Regularisation by regular noise

Springer Science and Business Media LLC - Tập 11 - Trang 714-729 - 2022
Máté Gerencsér1
1TU Wien, Vienna, Austria

Tóm tắt

We show that perturbing ill-posed differential equations with (potentially very) smooth random processes can restore well-posedness—even if the perturbation is (potentially much) more regular than the drift component of the solution. The noise considered is of fractional Brownian type, and the familiar regularity condition $$\alpha >1-1/(2H)$$ is recovered for all non-integer $$H>1$$ .

Tài liệu tham khảo

Butkovsky, O., Dareiotis, K., Gerencsér, M.: Approximation of SDEs: a stochastic sewing approach. Probab. Theory Relat. Fields 181(4), 975–1034 (2021). https://doi.org/10.1007/s00440-021-01080-2

Bellingeri, C., Friz, P.K., Gerencsér, M.: Singular paths spaces and applications. Stoch. Anal. Appl. (2021). https://doi.org/10.1080/07362994.2021.1988641

Cannizzaro, G., Chouk, K.: Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential. Ann. Probab. 46(3), 1710–1763 (2018). https://doi.org/10.1214/17-aop1213

Catellier, R., Gubinelli, M.: Averaging along irregular curves and regularisation of ODEs. Stoch. Process. Appl. 126(8), 2323–2366 (2016). https://doi.org/10.1016/j.spa.2016.02.002

Chaudru de Raynal, P.-E.: Strong existence and uniqueness for degenerate SDE with Hölder drift. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 53(1), 259–286 (2017). https://doi.org/10.1214/15-aihp716

Chaudru de Raynal, P.-E.: Weak regularization by stochastic drift: result and counter example. Discrete Contin. Dyn. Syst. A 38(3), 1269–1291 (2018). https://doi.org/10.3934/dcds.2018052

Chaudru de Raynal, P.-E., Honoré, I., Menozzi, S.A.: Strong regularization by Brownian noise propagating through a weak Hörmander structure. arXiv e-prints arXiv:1810.12225

Chaudru de Raynal, P.-É., Menozzi, S.: Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result. Trans. Am. Math. Soc. 375(01), 1–45 (2021). https://doi.org/10.1090/tran/7947

Delarue, F., Diel, R.: Rough paths and 1d SDE with a time dependent distributional drift: application to polymers. Probab. Theory Relat. Fields 165(1–2), 1–63 (2015). https://doi.org/10.1007/s00440-015-0626-8

Friz, P.K., Hairer, M.: A Course on Rough Paths. Springer (2020). https://doi.org/10.1007/978-3-030-41556-3

Kremp, H., Perkowski, N.: Multidimensional SDE with distributional drift and Lévy noise. arXiv e-prints (2020). arXiv:2008.05222

Lê, K.: A stochastic sewing lemma and applications. Electron. J. Probab. (2020). https://doi.org/10.1214/20-EJP442

Mandelbrot, B.B., van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968)

Nualart, D., Ouknine, Y.: Regularization of differential equations by fractional noise. Stoch. Process. Appl. 102(1), 103–116 (2002). https://doi.org/10.1016/S0304-4149(02)00155-2

Veretennikov, A.J.: Strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb. (N.S.) 153(3), 434–452 (1980)

Zvonkin, A.: A transformation of the phase space of a diffusion process that removes the drift. Math. USSR-Sbornik 22(1), 129–149 (1974). https://doi.org/10.1070/sm1974v022n01abeh001689

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Springer Science and Business Media LLC

Tập 11

714-729

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