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In the recent article [A. Jentzen, B. Kuckuck, T. M\"uller-Gronbach, and L.Yaroslavtseva, arXiv:1904.05963 (2019)] it has been proved that the solutionsto every additive noise driven stochastic differential equation (SDE) which hasa drift coefficient function with at most polynomially growing first orderpartial derivatives and which admits a Lyapunov-type condition (ensuring thethe existence of a unique solution to the SDE) depend in a logarithmicallyH\"older continuous way on their initial values. One might then wonder whetherthis result can be sharpened and whether in fact, SDEs from this classnecessarily have solutions which depend locally Lipschitz continuously on theirinitial value. The key contribution of this article is to establish that thisis not the case. More precisely, we supply a family of examples of additivenoise driven SDEs which have smooth drift coefficient functions with at mostpolynomially growing derivatives whose solutions do not depend on their initialvalue in a locally Lipschitz continuous, nor even in a locally H\"oldercontinuous way.

discrete and continuous dynamical systems. series b

Counterexamples to local Lipschitz and local H\"older continuity with  respect to the initial values for additive noise driven SDEs with smooth  drift coefficient functions with at most polynomially growing derivatives