Upper bounds for the density of solutions to stochastic differential equations driven by fractional brownian motions
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, p. 111-135

In this paper we study upper bounds for the density of solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H>1/3. We show that under some geometric conditions, in the regular case H>1/2, the density of the solution satisfies the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case H>1/3 and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper sub-Gaussian bound.

Dans ce papier nous étudions des bornes supérieures pour la densité d’une solution déquation différentielle conduite par un mouvement brownien fractionnaire d’indice de Hurst H>1/3. Nous montrons, que sous certaines conditions géomètriques, dans le cas régulier H>1/2, la densité de la solution satisfait l’inégalité de log-Sobolev, l’inégalité de concentration gaussienne et admet une borne supérieure gaullienne. Dans le cas H>1/3 et sous la même condition géomètrique, nous montrons que la densité est infiniment différentiable et admet une borne supérieure sous-gaussienne.

@article{AIHPB_2014__50_1_111_0,
     author = {Baudoin, Fabrice and Ouyang, Cheng and Tindel, Samy},
     title = {Upper bounds for the density of solutions to stochastic differential equations driven by fractional brownian motions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {1},
     year = {2014},
     pages = {111-135},
     doi = {10.1214/12-AIHP522},
     zbl = {1286.60051},
     mrnumber = {3161525},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_1_111_0}
}
Baudoin, Fabrice; Ouyang, Cheng; Tindel, Samy. Upper bounds for the density of solutions to stochastic differential equations driven by fractional brownian motions. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, pp. 111-135. doi : 10.1214/12-AIHP522. http://www.numdam.org/item/AIHPB_2014__50_1_111_0/

[1] F. Baudoin and L. Coutin. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Process. Appl. 117 (2007) 550-574. | MR 2320949 | Zbl 1119.60043

[2] F. Baudoin and M. Hairer. A version of Hörmander's theorem for the fractional Brownian motion. Probab. Theory Related Fields 139 (2007) 373-395. | MR 2322701 | Zbl 1123.60038

[3] F. Baudoin and C. Ouyang. Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions. Stochastic. Process. Appl. 121 (2011) 759-792. | MR 2770906 | Zbl 1222.60034

[4] F. Baudoin, E. Nualart, C. Ouyang and S. Tindel. Work in progress. Preprint, 2012.

[5] M. Besalú and D. Nualart. Estimates for the solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H(1 3,1 2). Stoch. Dyn. 11 (2011) 243-263. | MR 2836524 | Zbl 1231.60049

[6] M. Capitaine, E. Hsu and M. Ledoux. Martingale representation and logarithmic Sobolev inequality. Electron. Com. Probab. 2 (1997) 71-81. | MR 1484557 | Zbl 0890.60045

[7] T. Cass and P. Friz. Densities for rough differential equations under Hörmander condition. Ann. Math. To appear. | MR 2680405 | Zbl 1205.60105

[8] T. Cass, P. Friz and N. Victoir. Non-degeneracy of Wiener functionals arising from rough differential equations. Trans. Amer. Math. Soc. 361 (2009) 3359-3371. | MR 2485431 | Zbl 1175.60034

[9] T. Cass, C. Litterer and T. Lyons. Integrability estimates for Gaussian rough differential equations. Arxiv preprint, 2011. | Zbl 1278.60091

[10] A. Chronopoulou and S. Tindel. On inference for fractional differential equations. Arxiv preprint, 2011. | MR 3029332 | Zbl 1271.62197

[11] L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108-140. | MR 1883719 | Zbl 1047.60029

[12] R. C. Dalang and E. Nualart. Potential theory for hyperbolic SPDEs. Ann. Probab. 32(3A) (2004) 2099-2148. | MR 2073187 | Zbl 1054.60066

[13] A. Davie. Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express. 2007 (2007) abm009. | MR 2387018 | Zbl 1163.34005

[14] A. Deya, A. Neuenkirch and S. Tindel. A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 48(2) (2012) 518-550. | Numdam | MR 2954265 | Zbl 1260.60135

[15] P. Driscoll. Smoothness of density for the area process of fractional Brownian motion. Arxiv preprint, 2010. | Zbl 1268.60080

[16] P. Friz and N. Victoir. Multidimensional Stochastic Processes Seen as Rough Paths. Cambridge Univ. Press, Cambridge, 2010. | MR 2604669 | Zbl 1193.60053

[17] M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86-140. | MR 2091358 | Zbl 1058.60037

[18] M. Gubinelli and S. Tindel. Rough evolution equations. Ann. Probab. 38 (2010) 1-75. | MR 2599193 | Zbl 1193.60070

[19] M. Hairer. Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Probab. 33 (2005) 703-758. | MR 2123208 | Zbl 1071.60045

[20] M. Hairer and A. Ohashi. Ergodicity theory of SDEs with extrinsic memory. Ann. Probab. 35 (2007) 1950-1977. | MR 2349580 | Zbl 1129.60052

[21] M. Hairer and N. S. Pillai. Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths. Preprint, 2011. | MR 3112925 | Zbl 1288.60068

[22] E. Hsu. Stochastic Analysis on Manifolds. Graduate Series in Mathematics 38. Amer. Math. Soc., Providence, RI, 2002. | MR 1882015 | Zbl 0994.58019

[23] Y. Hu and D. Nualart. Differential equations driven by Hölder continuous functions of order greater than 1/2. Abel Symp. 2 (2007) 349-413. | MR 2397797 | Zbl 1144.34038

[24] S. Kou and X. Sunney-Xie. Generalized Langevin equation with fractional Gaussian noise: Subdiffusion within a single protein molecule. Phys. Rev. Lett. 93 (2004) 18.

[25] M. Ledoux. The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI, 2001. | MR 1849347 | Zbl 0995.60002

[26] T. Lyons and Z. Qian. System Control and Rough Paths. Oxford Univ. Press, Oxford, 2002. | MR 2036784 | Zbl 1029.93001

[27] A. Millet and M. Sanz-Solé. Large deviations for rough paths of the fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Stat. 42 (2006) 245-271. | Numdam | MR 2199801 | Zbl 1087.60035

[28] A. Neuenkirch, I. Nourdin, A. Rößler and S. Tindel. Trees and asymptotic developments for fractional diffusion processes. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 157-174. | Numdam | MR 2500233 | Zbl 1172.60017

[29] A. Neuenkirch, S. Tindel and J. Unterberger. Discretizing the Lévy area. Stochastic Process. Appl. 120 (2010) 223-254. | MR 2576888 | Zbl 1185.60076

[30] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Probability and Its Applications. Springer-Verlag, Berlin, 2006. | MR 2200233 | Zbl 0837.60050

[31] D. Nualart and A. Rǎşcanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55-81. | Zbl 1018.60057

[32] D. Nualart and B. Saussereau. Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Stochastic Process. Appl. 119 (2009) 391-409. | MR 2493996 | Zbl 1169.60013

[33] J. Szymanski and M. Weiss. Elucidating the origin of anomalous diffusion in crowded fluids. Phys. Rev. Lett. 103 (2009) 3.

[34] V. Tejedor, O. Bénichou, R. Voituriez, R. Jungmann, F. Simmel, C. Selhuber-Unkel, L. Oddershede and R. Metzle. Quantitative analysis of single particle trajectories: Mean maximal excursion method. Biophysical J. 98 (2010) 1364-1372.

[35] A. S. Üstünel. Analysis on Wiener space and applications. Arxiv preprint, 2010.

[36] M. Zähle. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 (1998) 333-374. | MR 1640795 | Zbl 0918.60037