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

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 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 the existence of a unique solution to the SDE) depend in a logarithmically H\"older 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 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 whose solutions do not depend on their initial value in a locally Lipschitz continuous, nor even in a locally H\"older continuous way.