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July  2022, 27(7): 3707-3724. doi: 10.3934/dcdsb.2021203

## Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven stochastic differential equations with smooth drift coefficient functions with at most polynomially growing derivatives

 1 Seminar for Applied Mathematics, Department of Mathematics, ETH Zurich, Zurich, Switzerland 2 Applied Mathematics: Institute for Analysis and Numerics, Faculty of Mathematics and Computer Science, University of Münster, Münster, Germany 3 School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen, China 4 Mathematical Institute, Faculty of Mathematics and Natural Sciences, University of Düsseldorf, Düsseldorf, Germany 5 Faculty of Computer Science and Mathematics, University of Passau, Passau, Germany

* Corresponding author

Received  December 2020 Revised  June 2021 Published  July 2022 Early access  August 2021

Fund Project: This work has been funded by DFG grant EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure

In the recent article [A. Jentzen, B. Kuckuck, T. Müller-Gronbach, and L. Yaroslavtseva, J. Math. Anal. Appl. 502, 2 (2021)] it has been proved that the solutions to every additive noise driven stochastic differential equation (SDE) which has a drift coefficient function with at most polynomially growing first order partial derivatives and which admits a Lyapunov-type condition (ensuring the existence of a unique solution to the SDE) depend in the strong sense in a logarithmically Hölder continuous way on their initial values. One might then wonder whether this result can be sharpened and whether in fact, SDEs from this class necessarily have solutions which depend in the strong sense locally Lipschitz continuously on their initial value. The key contribution of this article is to establish that this is not the case. More precisely, we supply a family of examples of additive noise driven SDEs, which have smooth drift coefficient functions with at most polynomially growing derivatives and whose solutions do not depend in the strong sense on their initial value in a locally Lipschitz continuous, nor even in a locally Hölder continuous way.

Citation: Arnulf Jentzen, Benno Kuckuck, Thomas Müller-Gronbach, Larisa Yaroslavtseva. Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven stochastic differential equations with smooth drift coefficient functions with at most polynomially growing derivatives. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3707-3724. doi: 10.3934/dcdsb.2021203
##### References:
 [1] X. Chen and X.-M. Li, Strong completeness for a class of stochastic differential equations with irregular coefficients, Electron. J. Probab., 19 (2014), no. 91, 34 pp. doi: 10.1214/EJP.v19-3293. [2] S. Cox, M. Hutzenthaler and A. Jentzen, Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations, arXiv: 1309.5595, 84 pages, to appear in Mem. Amer. Math. Soc.. [3] S. Fang, P. Imkeller and T. Zhang, Global flows for stochastic differential equations without global Lipschitz conditions, Ann. Probab., 35 (2007), 180-205.  doi: 10.1214/009117906000000412. [4] M. Hairer, M. Hutzenthaler and A. Jentzen, Loss of regularity for Kolmogorov equations, Ann. Probab., 43 (2015), 468-527.  doi: 10.1214/13-AOP838. [5] M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing., Ann. of Math. (2), 164 (2006), 993-1032.  doi: 10.4007/annals.2006.164.993. [6] A. Hudde, M. Hutzenthaler, A. Jentzen and S. Mazzonetto, On the Itô-Alekseev-Gröbner formula for stochastic differential equations, arXiv: 1812.09857, 29 pages, to appear in Ann. Inst. H. Poincaré Probab. Stat.. [7] A. Hudde, M. Hutzenthaler and S. Mazzonetto, A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations, Ann. Inst. Henri Poincaré Probab. Stat., 57 (2021), 603-626.  doi: 10.1214/20-aihp1064. [8] M. Hutzenthaler and A. Jentzen, On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients, Ann. Probab., 48 (2020), 53-93.  doi: 10.1214/19-AOP1345. [9] M. Hutzenthaler, A. Jentzen, F. Lindner and P. Pužnik, Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations, arXiv: 1911.01870, 60 pages. [10] A. Jentzen, B. Kuckuck, T. Müller-Gronbach and L. Yaroslavtseva, On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values, J. Math. Anal. Appl., 502 (2021), 125240, 23 pp. doi: 10.1016/j.jmaa.2021.125240. [11] A. Jentzen, T. Müller-Gronbach and L. Yaroslavtseva, On stochastic differential equations with arbitrary slow convergence rates for strong approximation, Commun. Math. Sci., 14 (2016), 1477-1500.  doi: 10.4310/CMS.2016.v14.n6.a1. [12] N. V. Krylov, On Kolmogorov's equations for finite-dimensional diffusions, in Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998), vol. 1715 of Lecture Notes in Math., Springer, Berlin, (1999), 1–63. doi: 10.1007/BFb0092417. [13] X.-M. Li, Strong p-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds, Probab. Theory Related Fields, 100 (1994), 485-511.  doi: 10.1007/BF01268991. [14] X.-M. Li and M. Scheutzow, Lack of strong completeness for stochastic flows, Ann. Probab., 39 (2011), 1407-1421.  doi: 10.1214/10-AOP585. [15] W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4. [16] M. Scheutzow and S. Schulze, Strong completeness and semi-flows for stochastic differential equations with monotone drift, J. Math. Anal. Appl., 446 (2017), 1555-1570.  doi: 10.1016/j.jmaa.2016.09.049. [17] X. Zhang, Stochastic flows and Bismut formulas for stochastic Hamiltonian systems, Stochastic Process. Appl., 120 (2010), 1929-1949.  doi: 10.1016/j.spa.2010.05.015.

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##### References:
 [1] X. Chen and X.-M. Li, Strong completeness for a class of stochastic differential equations with irregular coefficients, Electron. J. Probab., 19 (2014), no. 91, 34 pp. doi: 10.1214/EJP.v19-3293. [2] S. Cox, M. Hutzenthaler and A. Jentzen, Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations, arXiv: 1309.5595, 84 pages, to appear in Mem. Amer. Math. Soc.. [3] S. Fang, P. Imkeller and T. Zhang, Global flows for stochastic differential equations without global Lipschitz conditions, Ann. Probab., 35 (2007), 180-205.  doi: 10.1214/009117906000000412. [4] M. Hairer, M. Hutzenthaler and A. Jentzen, Loss of regularity for Kolmogorov equations, Ann. Probab., 43 (2015), 468-527.  doi: 10.1214/13-AOP838. [5] M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing., Ann. of Math. (2), 164 (2006), 993-1032.  doi: 10.4007/annals.2006.164.993. [6] A. Hudde, M. Hutzenthaler, A. Jentzen and S. Mazzonetto, On the Itô-Alekseev-Gröbner formula for stochastic differential equations, arXiv: 1812.09857, 29 pages, to appear in Ann. Inst. H. Poincaré Probab. Stat.. [7] A. Hudde, M. Hutzenthaler and S. Mazzonetto, A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations, Ann. Inst. Henri Poincaré Probab. Stat., 57 (2021), 603-626.  doi: 10.1214/20-aihp1064. [8] M. Hutzenthaler and A. Jentzen, On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients, Ann. Probab., 48 (2020), 53-93.  doi: 10.1214/19-AOP1345. [9] M. Hutzenthaler, A. Jentzen, F. Lindner and P. Pužnik, Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations, arXiv: 1911.01870, 60 pages. [10] A. Jentzen, B. Kuckuck, T. Müller-Gronbach and L. Yaroslavtseva, On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values, J. Math. Anal. Appl., 502 (2021), 125240, 23 pp. doi: 10.1016/j.jmaa.2021.125240. [11] A. Jentzen, T. Müller-Gronbach and L. Yaroslavtseva, On stochastic differential equations with arbitrary slow convergence rates for strong approximation, Commun. Math. Sci., 14 (2016), 1477-1500.  doi: 10.4310/CMS.2016.v14.n6.a1. [12] N. V. Krylov, On Kolmogorov's equations for finite-dimensional diffusions, in Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998), vol. 1715 of Lecture Notes in Math., Springer, Berlin, (1999), 1–63. doi: 10.1007/BFb0092417. [13] X.-M. Li, Strong p-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds, Probab. Theory Related Fields, 100 (1994), 485-511.  doi: 10.1007/BF01268991. [14] X.-M. Li and M. Scheutzow, Lack of strong completeness for stochastic flows, Ann. Probab., 39 (2011), 1407-1421.  doi: 10.1214/10-AOP585. [15] W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4. [16] M. Scheutzow and S. Schulze, Strong completeness and semi-flows for stochastic differential equations with monotone drift, J. Math. Anal. Appl., 446 (2017), 1555-1570.  doi: 10.1016/j.jmaa.2016.09.049. [17] X. Zhang, Stochastic flows and Bismut formulas for stochastic Hamiltonian systems, Stochastic Process. Appl., 120 (2010), 1929-1949.  doi: 10.1016/j.spa.2010.05.015.
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