# American Institute of Mathematical Sciences

• Previous Article
Weak convergence of delay SDEs with applications to Carathéodory approximation
• DCDS-B Home
• This Issue
• Next Article
Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control
doi: 10.3934/dcdsb.2021203
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021203
##### References:

show all references

##### References:
 [1] Kai Liu. On regularity of stochastic convolutions of functional linear differential equations with memory. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1279-1298. doi: 10.3934/dcdsb.2019220 [2] Huaiyu Jian, Xiaolin Liu, Hongjie Ju. The regularity for a class of singular differential equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1307-1319. doi: 10.3934/cpaa.2013.12.1307 [3] Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115 [4] Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87 [5] J. Colliander, A. D. Ionescu, C. E. Kenig, Gigliola Staffilani. Weighted low-regularity solutions of the KP-I initial-value problem. Discrete & Continuous Dynamical Systems, 2008, 20 (2) : 219-258. doi: 10.3934/dcds.2008.20.219 [6] Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817 [7] Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021002 [8] Yan Yong, Weiyuan Zou. Macroscopic regularity for the relativistic Boltzmann equation with initial singularities. Kinetic & Related Models, 2019, 12 (5) : 945-967. doi: 10.3934/krm.2019036 [9] Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure & Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353 [10] Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563 [11] Guji Tian, Xu-Jia Wang. Partial regularity for elliptic equations. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 899-913. doi: 10.3934/dcds.2010.28.899 [12] Ben-Yu Guo, Zhong-Qing Wang. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1029-1054. doi: 10.3934/dcdsb.2010.14.1029 [13] Huy Tuan Nguyen, Huu Can Nguyen, Renhai Wang, Yong Zhou. Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021030 [14] Hermen Jan Hupkes, Emmanuelle Augeraud-Véron. Well-posedness of initial value problems for functional differential and algebraic equations of mixed type. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 737-765. doi: 10.3934/dcds.2011.30.737 [15] Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 6133-6153. doi: 10.3934/dcds.2015.35.6133 [16] Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure & Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1 [17] Martí Prats. Beltrami equations in the plane and Sobolev regularity. Communications on Pure & Applied Analysis, 2018, 17 (2) : 319-332. doi: 10.3934/cpaa.2018018 [18] Tôn Việt Tạ. Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4507-4542. doi: 10.3934/dcds.2017193 [19] Angelo Favini, Rabah Labbas, Stéphane Maingot, Hiroki Tanabe, Atsushi Yagi. Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 973-987. doi: 10.3934/dcds.2008.22.973 [20] Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks & Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027

2020 Impact Factor: 1.327