in Hölder norm shows that w α,τ and τ 1 α w α,1 hae the same law. Then we can write Finally, P η. ij α > Mδ u B. < δ P B. < δ 1 P w. α c B. 1 α B. α > Mδ u, B. < δ. P η. ij α > Mδ u B. < δ P w. α c δ > Mδ u exp c α M δ 1 u by the independence of w and B., and by the gaussian inequality.3. iii We note another triial inequality: if z, z are α-hölder then z z is α-hölder and z z α z α z + z z α. In particular But B. i B. j α B. B. α. P B. B. α > Mδ u B. < δ = P B. α > Mδ u 1 B. < δ. 11,

12 The conclusion follows at once from i, ii and ξ. ij α η. ij α + 1 B.i B. j α η. ij α + B. B. α. i We apply Ito s formula seeral times using the usual conention that repeated indices are summed: fx s dξ ij s = fx t ξ ij t f l x s X l kx s ξ ij s db k s L s f x s ξs ij ds f l x s X l jx s B i s ds = I 1 + I + I 3 + I 4. Here L t is the generator of the diffusion x t and X l j denotes the l component of X j. It is sufficient to erify i for each I i, i = 1,, 3, 4. We readily see that a I 3, I 4 M α, u, because I 4 c B. and I 3 c ξ. ij, so, we consider only I 1 and I. Firstly, I 1 = fx ξ ij t + L s f x s ds ξ ij t + f l x s Xkx l s ξs ij dbs k ξ ij t = Again b I 1, I 11 M α, u, I 1 + I 11 + I 1. because I 1 α = c ξ. ij α, I 11 α c ξ. ij α. Setting α k = f l X l k, α k,m = α k x m I 1 = α k x t B k t ξ ij t we can write 1 Bs k L s α k x s ds ξ ij t

13 Bs k α k,m x s Xn m x s dbs n ξ ij t + α k,p x s X p k x s ds ξ ij t = I 11 + I 1 + I 13 + I 14. There is no problem to see that I 1 α c B. ξ. ij α and I 14 α c ξ. ij α and so, c I 1, I 14 M α, u. There exists a one-dimensional Brownian motion w, such that I 13 = wat ξ ij t, at = α k,m α k,m amm x s Bs k Bs k ds, where a ij = m k=1 Xk i X j k. We obtain P I 13 α > Mδ u B. < δ P ξ. ij α > M 1 δ B. < δ + P w. α, c B. c B. α ξ. ij α > Mδ u, ξ. ij α M 1 δ B. < δ. By iii, we hae to consider only the second term: P w. α > c M 1 δ u P B. < δ 1 exp This yields d I 13 M α, u. Then, α k x t ξ ij t B k t α k x s Bs k dξs ij I = α k x s ξ ij s db k s = α k,l x s X l mx s ξ ij s B k s db m s α j x s B i s ds c α M δ + c. u δ L s α k x s ξs ij Bs k ds ξ ij s α k,l x s X l mx s δ km ds 13

14 Clearly, B k s α k,l x s X l jx s B i s ds = J J 7. e I 11 + J 1 = and So, J 3 α c B. ξ. ij, J 5 α c B., J 6 α c ξ. ij, J 7 α c B.. f J 3, J 5, J 6, J 7 M α, u. By the same reasoning, J = wat, at = so, it suffices to estimate ξs ij α k,l α k,l all x s Bs k Bs k ds, P J α > Mδ u, ξ. ij M 1 δ B. < δ P w. α, c ξ. ij c B. ξ.ij α B. α > Mδ u, ξ. ij < M 1 δ B. < δ P w. α > c M 1 δ u P B. < δ 1 exp c α M δ + c. u δ Again g J M α, u. Finally we hae to study the martingale part of J 4, the bounded ariation being obiously controlled. We can write as aboe, Obiously, α k x s B k s B i s db j s = wat, at = α kx s B k s B i s ds. P α k x s Bs k Bs i dbs j α > Mδ u B. < δ 14

15 So, P w. α > c Mδ u P B. < δ 1 exp c α M δ u + c δ. h J 4 M α, u. Using formulas a-h we can conclude that fx s dξ ij s M α, u. We use the same idea, namely we shall apply Ito s formula seeral times. Firstly, denoting f = f x l l, fx s db i s = fx B i t + s dbs i But S 1 α c B. α and S = B i t s dbs i L u fx u du+ f l x u X l jx u db j u = S 1 + S + S 3. L s fx s ds where S 1 α c B. α and S α c B.. Clearly, S 1, S 1, S M α, u. Then, with the same notation as in i, S 3 = B i t By i, it is clear that α j x s db j s + S 3 = B i s L s fx s ds = S 1 + S, B i s α j x s db j s + S 31 + S 3 + S 33. α j x s dξ ij s M α, u, if i j. α j x s ds = For i = j we get a term with the same form as S 33, terms which are bounded in Hölder norm by a constant. To proe, it is sufficient to proe that S 31 M α u,. Note that S 31 = B i t B j t α j x B i t s dbs j 15 L u α j x u du

16 s Bt i dbs j But S 311 α c B. B. α and S 31 = B i t B j t α j,l x u X l kx u db k u = S S 31 + S 313. L s α j x s ds B i t B j s L s α j x s ds = S S 31, where S 311 α c B. α B. and S 31 α c B. α B.. Again S 311, S 311, S 31 M α u,. We denote β k x = α j,l xx l kx. Then, S 313 = B i t B j t β k x s db k s B i t B j s β k x s db k s B i t β k x s ds = S S S Arguing as for S 3, S 33 we see that S 313 = Bt i β kx s dξs jk, j k and S 3133 are in M α u,. We repeat with S 3131 the computations which we already performed for S 31 and we see that with clear notations S 31311, S 31311, S 3131, S M α, u. Then S = BtB i j t γ lx s dξs kl, l k, where γ l = β m xxl m S satisfies as aboe. To control the Hölder norm of S we can write x, so S = B i tb j t B k t γ l x s db l s = B i tb j t B k t wat, at = where w is a one-dimensional Brownian motion. So, γ l x s ds, P S α > Mδ u B. < δ P B. α > M 1 δ u 1 B. < δ + P w. α c B. α B. > Mδ u, B. α M 1 δ u 1 B. < δ P B. α > M 1 δ u 1 B. < δ + exp c α M + c. δ 3 δ From this we can conclude that S satisfies. The proof of the lemma is complete. 16

17 q.e.d. 4. Support theorem in Hölder norm Now we are able to extend the support theorem of Stroock-Varadhan for α-hölder topology. Let us denote by Φ x the mapping which associates to h L = L [, 1], IR m the solution of the differential equation m 4.1 dy t = X j t, y t h j t dt + X t, y t dt, y = x. j=1 4. Theorem. Let α [, 1 [. For the α-topology, the support of the probability P x coincide with the closure of Φ x L, that is, 4.3 supp α P x = Φ x L α. Proof. To begin with, we note that, for eery ε > and δ = ε 1 n u, u ], 1 α[, n > integer, P X k s, x s dbs k > ε B. < δ = α P X k s, x s dbs k > n δ u B. < δ α sup P X k s, x s dbs k > n η u B. < η. <η 1 α Letting n, by of the Lemma 3.4, we obtain, for eery ε >, 4.4 lim P δ Then we proe that, for eery ε >, X k s, x s dbs k > ε B. < δ =. α 4.5 lim δ P x. Φ x α < ε B. < δ = 1, using 4.4 and the following ariant of Gronwall s lemma: 17

18 4.6 Lemma. For m and l two functions, put z t = z + mt + lz s ds, z t = z + l z s ds. Suppose that m α η, m = and that l is a Lipschitz continuous function with constant L. Then z z α 1 + L e L η. Proof. By Gronwall s lemma we can immediately write z z η e L. Then, z z α, t η + lz u l z u du α, t L p η + p q z α u z u du q max p < q t L p η + max p < q t p q α z q z q + u q α z z α, u du q η + L z z + L z z α, u du. Gronwall s lemma ends up the proof of the Lemma 4.6. q.e.d. We apply this with z = Φ x B., z = Φ x, mt = X ks, x s db k s and lx s = X s, x s. So, there exists a positie constant K, such that Φ x B. Φ x α < K ε, proided Thus we obtain X k s, x s dbs k ε. α P x. Φ x α > ε B. < δ = 18

19 P x. Φ x α > ε P X k s, x s dbs k α > ε K X k s, x s dbs k α > ε K B. < δ. Now 4.5 is a clear consequence of 4.4. Finally, Girsano s formula gies, for any h L and ε >, B. < δ 4.7 lim δ P Φ x B. Φ x h. α < ε B. h. < δ = 1 as in [S-V], p But, 4.7 implies 4.8 P Φ x B. Φ x h. α < ε >, for eery ε >. and, consequently, we obtain the inclusion 4.9 supp α P x Φ x L α. The conerse inclusion is easily obtained using the polygonal approximation of the Brownian motion. For each n and t, we consider t n = [n ] n, t+ n = [n ] + 1 n, Ḃ n t = n B t + n B tn. Let x n t be the solution of the equation 4.1 with Ḃnk t denote P x n the law of this solution, it is obious that x. n Φ x L and P n x Φ x L α = 1. instead h k t. If we It suffices to show that P x is the weak limit of P x n or, that P n is relatiely weakly compact with respect to α -topology. By classical estimates, for eery p, there exists a positie constant c p, such that, for eery positie integer n and for eery s, t [, 1 ], E x n t x n s p c p t s p see for instance [Bi], p. 4. It is easy to see that sup E x. n p α < c, if n α < p 1 p. 19 x

20 If we choose p large enough so that α < p 1, and if p α ]α, p 1 [, it is then p clear that the set Kc = { z : z α < c } is compact in α -topology, and that, for eery ε >, there exists a positie constant c ε, such that, sup P x n Kc ε < ε. n So, P x n is tight. The proof of the Theorem 4. is complete. q.e.d. Appendix We gie now another proof of a ariant of 1.1 or 1.19, when β =, which does not require the use of Ciesielski s theorem that is 1.4 and 1.5 nor the correlation inequality. A.1 Theorem. Let r, R be a couple of real positie numbers. For eery a < a and b > b, there exists a constant c, such that, if Ra r b A. for β < α < 1. P w α > R w β < r exp 1 R 1 β r 1 α > c, then, Proof. Put η = r R 1. Then, if w β < r, Thus we obtain sup s<t s+η w t w s sup s<t,t s>η t s R. α w α > R w β < r w t w s t s R α sup s<t s+η w t w s t s α R sup w t < r t = sup X α R. D

21 Here = s, t, D = { : s < t s + η} and X α gaussian ariable. Now, we can estimate = wt ws t s α is a two-parameter P w α > R w β < r P sup X α R exp D R M α X α where the last inequality is alid when R M α see [L-T], p. 57. Here and So, we get write < M α = E sup X α E w α < D Xα = sup E X α = η 1 α. D P w α > R w β < r exp R = exp 1 R 1 β, β. η 1 α r 1 α The restriction R M α may be weakened as follows. Take α > α and sup s<t s+η sup s<t s+η Now, we need only w t w s t s R = α sup s<t s+η w t w s R η α α = t s α w t w s t s α α R t s α sup s<t s+η w t w s R α β t s α r α α,. R α β r α α > E w α = M α, and the proof of the theorem is complete. q.e.d. 1

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