Positivity theorem for a general manifold

Statistics & Operations Research Transactions SORT 29 (1) January-June 2005, 11-26 Statistics & Operations Research Transactions Positivity theorem for a general manifold Hola c Institut d’Estad´ ıstica de Catalunya [email protected] ISSN: 1696-2281 www.idescat.net/sort R´ emi L´ eandre∗ Universit´e de Bourgogne Abstract We give a generalization in the non-compact case to various positivity theorems obtained by Malliavin Calculus in the compact case. MSC: 60H07 Keywords: Malliavin calculus. Density 1 Introduction For background about differential geometry, the reader can see the appendix. Let us consider X0,X1,...,Xmm+1 smooth vector fields on Rd. Let us consider Bi t mindependent Brownian motions. We consider the equation in Stratonovitch sense on Rd: dxt(x)=X0(xt(x))dt + i>0 Xi(xt(x))dBi t(1) issued of x∈Rd. If we perform a change of coordinates through a diffeomorphism of Rd, the vector fields are transformed according this change of coordinates, and since in Itˆ o-Stratonovitch Calculus, the Itˆ o formula is the traditional one, equation (1) has a meaning independent of the system of coordinates chosen. This means that we can look at (1) on a manifold. On Rd, we can consider the quadratic form ∗Address for correspondence: Institut de Math´ ematiques. Facult´ e des Sciences. Universit´ e de Bourgogne. 21000. Dijon. FRANCE. e-mail: [email protected]. Received: September 2003 Accepted: December 2004 12 Positivity theorem for a general manifold g(x)−1:ξ→i>0<Xi(x),ξ > 2. This quadratic form depends smoothly on x.We say that we are in an elliptic situation if this quadratic form is non-degenerated. We can introduce the measure on Rddµ(x)=detg(x)1/2dx where dx is the Lebesgue measure on Rd.dµis transformed intrinsically under a change of coordinates. We say that Rd endowed with the family of quadratic forms g(x) is a Riemannian manifold endowed with the Riemannian measure dµ. These considerations lead to the definition of a general (curved) Riemannian manifold. If Xis a vector field on Rd, we can define its divergence with respect of the measure dµby the following integration by parts formula: Rd Xfdµ=Rd fdivXdµ(2) If fis a function on Rd, the differential df(x) can be assimilated via the non-degenerated quadratic form g(x)toavectorfieldgrad f (We suppose that we are in an elliptic situation). The Laplace-Beltrami operator is therefore ∆=div grad. Moreover, (1) generates a Markov process whose generator is L=X0+1/2i>1X2 i. We can find a drift ˜ X0such that −L=˜ X0+∆ (3) All these considerations are invariant under a change of coordinates on Rd, and apply to a general manifold. Let us consider a general Riemannian manifold M, endowed with its Riemannian measure dy. On the space C∞ K(M) of smooth functions fwith compact support on M, there is a canonical second order operator ∆called the Laplace-Beltrami operator. ∆is symmetric positive: M g(y)∆f(y)dy =M f(y)∆g(y)dy (4) By analytical methods (see Gilkey(1995) for instance), we can solve the parabolic differential equation: ∂ ∂tut=−∆u;u0=f(5) where fis smooth with compact support. We call ut=exp[−t∆]f. By classical analytical technics (see Gilkey, 1995), the semi-group has smooth heat-kernels: ut(x)=M pt(x,y)f(y)dy (6) where (t,x,y)→pt(x,y) is smooth from R+∗×M×Minto R+∗. The fact that pt(x,y)>0 can be proved by analytical methods based upon the maximum principle. Let us suppose that Mis compact. By using a finite cover by small balls and some suitable partition of unity, we can write ∆under Hoermander’s form: −∆=X0+1/2X2 i(7) R´ emi L´ eandre 13 for some suitable smooth vector fields on M(this decomposition is still true over each relatively compact open subset of Min the non compact case). Reciprocally, we can consider an Elliptic Hoermander’s type operator Lon M: the vector fields Xii0span in all points the tangent space of Mif L=X0+1/2i>0X2 i. In such a case, we can introduce a metric on Mand a drift ˜ X0such that −L=˜ X0+∆ (8) where ∆is the Laplace-Beltrami operator associated to the Riemannian metric. We can solve the linear parabolic equation: ∂ ∂tut=Lut;u0=f(9) We get a semi-group exp[tL] and we can show that there exists a smooth in (x,y) strictly positive heat-kernel such that: exp[tL]f(x)=M pt(x,y)f(y)dy (10) The proofs of these results are based upon elliptic theory and the maximum principle (Gilkey, 1995). The natural geometrical object associated with these operators is the Riemannian metric. There are other distances, called Carnot-Caratheodory distances, which are associated with non-integrable systems of subspaces of the tangent space of the manifold. It is a natural object in Sub-Riemannian geometry. The big difference is the following: if the Riemannian distance is Lipschitz, a Sub-Riemannian distance is in general only Hoelder. Hoermander type operators are associated to Sub-Riemannian geometry. They are of the shape L=1/2X2 i.Ifinallx, the Lie algebra spanned by the Xiis equal to the tangent space of M(Strong Hoermander’s hypothesis), the semigroup exp[tL] has a smooth strictly positive heat-kernel pt(x,y) (Hoermander, 1967). We are motivated by an extension of this theorem to Hoermander’s type operator with drift L=X0+1/2X2 i. Let us consider a general manifold Mand some smooth vector fields Xi,i=0,...,m. In x, let us introduce the Lie ideal spanned by the vector fields Xi,X0excluded in the Lie algebra spanned by all the vector fields Xi,X0included. It is constructed as follows: we consider the space F0(x) spanned by the vector fields Xi,i0inx. We define inductively Fn(x) the linear space spanned by the Lie Brackets between an element of Fn−1(x)and a vector Xii=0,...,m. We suppose that in x,∪Fn(x)=Tx(M). This Hypothesis is called weak Hoermander’s hypothesis in x. Let us consider a Hoermander’s type operator L=X0+1/2i>0X2 i. Hoermander’s theorem (Hoermander, 1967) states that the semi-group generated by Lhas a smooth density pt(x,y) if the weak Hoermander hypothesis is checked in all x. We want to know when p1(x,y)>0. For that, we consider 14 Positivity theorem for a general manifold the control equation: dxt(h)=X0(xt(h))dt + i>0 Xi(xt(h))hi tdt (11) starting from x, where hi tbelongs to L2([0,1]). In Rd, when Hoermander’s condition is satisfied in xand when the vector fields Xiare bounded with bounded derivatives of all orders, Ben Arous and L´ eandre (1991) have given the following criterion: p1(x,y)>0if and only if there exists a hsuch that x1(h)=yandsuchthath→x1(h) is a submersion in h. This last condition is called Bismut condition (Bismut, 1984). The boundedness assumption in this theorem can be seen as a compactness assumption. Let us namely compactify Rnby adding a point at the infinity. We get the sphere Sn. The vector fields Xibounded with bounded derivatives can be extended into smooth vector fields over Snequal to 0 at the infinity. Tools used by Ben Arous and L´ eandre were Malliavin Calculus. This theorem was generalized by L´ eandre (1990) for jump processes. An abstract version for diffusions was given by Aida, Kusuoka and Stroock (1993). Bally and Pardoux (1998) have given a version of this theorem for the case of a stochastic heat equation. A. Millet and M. Sanz Sol´ e(1997) have given a positivity theorem for the case of a stochastic wave equation. Fournier (2001) has generalized the theorem of L´ eandre (1990) for the case of a nonlinear jump process associated to the Boltzmann equation. L´ eandre (2003a) has studied the case of a delay equation on a manifold. By using the mollifier in Malliavin sense introduced by Jones and L´ eandre (1997) and L´ eandre (1994), our goal is to remove the boundedness assumption in the theorem of Ben Arous and L´ eandre, and to generalize it to a general manifold Mnot necessarily complete. Our theorem is the following: Main Theorem. Let us suppose that in all points x of the manifold M, the weak Hoermander’s hypothesis is checked. Then p1(x,y)>0if and only if there exists an h such that x1(h)=y and such that h→x1(h)is a submersion in h. We refer for more details on Malliavin Calculus to the review of Meyer (1984), to the surveys of L´ eandre (1988), L´ eandre (1990), Kusuoka (1992) and Watanabe (1992) for the application of Malliavin Calculus to heat kernels in the compact or the bounded case. In the first part, we give a proof of the main theorem. In the second part, we give some extensions to other processes than diffusions. 2 Proof of the main theorem Let us show that the condition is sufficient. R´ emi L´ eandre 15 Let us introduce the solution of the stochastic differential equation in Stratonovitch sense, where Bi tare some independent Brownian motions: dxt(x)=X0(xt(x))dt + i>0 Xi(xt(x))dBi t(12) starting from x. Let us introduce the exit time τof the manifold. If fis a smooth function on M, we have classically (see Ikeda-Watanabe (1981), Nualart (1995)): p1(x,y)f(y)dy =E[f(x1(x))1τ>1] (13) where pt(x,y) is the heat-kernel associated to the heat semi-group associated to the Hoermander’s type operator L=X0+1/2i>0X2 i. In general, we cannot apply Malliavin Calculus to the diffusion xt(x). In order to be able to apply Malliavin Calculus, we introduce the mollifiers of Jones-L´ eandre (1997) and L´ eandre (1994). We consider a smooth function dfrom Minto R+, equal to 0 only in xand which tends to ∞when y tends to infinity, the one compactification point of M. We consider a smooth function over ] −k,k[(k∈R+), equal to 1 over [−k/2,k/2] and which behaves as 1 (k−y)rwhen y→k−. Outside ] −k,k[, this function, called gk(y) is equal to ∞. We suppose that gk≥1. We choose a big integer r. We choose a smooth function from [1,∞[into[0,1], with compact support, equal to 1 in 1 and which decreases. The mollifier functional of Jones-L´ eandre (1997) is Fk=h(1 0 gk(d(xs(x)))ds) (14) Lemma. Fkbelongs to all the Sobolev spaces in the sense of Malliavin Calculus if r is big enough, and is equal to 1 if supsd(xs(x)) ≤k/2, is smaller than 1 if supsd(xs(x)) >k/2and is equal to 0 almost surely if supsd(xs(x)) ≥k. Moreover, Fk≥0. Proof of the Lemma. The support property of Fkcomes from the fact that the paths of the diffusion s→xs(x) are in fact almost surely Hoelder with a Hoelder exponent strictly smaller than 1/2, instead of being only continuous. Let us show that Fkbelongs to all the Sobolev spaces. Let us introduce some smooth vector fields Xk iwhich are equal to Xifor d≤kand which are equal to 0 if d≥k+1. We consider the stochastic differential equation in Stratonovitch sense starting from x: dxk t(x)=Xk 0(xk t(x))dt + i>0 Xk i(xk t(x))dBi t(15) 16 Positivity theorem for a general manifold Since we consider a Stratonovitch equation, its solution is the limit in all the Lp of the solution of the random ordinary differential equation got when we replace the Stratonovich differential dBi tby the random ordinary differential of the polygonal approximation of the leading Brownian motion. It is called Wong-Zakai approximation (Ikeda-Watanabe (1981)). This explains, as we will see later, that the rules of computations with this equation are formally the same as for the solution of an ordinary differential equation, unlike an Itˆ o equation. We put ˜ Fk=h(1 0 gk(d(xk s(x)))ds) (16) We get clearly ˜ Fk=Fk. The interest to use the diffusion xk t(x) instead of the initial diffusion is that we can apply Malliavin Calculus to it. Let us recall quickly how we proceed (see Meyer (1984) for a detailed exposition). Since the vector fields Xk ihave compact support, we can exhibit a smooth version of x→xt(x)(SeeIkedaWatanabe(1981) and Meyer (1981)). We put φk t(x)=∂ ∂xxk t(x) (17) which is the solution of the linear equation in Stratonovitch sense: dφk t(x)=∂ ∂xXk 0(xk t(x))φk t(x)dt + i>0 ∂ ∂xXk i(xk t(x))φk t(x)dBi t(18) If we perturb dBi tinto dBi t+λhi tdt, we get by Ikeda-Watanabe (1981) and Meyer (1981) a smooth version of the solution xk t(λ, x). Moreover , ∂ ∂λ xk t(0,x) is solution of the stochastic differential equation with second member which is deduced from the first one by taking formally the derivative of the equation of xk t(λ, x). These formal considerations are justified because the vector fields have compact supports (see IkedaWatanabe (1981) and Meyer (1981)). We get, in Stratonovitch sense: d∂ ∂λ xk t(0,x)=∂ ∂xXk 0(xk t(x)) ∂ ∂λ xk t(0,x)dt + i>0 ∂ ∂xXk i(xk t(x)) ∂ ∂λ xk t(0,x)dBi t+ i>0 Xk i(xk t(x))hi tdt (19) Since we consider Stratonovitch differential, we can solve (19) by the method of variation of constant. We get: ∂ ∂λ xk t(0,x)=Dhxk t(x)=φk t(x)t 0 (φk s(x))−1Xk i(xk t(x))hi tdt (20) R´ emi L´ eandre 17 Therefore the random kernel of Dxk t(x)isgivenby Dxk t(x)(s)=φk t(x)(φk s(x))−1Xk i(xk s(x)) for s≤t. Since the vector fields have compact supports, φk t(x) as well as its inverse are bounded in all Lpfor finite p. So the kernel of Dxk t(x) are bounded in all the Lp(see Meyer (1984)). Moreover, the path t→xk t(x) is Hoelder with Hoelder exponent strictly smaller than 1/2. By Kolmogorov lemma (see Meyer (1981)), the Hoelder norm of the diffusion t→xk t(x),t≤1 belongs to all the Lp. We deduce that for rbig enough (see JonesL´ eandre (1997) (2.14)) P{sup t 1 k−d(xk t(x)+>1 ;1 0 dt (k−d(xk t(x)))+r<C}<C(p)p(21) for all p. The kernel of the first derivative of ˜ Fkis not 0 only when sup d(xk t(x)) ≤k.Itisgiven by h1 0 gk(d(xk t(x))dt1 0 g k(d(xk t(x)))d(xk t(x))Dxk t(x)(s)dt (22) It remains to use the inequality |1 0 g k(d(xk t(x)))d(xk t(x))Dxk t(x)(s)dt| ≤1 0 (g k(d(xk t(x))2 dt)1/21 0 (d(xk t(x))Dxk t(s)(s))2dt1/2 (23) and to use (21) in order to deduce that D˜ Fk(s) is bounded in all the Lp. The same holds for the derivatives of higher order of ˜ Fk. We introduce the auxiliary measure µk: µk:f→E[Fkf(x1(x))] (24) To the measure µk, we can apply Malliavin Calculus. Namely, µk[f]=E[˜ Fkf(xk 1(x))]. In particular µkhas a density qksmaller than p1(x,y). In particular, if there exists a h such that x1(h)=yand h→x1(h) is a submersion in h, we can find klarge enough such that qk(y)>0, by the positivity theorem of Ben Arous and L´ endre (1991) in the compact case with the extra-condition that ˜ Fkhas to be strictly positive. This shows that the condition is sufficient. 18 Positivity theorem for a general manifold In order to show that the condition is necessary, we remark that if p1(x,y)>0iny, qk(y) is still strictly positive for klarge enough , because for kenough large, for small |E[(1τ>1−Fk)f(x1(x))]|≤f∞(25) where f∞denotes the uniform norm of f. Therefore, it is enough to apply the Ben Arous-L´ eandre result in the other sense. Remark: Let us suppose that Hoermander’s condition is satisfied only in x. We can suppose that his decreasing and that gkdecreases to 1, such that Fkincreases to 1τ>1.By Malliavin Calculus, µkhas a density qk, which increases. Let us consider the function f=1Afor a set Aof measure 0 for the Lebesgue measure over M.Wehave: µk[f]=0 (26) But µk[f]=E[Fkf(x1(x))] =0 (27) anf Fkf(x1(x)) increases and tends to 1τ>1f(x1(x)), which is in L1. We deduce that E[1τ>1f(x1(x))] =0 (28) This means that the the law of x1(x) has a density without to suppose that Hoermander’s hypothesis is satisfied in all points. Remark: The localization procedure given in this work is a localization procedure of all the paths between 0 and x1(x), when we cannot apply Malliavin Calculus to all the diffusions xt(x). It is different of various localization procedures, developped in L´ eandre (1988) for instance, in order to get some estimates of hypoelliptic heat-kernels in small time, which were used when we can apply the machinery of the Malliavin Calculus to all the diffusion xt(x): namely, in L´ eandre (1988), we consider vector fields Xion Rn with bounded derivatives of all orders in order to apply Malliavin Calculus. This allows to get a rough estimate of the heat kernel. Nash inequality (Carlen-Kusuoka-Stroock (1987)) allows to get rough estimates of the heat kernel: in L´ eandre (2002) we mix the localization procedures developped in this part and the Nash inequality, in order to localize the estimates which were got previously by Malliavin Calculus (see Kusuoka (1992), L´ eandre (1988), L´ eandre (1990), Watanabe (1992)) under the restrictions of Malliavin Calculus, and to avoid the classical boundedness assumption of Malliavin Calculus. Remark: Since the Laplace-Beltrami operator is an elliptic Hoermander’s type operator on each locally compact open subset of the manifold, we can apply the previous localization method to show that the heat-kernel associated to the Laplace-Beltrami operator on a Riemannian manifold is strictly positive. R´ emi L´ eandre 19 Remark: If the drift X0is identically equal to 0, this theorem recovers the fact that the heat kernel associated to the operator 1/2X2 iunder the strong Hoermander’s hypothesis is strictly positive, by using the technics of L´ eandre (1988) Theorem II.1. 3 Extensions The main novelty of the Malliavin Calculus with respect to its preliminary forms (See works of Hida, Elworthy, Albeverio, Fomin, Berezanskii..) is the following: it can be applied to diffusions, and can differentiate some functionals which are only almost surely defined. There are other examples of Wiener functionals, almost surely defined, where we can apply the Malliavin Calculus and where we can get some positivity theorems. We sketch the proof only. Nualart-Sanz (1985) consider some smooth vector fields Xi,i=0,...,don Rnwith derivatives at each order bounded. They consider dindependent Brownian sheets Bi(s,t) s≥0,t≥0. Let us recall that it is a Gaussian process indexed by R+×R+defined by: E[B(s,t)B(s,t)] =(s∧s)(t∧t) (29) and (3.2) E[B(s,t)] =0 They consider the Cairoli equation (δdenotes the Itˆ o integral): x(s,t)=x+ i>0[0,s]×[0,t] Xi(x(u,v))δBi(u,v)+[0,s]×[0,t] X0(x(u,v))dudv (30) By Malliavin Calculus, Nualart-Sanz (1985) can show if st >0, that x(s,t)has a law having a smooth density with respect of the Lebesgue measure on Rnif in all x,the vector fields Xii0spanRn(Nualart-Sanz (1985) study in fact a more degenerated situation). Millet-Sanz (1997) have shown that this density is strictly positive under this non degenerate assumption (They establish in fact under a more general assumption a necessary and sufficient condition for this density to be strictly positive). By using the fact that the path (s,t)→x(s,t)is Hoelder, we can remove the hypothesis that the derivative at each order of the vector fields are bounded. If we remove these hypothesis, the two-parameter diffusion can blow up. We introduce Othe measurable set where x(u,v)does not blow up on [0,s]×[0,t]. By using the technics of Part III, we can prove the following theorem: Theorem III.1. Let us suppose that the vector fields are smooth, and that in all x the vector fields Xii>0span Rn. Let us consider the measure: f →E[1Of(x(s,t))].This