Geodesic
Solution theory of fractional SDEs in complete subcritical regimes
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e12

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We consider stochastic differential equations (SDEs) driven by a fractional Brownian motion with a drift coefficient that is allowed to be arbitrarily close to criticality in a scaling sense. We develop a comprehensive solution theory that includes strong existence, path-by-path uniqueness, existence of a solution flow of diffeomorphisms, Malliavin differentiability and $\rho $-irregularity. As a consequence, we can also treat McKean-Vlasov, transport and continuity equations. 
Galeati, Lucio; Gerencsér, Máté. Solution theory of fractional SDEs in complete subcritical regimes. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e12. doi: 10.1017/fms.2024.136
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[1] Ambrosio, L., ‘Transport equation and Cauchy problem for non-smooth vector fields’, in Calculus of Variations and Nonlinear Partial Differential Equations (Springer, 2008), 1–41. Google Scholar | DOI

[2] Anzeletti, L., Richard, A. and Tanré, E., ‘Regularisation by fractional noise for one-dimensional differential equations with distributional drift’, Electron. J. Probab. 28 (2023), Paper No. 135, 49. Google Scholar | DOI

[3] Athreya, S., Butkovsky, O., Lê, K. and Mytnik, L., ‘Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation’, Comm. Pure Appl. Math. 77(5) (2024), 2708–2777. Google Scholar | DOI

[4] Ayache, A., Shieh, N.-R., and Xiao, Y., ‘Multiparameter multifractional Brownian motion: local nondeterminism and joint continuity of the local times’, Ann. Inst. Henri Poincaré Probab. Stat. 47(4) (2011), 1029–1054. Google Scholar | DOI

[5] Bahouri, H., Chemin, J.-Y. and Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations vol. 343 (Springer, Heidelberg, 2011). Google Scholar | DOI

[6] Bass, R. F. and Chens, Z.-Q., ‘Stochastic differential equations for Dirichlet processes’, Probab. Theory Related Fields 121(3) (2001), 422–446. Google Scholar | DOI

[7] Bechtold, F. and Hofmanová, M., ‘Weak solutions for singular multiplicative SDEs via regularization by noise’, Stochastic Process. Appl. 157 (2023), 413–435. Google Scholar | DOI

[8] Beck, L., Flandoli, F., Gubinelli, M. and Maurelli, M., ‘Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness’, Electron. J. Probab. 24 (2019), 1–72. Google Scholar | DOI

[9] Benassi, A., Jaffard, S. and Roux, D., ‘Elliptic Gaussian random processes’, Rev. Mat. Iberoamericana 13(1) (1997), 19–90. Google Scholar | DOI

[10] Brué, E., Colombo, M. and De Lellis, C., ‘Positive solutions of transport equations and classical nonuniqueness of characteristic curves’, Arch. Ration. Mech. Anal. 240(2) (2021), 1055–1090. Google Scholar | DOI

[11] Brué, E. and Nguyen, Q.-H., ‘Sharp regularity estimates for solutions of the continuity equation drifted by Sobolev vector fields’, Anal. PDE 14(8) (2021), 2539–2559. Google Scholar | DOI

[12] Butkovsky, O., Dareiotis, K. and Gerencsér, M., ‘Approximation of SDEs: a stochastic sewing approach’, Probab. Theory Related Fields (2021). Google Scholar | DOI

[13] Butkovsky, O., Dareiotis, K. and Gerencsér, M., ‘Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise’, to appear in Ann. Inst. H. Poincaré Probab. Statist., 2022, . Google Scholar | arXiv

[14] Butkovsky, O. and Gallay, S., ‘Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime’, Preprint, 2023, . Google Scholar | arXiv

[15] Butkovsky, O., Lê, K. and Matsuda, T, ‘Regularization of differential equations with integrable drifts by fractional noise’, in preparation, 2024. Google Scholar

[16] Butkovsky, O., Lê, K. and Mytnik, L., ‘Stochastic equations with singular drift driven by fractional Brownian motion’, Preprint, 2023, . Google Scholar | arXiv

[17] Butkovsky, O. and Mytnik, L., ‘Weak uniqueness for singular stochastic equations’, Preprint, 2024, . Google Scholar | arXiv

[18] Cannizzaro, G. and Chouk, K., ‘Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential’, Ann. Probab. 46(3) (2018). Google Scholar | DOI

[19] Cass, T., Friz, P. and Victoir, N., ‘Non-degeneracy of Wiener functionals arising from rough differential equations’, Trans. Amer. Math. Soc. 361(6) (2009), 3359–3371. Google Scholar | DOI

[20] Catellier, R. and Gubinelli, M., ‘Averaging along irregular curves and regularisation of ODEs’, Stochastic Process. Appl. 126(8) (2016), 2323–2366. Google Scholar | DOI

[21] Catellier, R., ‘Rough linear transport equation with an irregular drift’, Stoch. Partial Differ. Equ. Anal. Comput. 4(3) (2016), 477–534. Google Scholar

[22] Chaudru De Raynal, P.-É.. ‘Weak regularization by stochastic drift: Result and counter example’, Discrete Contin. Dyn. Syst. 38(3) (2018), 1269–1291. Google Scholar | DOI

[23] Chaudru De Raynal, P.-É, Honoré, I. and Menozzi, S., ‘Strong regularization by Brownian noise propagating through a weak Hörmander structure’, Probab. Theory Related Fields 184(1–2) (2022), 1–83. Google Scholar | DOI

[24] Chaudru De Raynal, P.-É, Honoré, I. and Menozzi, S., ‘Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result’, Trans. Amer. Math. Soc. 375(01) (2021), 1–45. Google Scholar | DOI

[25] Chaudru De Raynal, P-É, Menozzi, S. and Priola, E., ‘Schauder estimates for drifted fractional operators in the supercritical case’, J. Funct. Anal. 278(8) (2020), 108425. Google Scholar | DOI

[26] Chen, X. and Li, X.-M., ‘Strong completeness for a class of stochastic differential equations with irregular coefficients’, Electron. J. Probab. 19 (2014). Google Scholar | DOI

[27] Cheridito, P., ‘Mixed fractional Brownian motion’, Bernoulli 7 (2001), 913–934. Google Scholar | DOI

[28] Chouk, K. and Gubinelli, M., ‘Nonlinear PDEs with modulated dispersion I: Nonlinear Schrödinger equations’, Comm. Partial Differential Equations 40(11) (2015), 2047–2081. Google Scholar | DOI

[29] Chouk, K. and Gess, B., ‘Path-by-path regularization by noise for scalar conservation laws’, J. Funct. Anal. 277(5) (2019), 1469–1498. Google Scholar | DOI

[30] Chouk, K., Gubinelli, M., Li, G., Li, J. and Oh, T., ‘Nonlinear PDEs with modulated dispersion II: Korteweg–de Vries equation’, Preprint, 2024, (v2). Google Scholar | arXiv

[31] Dareiotis, K. and Gerencsér, M., ‘Path-by-path regularisation through multiplicative noise in rough, Young, and ordinary differential equations’, Ann. Probab. 52(5) (2024), 1864–1902. Google Scholar | DOI

[32] Davie, A. M., ‘Uniqueness of solutions of stochastic differential equations’, Int. Math. Res. Not. IMRN 24 (2007), Art. ID rnm124, 26. Google Scholar

[33] Debussche, A., Glatt-Holtz, N. and Temam, R., ‘Local martingale and pathwise solutions for an abstract fluids model’, Phys. D 240(14–15) (2011), 1123–1144. Google Scholar | DOI

[34] Decreusefond, L., ‘Stochastic integration with respect to Volterra processes’, Ann. Inst. H. Poincaré Probab. Statist. 41(2) (2005), 123–149. Google Scholar | DOI

[35] Delarue, F. and Diel, R., ‘Rough paths and 1d SDE with a time dependent distributional drift: application to polymers’, Probab. Theory Related Fields 165(1–2) (2015), 1–63. Google Scholar | DOI

[36] Diperna, R. J. and Lions, P.-L., ‘Ordinary differential equations, transport theory and Sobolev spaces’, Invent. Math. 98(3) (1989), 511–547. Google Scholar | DOI

[37] Fedrizzi, E. and Flandoli, F., ‘Hölder flow and differentiability for SDEs with nonregular drift’, Stoch. Anal. Appl. 31(4) (2013), 708–736. Google Scholar | DOI

[38] Fedrizzi, E. and Flandoli, F., ‘Noise prevents singularities in linear transport equations’, J. Funct. Anal. 264(6) (2013), 1329–1354. Google Scholar | DOI

[39] Flandoli, F., Random Perturbation of PDEs and Fluid Dnamic Models: École d’été de Pobabilités de Saint-Flour XL–2010 vol. 2015 (Springer Science & Business Media, 2011). Google Scholar | DOI

[40] Flandoli, F., Gubinelli, M. and Priola, E., ‘Well-posedness of the transport equation by stochastic perturbation’, Invent. Math. 180(1) (2010), 1–53. Google Scholar | DOI

[41] Flandoli, F., Issoglio, E. and Russo, F., ‘Multidimensional stochastic differential equations with distributional drift’, Trans. Amer. Math. Soc. 369(3) (2016), 1665–1688. Google Scholar | DOI

[42] Friz, P. and Victoir, N., ‘A variation embedding theorem and applications’, J. Funct. Anal. 239(2) (2006), 631–637. Google Scholar | DOI

[43] Friz, P. K. and Hairer, M., A Course on Rough Paths (Springer International Publishing, 2020). Google Scholar | DOI

[44] Friz, P. K., Hocquet, A. and Lê, K., ‘Rough stochastic differential equations’, Preprint, 2021, . Google Scholar | arXiv

[45] Friz, P. K. and Victoir, N.B., Multidimensional Stochastic Processes as Rough Paths (Cambridge University Press, 2009). Google Scholar

[46] Galeati, L., ‘Nonlinear Young differential equations: a review’, J. Dynam. Differential Equations 35(2) (2023), 985–1046. Google Scholar | DOI

[47] Galeati, L., ‘A note on weak existence for singular SDEs’, Stoch. Dyn. 24(3) (2024), Paper No. 2450025. Google Scholar | DOI

[48] Galeati, L. and Gubinelli, M., ‘Prevalence of -irregularity and related properties’, Ann. Inst. H. Poincaré Probab. Statist. 60(4) 2415–2467, November 2024. https://doi.org/10.1214/23-AIHP1399. Google Scholar | DOI

[49] Galeati, L. and Gubinelli, M., ‘Noiseless regularisation by noise’, Revista Mat. Iberoam. 38(2) (2021), 433–502. Google Scholar | DOI

[50] Galeati, L. and Gubinelli, M., ‘Mixing for generic rough shear flows’, SIAM J. Math. Anal. 55(6) (2023), 7240–7272. Google Scholar | DOI

[51] Galeati, L., Harang, F. A. and Mayorcas, A., ‘Distribution dependent SDEs driven by additive fractional Brownian motion’, Probab. Theory Related Fields 185(1–2) (2023), 251–309. Google Scholar | DOI

[52] Gatheral, J., Jaisson, T. and Rosenbaum, M., ‘Volatility is rough’, Quant. Finance 18(6) (2018), 933–949. Google Scholar | DOI

[53] Gerencsér, M., ‘Regularisation by regular noise’, Stoch. Partial Differ. Equ. Anal. Comput. 11(2) (2023), 714–729. Google Scholar

[54] Goudenège, L., Haress, E. M. and Richard, A., ‘Numerical approximation of SDEs with fractional noise and distributional drift’, Stochastic Process. Appl., (2023). https://doi.org/10.1016/j.spa.2024.104533. Google Scholar | DOI

[55] Gräfner, L. and Perkowski, N., ‘Weak well-posedness of energy solutions to singular SDEs with supercritical distributional drift’, Preprint, 2024, . Google Scholar | arXiv

[56] Han, Y., ‘Solving McKean–Vlasov SDEs via relative entropy’, to appear in Ann. Appl. Probab. (2022). Google Scholar | arXiv

[57] Hao, Z. and Zhang, X., ‘SDEs with supercritical distributional drifts’, Preprint, 2023, . Google Scholar | arXiv

[58] Harang, F. A. and Perkowski, N., ‘-regularization of ODEs perturbed by noise’, Stoch. Dyn. 21(08) (2021). Google Scholar | DOI

[59] Hurst, H. E., Black, R. P. and Simaika, Y. M., Long-Term Storage: An Experimental Study (Constable, 1965). Google Scholar

[60] Kinzebulatov, D. and Madou, K. R., ‘Strong solutions of SDEs with singular (form-bounded) drift via Roeckner-Zhao approach’, Preprint, 2023, . Google Scholar | arXiv | DOI

[61] Kinzebulatov, D. and Semënov, Y. A., ‘Sharp solvability for singular SDEs’, Electron. J. Probab. 28 (2023), Paper No. 69, 15. Google Scholar | DOI

[62] Kolmogorov, A. N, ‘Wienersche spiralen und einige andere interessante kurven in hilbertscen raum’, Acad. Sci. URSS (NS) 26 (1940), 115–118. Google Scholar

[63] Kremp, H. and Perkowski, N., ‘Rough weak solutions for singular Lévy SDEs’, Preprint, 2023, . Google Scholar | arXiv

[64] Krylov, N. V., ‘On time inhomogeneous stochastic Itô equations with drift in ’, Ukraïn. Mat. Zh. 72(9) (2020), 1232–1253. Google Scholar | DOI

[65] Krylov, N. V., ‘On strong solutions of Itô’s equations with and ’, Ann. Probab. 49(6) (2021), 3142–3167. Google Scholar | DOI

[66] Krylov, N. V., ‘On strong solutions of time inhomogeneous Itô’s equations with Morrey diffusion gradient and drift. A supercritical case, Preprint, 2022, . Google Scholar | arXiv

[67] Krylov, N. V., ‘On weak and strong solutions of time inhomogeneous Itô’s equations with VMO diffusion and Morrey drift’, Stochastic Process. Appl. 179 (2025), Paper No. 104505. Google Scholar | DOI

[68] Krylov, N. V. and Röckner, M., ‘Strong solutions of stochastic equations with singular time dependent drift’, Probab. Theory Related Fields 131(2) (2005), 154–196. Google Scholar | DOI

[69] Kunita, H., ‘Stochastic differential equations and stochastic flows of diffeomorphisms’, in Ecole d’été de probabilités de Saint-Flour XII-1982 (Springer, 1984), 143–303. Google Scholar

[70] Kusuoka, S., ‘On the regularity of solutions to SDEs’, in Asymptotic Problems in Probability Theory: Wiener Functionals and Symptotics (Longman Science & Technical, 1993), 90–103. Google Scholar

[71] Lê, K., ‘A stochastic sewing lemma and applications’, Electron. J. Probab. 25 (2020). Google Scholar | DOI

[72] Lê, K., ‘Stochastic sewing in Banach spaces’, Electron. J. Probab. 28 (2023), 1–22. Google Scholar | DOI

[73] Liu, C., Prömel, D. J. and Teichmann, J., ‘Characterization of nonlinear Besov spaces’, Trans. Amer. Math. Soc. 373(1) (2020), 529–550. Google Scholar | DOI

[74] Mandelbrot, B. B. and Van Ness, J. W., ‘Fractional Brownian motions, fractional noises and applications’, SIAM Review 10(4) (1968), 422–437. Google Scholar | DOI

[75] Marinucci, D. and Robinson, P. M.., ‘Mixed fractional Brownian motion’, J. Statist. Plann. Inference 80(1–2) (1999), 111–122. Google Scholar | DOI

[76] Menoukeu-Pamen, O., Meyer-Brandis, T., Nilssen, T., Proske, F. and Zhang, T., ‘A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s’, Math. Ann. 357(2) (2013), 761–799. Google Scholar | DOI

[77] Modena, S. and Székelyhidi, L., ‘Non-uniqueness for the transport equation with Sobolev vector fields’, Ann. PDE 4(2) (2018), 1–38. Google Scholar PubMed | DOI

[78] Mohammed, S.-E. A., Nilssen, T. K. and Proske, F. N., ‘Sobolev differentiable stochastic flows for SDEs with singular coefficients: applications to the transport equation’, Ann. Probab. 43(3) (2015), 1535–1576. Google Scholar | DOI

[79] Nilssen, T., ‘Rough linear PDEs with discontinuous coefficients – existence of solutions via regularization by fractional Brownian motion’, Electron. J. Probab. 25 (2020), 1–33. Google Scholar | DOI

[80] Nualart, D., The Malliavin Calculus and Related Topics (Probability and its Applications), second edn. (Springer-Verlag, Berlin, 2006). Google Scholar

[81] Nualart, D. and Ouknine, Y., ‘Regularization of differential equations by fractional noise’, Stochastic Process. Appl. 102(1) (2002), 103–116. Google Scholar | DOI

[82] Nualart, D. and Sönmez, E., ‘Regularization of differential equations by two fractional noises’, Stoch. Dyn. 22(6) (2022), Paper No. 2250029, 19. Google Scholar | DOI

[83] Panja, D., ‘Generalized langevin equation formulation for anomalous polymer dynamics’, J. Stat. Mech. Theory Exp. 2010(02) (2010), L02001. Google Scholar

[84] Pardoux, É., ‘Equations aux derivees partielles stochastiques non lineaires monotones etude de solutions fortes de type Itô’,1975. Google Scholar

[85] Peltier, R.-F. and Véhel, J. L., ‘Multifractional Brownian motion: definition and preliminary results’, PhD thesis, INRIA, 1995. Google Scholar

[86] Perrin, E., Harba, R., Berzin-Joseph, C., Iribarren, I. and Bonami, A., ‘th-order fractional Brownian motion and fractional Gaussian noises’, IEEE Trans. Signal Process. 49(5) (2001), 1049–1059. Google Scholar | DOI

[87] Picard, J., Representation Formulae for the Fractional Brownian Motion (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011), 3–70. Google Scholar

[88] Röckner, M. and Zhao, G., ‘SDEs with critical time dependent drifts: strong solutions’, Preprint, 2021, . Google Scholar | arXiv

[89] Romito, M. and Tolomeo, L., ‘Yet another notion of irregularity through small ball estimates’, Preprint, 2022, . Google Scholar | arXiv

[90] Russo, F. and Tudor, C. A., ‘On bifractional Brownian motion’, Stochastic Process. Appl. 116(5) (2006), 830–856. Google Scholar | DOI

[91] Shaposhnikov, A. V., ‘Some remarks on Davie’s uniqueness theorem’, Proc. Edinb. Math. Soc. 59(4) (2016), 1019–1035. Google Scholar | DOI

[92] Shaposhnikov, A. and Wresch, L., ‘Pathwise vs. path-by-path uniqueness’, Ann. Inst. Henri Poincaré Probab. Stat. 58(3) (2022), 1640–1649. Google Scholar | DOI

[93] Simon, J., ‘Sobolev, Besov and Nikolskii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval’, Ann. Mat. Pura Appl. (4) 157 (1990), 117–148. Google Scholar | DOI

[94] Szymanski, J. and Weiss, M., ‘Elucidating the origin of anomalous diffusion in crowded fluids’, Phys. Rev. Lett. 103(3) (2009), 038102. Google Scholar PubMed | DOI

[95] Tudor, C. A. and Xiao, Y., ‘Sample path properties of bifractional Brownian motion’, Bernoulli 13(4) (2007), 1023–1052. Google Scholar | DOI

[96] Tudor, C. A. and Xiao, Y., ‘Sample paths of the solution to the fractional-colored stochastic heat equation’, Stoch. Dyn. 17(1) (2017), 1750004, 20. Google Scholar | DOI

[97] Veretennikov, A. Y., ‘On strong solutions and explicit formulas for solutions of stochastic integral equations’, Sb. Math. 39 (1981), 387–403. Google Scholar | DOI

[98] Wei, J., Hu, J. and Yuan, C., ‘Stochastic equations with low regularity drifts’, Preprint, 2023, . Google Scholar | arXiv

[99] Xia, P., Xie, L., Zhang, X. and Zhao, G., ‘-theory of stochastic differential equations’, Stochastic Process. Appl. 130(8) (2020), 5188–5211. Google Scholar | DOI

[100] Yamada, T. and Watanabe, S., ‘On the uniqueness of solutions of stochastic differential equations’, Kyoto J. Math. 11(1) (1971). Google Scholar | DOI

[101] Zhang, X., ‘Stochastic flows of SDEs with irregular coefficients and stochastic transport equations’, Bull. Sci. Math. 134(4) (2010), 340–378. Google Scholar | DOI

[102] Zhang, X., ‘Stochastic differential equations with Sobolev diffusion and singular drift and applications’, Ann. Appl. Probab. 26(5) (2016), 2697–2732. Google Scholar | DOI

[103] Zhang, X. and Zhao, G., ‘Stochastic Lagrangian path for Leray’s solutions of D Navier–Stokes equations’, Comm. Math. Phys. 381(2) (2021), 491–525. Google Scholar | DOI

[104] Zvonkin, A. K., ‘A transformation of the phase space of a diffusion process that removes the drift’, Math. Sb. 22(1) (1974), 129. Google Scholar | DOI

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