Geodesic
Stochastic differential equations driven by processes generated by divergence form operators II : convergence results
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 387-411.

Voir la notice de l'article provenant de la source Numdam

We have seen in a previous article how the theory of “rough paths” allows us to construct solutions of differential equations driven by processes generated by divergence form operators. In this article, we study a convergence criterion which implies that one can interchange the integral with the limit of a family of stochastic processes generated by divergence form operators. As a corollary, we identify stochastic integrals constructed with the theory of rough paths with Stratonovich or Itô integrals already constructed for stochastic processes generated by divergence form operators by using time-reversal techniques.

DOI : 10.1051/ps:2007040
Classification : 60H10, 60J60
Keywords: rough paths, stochastic differential equations, stochastic process generated by divergence form operators, condition UTD, convergence of stochastic integrals 
@article{PS_2008__12__387_0,
     author = {Lejay, Antoine},
     title = {Stochastic differential equations driven by processes generated by divergence form operators {II} : convergence results},
     journal = {ESAIM: Probability and Statistics},
     pages = {387--411},
     publisher = {EDP-Sciences},
     volume = {12},
     year = {2008},
     doi = {10.1051/ps:2007040},
     mrnumber = {2437716},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps:2007040/}
}
TY  - JOUR
AU  - Lejay, Antoine
TI  - Stochastic differential equations driven by processes generated by divergence form operators II : convergence results
JO  - ESAIM: Probability and Statistics
PY  - 2008
SP  - 387
EP  - 411
VL  - 12
PB  - EDP-Sciences
UR  - http://geodesic.mathdoc.fr/articles/10.1051/ps:2007040/
DO  - 10.1051/ps:2007040
LA  - en
ID  - PS_2008__12__387_0
ER  - 
%0 Journal Article
%A Lejay, Antoine
%T Stochastic differential equations driven by processes generated by divergence form operators II : convergence results
%J ESAIM: Probability and Statistics
%D 2008
%P 387-411
%V 12
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/articles/10.1051/ps:2007040/
%R 10.1051/ps:2007040
%G en
%F PS_2008__12__387_0
Lejay, Antoine. Stochastic differential equations driven by processes generated by divergence form operators II : convergence results. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 387-411. doi : 10.1051/ps:2007040. http://geodesic.mathdoc.fr/articles/10.1051/ps:2007040/

[1] R. Adams, Sobolev spaces. Academic Press (1975). | Zbl | MR

[2] F. Baudoin, An introduction to the geometry of stochastic flows. Imperial College Press, London (2004). | Zbl | MR

[3] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland (1978). | Zbl | MR

[4] P. Billingsley, Convergence of Probability Measures. Wiley (1968). | Zbl | MR

[5] C.J.K. Batty, O. Bratteli, P.E.T. Jørgensen and D.W. Robinson, Asymptotics of periodic subelliptic operators. J. Geom. Anal. 5 (1995) 427-443. | Zbl | MR

[6] L. Capogna, D. Danielli, S.D. Pauls and J.T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem. Progress in Mathematics, Vol. 259. Birkhäuser (2007). | Zbl | MR

[7] L. Coutin, P. Friz and N. Victoir, Good Rough Path Sequences and Applications to Anticipating Stochastic Calculus. Ann. Prob. 35 (2007) 1172-1193. | Zbl | MR

[8] L. Coutin and A. Lejay, Semi-martingales and rough paths theory. Electron. J. Probab. 10 (2005) 761-785. | Zbl | MR

[9] F. Coquet and L. Słomiński, On the convergence of Dirichlet processes. Bernoulli 5 (1999) 615-639. | Zbl | MR

[10] S.N. Ethier and T.G. Kurtz, Markov Processes, Characterization and Convergence. Wiley (1986). | Zbl | MR

[11] H. Föllmer, Dirichlet processes, in Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980), Lecture Notes in Math. 851 476-478. Springer, Berlin (1981). | Zbl | MR

[12] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Process. De Gruyter (1994). | Zbl | MR

[13] P. Friz and N. Victoir, A note on the notion of geometric rough path. Probab. Theory Related Fields 136 (2006) 395-416. | Zbl | MR

[14] P. Friz and N. Victoir, On Uniformly Subelliptic Operators and Stochastic Area. Preprint Cambridge University (2006). <arXiv:math.PR/0609007>. | Zbl | MR

[15] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer-Verlag (1994). | Zbl | MR

[16] T.G. Kurtz and P. Protter, Weak Convergence of Stochastic Integrals and Differential Equations, in Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme, 1995, Talay D. and Tubaro L. Eds., Lecture Notes in Math. 1629 1-41. Springer-Verlag (1996). | Zbl | MR

[17] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag, 2nd edition (1991). | Zbl | MR

[18] A. Lejay, Méthodes probabilistes pour l'homogénéisation des opérateurs sous forme-divergence : cas linéaires et semi-linéaires. Ph.D. thesis, Université de Provence, Marseille, France (2000). url: http://www.iecn.u-nancy.fr/~lejay/.

[19] A. Lejay, A Probabilistic Approach of the Homogenization of Divergence-Form Operators in Periodic Media. Asymptot. Anal. 28 (2001) 151-162. | Zbl | MR

[20] A. Lejay, On the convergence of stochastic integrals driven by processes converging on account of a homogenization property. Electron. J. Probab. 7 1-18 (2002). | Zbl | MR

[21] A. Lejay, An introduction to rough paths, in Séminaire de probabilités, XXXVII, Lect. Notes Math. 1832 1-59, Springer, Berlin (2003). | Zbl | MR

[22] A. Lejay, Stochastic Differential Equations driven by processes generated by divergence form operators I: a Wong-Zakai theorem. ESAIM: PS 10 (2006) 356-379. | MR | mathdoc-id

[23] A. Lejay, Yet another introduction to rough paths. Preprint, Institut Élie Cartan, Nancy (2006). <http://hal.inria.fr/inria-00107460>.

[24] A. Lejay and T.J. Lyons, On the Importance of the Lévy Area for Systems Controlled by Converging Stochastic Processes. Application to Homogenization, in New Trend in Potential Theory, D. Bakry, L. Beznea, Gh. Bucur and M. Röckner Eds., The Theta Foundation (2006).

[25] A. Lejay and N. Victoir, On (p,q)-rough paths. J. Diff. Equ. 225 (2006) 103-133. | Zbl | MR

[26] T. Lyons and Z. Qian, System Control and Rough Paths. Oxford Mathematical Monographs. Oxford University Press (2002). | Zbl | MR

[27] T.J. Lyons and L. Stoica, The limits of stochastic integrals of differential forms. Ann. Probab. 27 (1999) 1-49. | Zbl | MR

[28] T.J. Lyons, Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215-310. | Zbl | MR

[29] P. Marcellini, Convergence of Second Order Linear Elliptic Operator. Boll. Un. Mat. Ital. B (5) 16 (1979) 278-290. | Zbl | MR

[30] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs 91. American Mathematical Society, Providence, RI (2002). | Zbl | MR

[31] A. Rozkosz, Stochastic Representation of Diffusions Corresponding to Divergence Form Operators. Stochastic Process. Appl. 63 (1996) 11-33. | Zbl | MR

[32] A. Rozkosz, Weak Convergence of Diffusions Corresponding to Divergence Form Operator. Stochastics Stochastics Rep. 57 (1996) 129-157. | Zbl | MR

[33] A. Rozkosz and L. Slomiński, Extended Convergence of Dirichlet Processes. Stochastics Stochastics Rep. 65 (1998) 1-2, 79-109. | Zbl | MR

[34] D.W. Stroock, Diffusion Semigroups Corresponding to Uniformly Elliptic Divergence Form Operator, in Séminaire de Probabilités XXII, Lecture Notes in Math. 1321 316-347. Springer-Verlag (1988). | Zbl | MR | mathdoc-id

Cité par Sources :