The Stochastic Flow Theorem for an Operator of Order Four

28/08/2013
Auteurs : Rémi Léandre
OAI : oai:www.see.asso.fr:2552:5069
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The Stochastic Flow Theorem for an Operator of Order Four

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The Stochastic Flow Theorem for an Operator of Order Four R´emi L´eandre Laboratoire de Math´ematiques. Universit´e de Franche-Comt´e. 25030. Besan¸con. Cedex. France July 8, 2013 Dedicated to the academician J.M. Bis- mut for his retirement Some basic tools of stochastic analysis were translated recently in semi-group the- ory by L´eandre, by avoiding the theory of stochastic processes. We refer to the re- view[5], [6]. This holds for the Girsanov formula, the Itˆo formula,the Wentzel-Freidlin estimates, the theorem of stochastic flow of Malliavin, the Burkholder-Davies-Gundy inequalities, the Malliavin Calculus of Bis- mut type as well as the Bismutian proce- dure. Classically, following Hunt theory, there is a stochastic process associated to a semi- 1 group when the generator satisfies the max- imum principle. It becomes interesting to enlarge stochas- tic tools for a parabolic equation when the generator does not satisfy the maximum principle. A beginning of works in this di- rection was done by L´eandre. This works for the Itˆo formula, the classical martin- gale problems of stochastic analysis, the generalization of the Brownian sheet as well as the Airault-Malliavin-Baxendale equa- tion and the Girsanov formula. We refer to the review papers of L´eandre [7], [8], [9] and the two papers [10], [11]. The object of this communication is to give an analog of the stochastic flow of Malliavin and of the Burkholder-Davies- Gundy inequality to the case of a semi- group generated by an operator of order four. Let us remark that to state the Burkholder- Davies-Gundy inequality, we use the stan- 2 dard Davies method of harmonic analysis [1],[13]. Let us recall that the introduction of the Davies method in stochastic analy- sis was done by ourself in [4]. For a review papers on analytical meth- ods to semi-group generated by big order generator, we refer to the review of Davies [2]. 1 An extension of the theorem of Malliavin Let dµ = 1/Z exp[b(x)]dx be a proba- bility measure on R where b is supposed smooth with bounded derivatives at each order. Let X = a(x) d dx be a vector field on R without any divergence with respect of dµ. We suppose that a is smooth with bounded derivatives at each order. We consider the symmetric positive densely defined generator on L2 (dµ) L = X4 . It has a self-adjoint extention and therefore generates a semi-group of contractions exp[−tL] 3 on L2 (dµ): ∂ ∂t exp[−tL] = −L exp[−tL]. On R×R with generic element (x, u) we consider the measure dµt = dµ⊗|u|−1 du. We consider the vector field on R × R Xt = (X(x), a (x)u ∂ ∂u ) (1) Xt is without divergence with respect of dµt . We consider the densely defined sym- metric operator on L2 (dµt ) Lt = (Xt )4 . Lt has a self-adjoint extension which gen- erates a semi-group of contraction exp[−tLt ] on L2 (dµt ): ∂ ∂t exp[−tLt ] = −Lt exp[−tLt ]. We consider the Sobolev space Wk,2 k > 0 of function f such that the below quan- tity R |d/dxf(x)|k dµ is finite. We con- sider the intersection of these Sobolev space W∞−,2 and its topological dual (the space of distribution) W−∞,2. The main theorem is the following: Theorem 1 Let us consider f ia smooth function on R with bounded derivatives at each order. Then: 4 -i)We can extend exp[−tLt ] to the test function uf such that the quantity exp[−tLt ][uf ](x, 1) belongs to L2 (dµ). -ii)x → exp[−tL][f](x) has a distribu- tional derivative in x such that d dx exp[−tL][f](x) = exp[−tLt ][uf ](x, 1) (2) Remark:Let us explain heuristically this theorem. Let us consider the driving op- erator L1 = d4 dx4. Let us consider a formal path measure dP on a convenient path space of function from [0, 1] into R t → bt starting from 0 such that exp[−tL1][f](0) = f(bt)dP (3) Let us introduce a formal Stratonovich dif- ferential equation: dxt(x) = a(xt(x))dbt ; x0(x) = x (4) such that exp[−tL][f](x) = f(xt(x))dP (5) 5 If x → xt(x) is a flow, d dx exp[−tL][f](x) = f (xt(x)) d dx xt(x)dP (6) where d dxxt(x) satisfies the formal linear Stratonovich equation dt d dx xt(x) = a (xt(x)) d dx xt(x)dbt (7) starting from 1. The couple (xt(x), d dxxt(x)) should represent the semi-group exp[−tLt ] In the case where bt is a standard Brow- nian motion, this result is the standard stochastic flow of Malliavin [3] [12]. 2 A formal proof of the theorem Let us write the generator in Itˆo form: Xf = af (8) X2 f = a2 f(2) + aa f = A2,2f(2) + A2,1f (9) 6 X3 f = a3 f(3) +3a2 a f(2) +(a(a )2 +a2 a(2) )f = A3,3f(3) + A2,3f(2) + A1,3f (10) X4 f = a4 f(4) + 6a3 a f(3) + (4a3 a(2) +7a2 (a(2) )2 )f(2) +(a(a )3 +4a2 a a(2) +a3 a(3) )f = A4,4f(4) + A3,4f(3) + A2,4f(2) + A1,4f (11) ut(x) = exp[−tL][f](x) is the solution of the parabolic equation in the distribu- tional sense d/dtut(x) = −A4,4(x)u (4) t (x) −A3,4(x)u (3) t −A2,4(x)u (2) t (x)−A1,4(x)u (1) t (x) (12) starting from f. Therefore gt(x, h) = ut(x,h)−ut(x) h is the solution of the parabolic equation in 7 distribution sense: d/dtgt(x, h) = −A4,4(x+h)g (4) t (x, h)−A3,4(x+h)g (3) t (x, h) −A2,4(x+h)g (2) t (x, h)−A1,4(x, h)g (1) t (x, h) + −A4,4(x) + A4,4(x + h) h u (4) t (x) + −A3,4(x) + A3,4(x + h) h u (3) t (x)+ + −A2,4(x) + A2,4(x + h) h u (2) t (x)+ + −A1,4(x) + A1,4(x + h) h u (1) t (x) (13) We write this equation in distributional sense d/dtgt(x, h) = −X4 (x + h)gt(x, h) + αh (14) where αh tends in distributional sense to α0(x) = A4,4(x)u (4) t + A3,4(x)u (3) t (x)+ A2,4(x)u (2) t (x) + A1,4(x)u (1) t (x) (15) 8 We put fh(x) = f(x+h)−f(x) h . fh tends in distributional sense to f . By the method of variation of constant gt(., h) = t 0 exp[(t−s)L][αh]ds+exp[−tL][fh] (16) which tends in distributional sense to gt(0, h) = t 0 exp[(t−s)L][α0]ds+exp[−tL][f ] (17) This quantity is solution in the distribu- tional sense of d/dtψt(x) = −X4 ψt(x) − A4,4ψ (3) t (x)− A3,4ψ (2) t (x) − A2,4ψ (1) t (x) − A1,4ψt(x) (18) On the other hand Xj f = Ai,jf(i) (19) 9 such that Xj+1 f = (aAi,jf(i) + aAi,jf(i+1) ) (20) Therefore Ai,j+1 = a(Ai,j + Ai−1,j) (21) Let us write φt(x, u0) = exp[−tLt ][uf ](x, u0) (22) Let us suppose by induction that (Xt )j φt(x, u0) == Ai,jφ (i) t (x, u0)+ Ai,jφ (i−1) t (x, u0) (23) The main remark is that φt is linear in u0 and that the derivatives in u0 of φ (i) t (x, u0) are equal to this quantity. The result holds therefore from the recursion formula (21). 3 A Burkholder-Davies-Gundy inequality Theorem 2 (Burkholder-Davies-Gundy inequality)Let |f(x, u)| ≤ (|u|p + |u|). Then exp[−tLt ] can be extended to f(x, u) 10 Proof: We consider g(u) = C(|u|2r + 1) where r is a big positive integer. Accord- ing the work of Davies, we consider the operator Xp = g(u)−1 Xt g(u) = (X, a (x)(u ∂ ∂u +C(u)) (24) where C(u) is bounded with bounde deriva- tives at each order. Lp = (Xp )4 is a pos- itive symmetric densely defined operator on L2 (dµt ). It has therefore a positive self- adjoint extension on L(dµt ), which gener- ates a contraction semi group on L2 (dµt ). Moreover exp[−tLp ] = g−1 exp[−tLt ]g (25) We consider r big enough such that h = fg−1 belongs to L2 (dµt ). Therefore exp[−tLp ][h](x, u) = g(u)−1 exp[−tLt ][f](x, u) (26) belongs to L2 (dµt ). ♦ 11 End of the proof of theorem 1: exp[−tLt ][uf ](x, u0) is linear in u0. There- fore g(u0)−1 exp[−tLt ][uf ](x, u0) = l(u0) exp[−tLt ][uf ](x, 1) (27) belongs clearly to L2 (dµt ). ♦ References [1] Davies E.B.: heat kernels and spectral theory. Cambridge University Press (1989) [2] Davies E.B.: Lp spectral theory of higher-order elliptic differential opera- tors. Bull. Lond. Math. Soc. 29 , 513– 546 (1997). [3] Kunita H.: Stochastic flows and stochastic differential equations. Cambridge University Press (1990). 12 [4] L´eandre R: A simple prof of a large deviation theorem. In David Nu- alart and al (eds). Barcelona Semi- nar on Stochastic Analysis (St. Fe- liu de Guixols). Progress. Probab 32, Birkhauser, 72–76 (1993). [5] L´eandre R.: Applications of the Malliavin Calculus of Bismut type without probability. In Ana Maria Madureira and al (eds) 6th WSEAS Conf Simulation, Modelling and Op- timization (Lisboa). C.D. W.S.E.A.S. (2006) Copy in WSEAS Trans Math 5, 1205-1210 (2006). [6] L´eandre R.: Malliavin Calculus of Bismut type in semi-group theory. Far. East. J. Math. Sci 30, 1–26 (2008). [7] L´eandre R.: Stochastic analysis with- out probability: study of some basic tools. Journal Pseudo Differential Op- 13 erators and Applications 1, 389–410 (2010). [8] L´eandrev R.: A generalized Fock space associated to a Bilaplacian. In Victor Jin (eds). 2011 World Congress Engineering Technology (Shanghai), 68–72. C.D. I.E.E.E. (2011). [9] L´andre R.K: Stochastic analysis for a non-markovian generator: an intro- duction. [10] L´eandre R.: A path-integral approach to the Cameron-Martin-Maruyama- Girsanov formula associated to a Bi- laplacian. In Ti-Jun Xiao and al (eds) Integral and differential systems in function spaces and related problems. Journal of function spaces and appli- cations (Open access), Article 458738 (2012). [11] L´eandre R.: A Girsanov formula associated to a big order pseudo- 14 differential operator. [12] Meyer P.A.: Flot d’une ´equation diff´erentielle stochastique (d’apres Malliavin, Bismut, Kunita). In J. Az´ema and al eds. S´eminaire de Probabilit´es XV. Lectures Notes of Math 850, Springer (1981), 103–117. [13] Varopoulos N.Th. Saloff-Coste L. and Coulhon T.: Analysis and geome- try on groups . Cambridge University Press (1992) 15