The Stochastic Flow Theorem for an Operator of Order Four

Auteurs : Rémi Léandre
DOI : You do not have permission to access embedded form.


The Stochastic Flow Theorem for an Operator of Order Four


142.47 Ko


Creative Commons Aucune (Tous droits réservés)


Sponsors scientifique


Sponsors financier


Sponsors logistique

Séminaire Léon Brillouin Logo
<resource  xmlns:xsi=""
        <identifier identifierType="DOI">10.23723/2552/5069</identifier><creators><creator><creatorName>Rémi Léandre</creatorName></creator></creators><titles>
            <title>The Stochastic Flow Theorem for an Operator of Order Four</title></titles>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><dates>
	    <date dateType="Created">Sat 5 Oct 2013</date>
	    <date dateType="Updated">Mon 25 Jul 2016</date>
            <date dateType="Submitted">Sat 17 Feb 2018</date>
	    <alternateIdentifier alternateIdentifierType="bitstream">1051d85e63c96f02332b02d489aee385bf6f9984</alternateIdentifier>
            <description descriptionType="Abstract"></description>

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