Comparison of classical and path-by-path solutions to SDEs

We consider the Stochastic Differential Equation [Formula: see text], in [Formula: see text]. We give an example of a drift [Formula: see text] such that there does not exist a weak solution, but there exists a solution for almost every realization of the Brownian motion [Formula: see text]. We also give an explicit example of a drift such that the SDE has a pathwise unique weak solution, but path-by-path uniqueness (i.e. uniqueness of solutions to the ODE for almost every realization of the Brownian motion) is lost. These counterexamples extend the results obtained in Shaposhnikov and Wresch [Pathwise versus path-by-path uniqueness, Ann. Inst. Henri Poincaré Probab. Stat. 58(3) (2022) 1640–1649] to dimension [Formula: see text].