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

We consider the Stochastic Differential Equation $X_t = X_0 + \int_0^t b(s,X_s) ds + B_t$,in$$\mathbb{R}$^d$\mathrm{.}Wegiveanexampleofadrift$b$ such that there does not exist a weak solution, but there exists a solution for almost every realization of the Brownian motion $B$. 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 arXiv:2001.02869 to dimension $d=1$.