Fan, Xiliang Logarithmic Sobolev inequalities for fractional diffusion. (English) Zbl 1397.60050 Stat. Probab. Lett. 106, 165-172 (2015). Summary: In this note, logarithmic Sobolev inequalities are established on the path space for the fractional Brownian motion with drift. The reference distance on the path space is the \(L^2\)-norm of the gradient along paths. Cited in 1 Document MSC: 60E15 Inequalities; stochastic orderings 60H07 Stochastic calculus of variations and the Malliavin calculus 60G22 Fractional processes, including fractional Brownian motion Keywords:logarithmic Sobolev inequality; fractional Brownian motion; path space PDF BibTeX XML Cite \textit{X. Fan}, Stat. Probab. Lett. 106, 165--172 (2015; Zbl 1397.60050) Full Text: DOI References: [1] Alòs, E.; Mazet, O.; Nualart, D., Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29, 766-801, (2001) · Zbl 1015.60047 [2] Baudoin, F.; Ouyang, C., Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions, Stochastic Process. Appl., 121, 759-792, (2011) · Zbl 1222.60034 [3] Baudoin, F.; Ouyang, C.; Tindel, S., Upper bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions, Ann. Inst. H. Poincaré Probab. Statist., 50, 111-135, (2014) · Zbl 1286.60051 [4] Cattiaux, P.; Guillin, A.; Wu, L., A note on talagrand’s transportation inequality and logarithmic Sobolev inequality, Probab. Theory Related Fields, 148, 285-304, (2010) · Zbl 1210.60024 [5] Coutin, L.; Qian, Z., Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields, 122, 108-140, (2002) · Zbl 1047.60029 [6] Decreusefond, L.; Üstünel, A. S., Stochastic analysis of the fractional Brownian motion, Potential Anal., 10, 177-214, (1998) · Zbl 0924.60034 [7] Fan, X. L., Harnack inequality and derivative formula for SDE driven by fractional Brownian motion, Sci. China-Math., 561, 515-524, (2013) · Zbl 1273.60068 [8] Fan, X. L., Harnack-type inequalities and applications for SDE driven by fractional Brownian motion, Stoch. Anal. Appl., 32, 602-618, (2014) · Zbl 1304.60072 [9] Gourcy, M.; Wu, L., Logarithmic Sobolev inequalities of diffusions for the \(L^2\) metric, Potential Anal., 25, 77-102, (2006) · Zbl 1098.60027 [10] Gross, L., Logarithmic Sobolev inequalities, Amer. J. Math., 97, 1061-1083, (1975) · Zbl 0318.46049 [11] Guo, D. J., Non-linear functional analysis (cininese), (2002), Science and Technology Press Shandong [12] Hairer, M.; Ohashi, A., Ergodic theory for SDEs with extrinsic memory, Ann. Probab., 35, 1950-1977, (2007) · Zbl 1129.60052 [13] Hairer, M.; Pillai, N. S., Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion, Ann. Inst. H. Poincaré Probab. Statist., 47, 601-628, (2011) · Zbl 1221.60083 [14] Ledoux, M., Concentration of measure and logarithmic Sobolev inequalities, (Séminaire de Probabilités XXXIII, Lecture Notes in Math., vol. 1709, (1999), Springer Berlin), 120-216 · Zbl 0957.60016 [15] Lyons, T., Differential equations driven by rough signals, Rev. Mat. Iberoam., 14, 215-310, (1998) · Zbl 0923.34056 [16] Nikiforov, A. F.; Uvarov, V. B., Special functions of mathematical physics, (1988), Birkhäuser Boston · Zbl 0694.33005 [17] Nourdin, I.; Simon, T., On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion, Statist. Probab. Lett., 76, 907-912, (2006) · Zbl 1091.60008 [18] Nualart, D., The Malliavin calculus and related topics, (2006), Springer-Verlag Berlin · Zbl 1099.60003 [19] Nualart, D.; Răşcanu, A., Differential equations driven by fractional Brownian motion, Collect. Math., 53, 55-81, (2002) · Zbl 1018.60057 [20] Nualart, D.; Saussereau, B., Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stochastic Process. Appl., 119, 391-409, (2009) · Zbl 1169.60013 [21] Otto, F.; Villani, C., Generalization of an inequality by talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173, 361-400, (2000) · Zbl 0985.58019 [22] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives, theory and applications, (1993), Gordon and Breach Science Publishers Yvendon · Zbl 0818.26003 [23] Saussereau, B., Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion, Bernoulli, 18, 1-23, (2012) · Zbl 1242.60056 [24] Villani, C., (Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, (2003), American Mathematical Society Providence, RI) · Zbl 1106.90001 [25] Wang, F. Y., Logarithmic Sobolev inequalities: conditions and counterexamples, J. Oper. Theory, 46, 183-197, (2001) · Zbl 0993.58019 [26] Wang, F. Y., Log-Sobolev inequalities: different roles of ric and Hess, Ann. Probab., 37, 1587-1604, (2009) · Zbl 1187.60061 [27] Zähle, M., Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields, 111, 333-374, (1998) · Zbl 0918.60037 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.