There are many results on the existence and uniqueness of weak solutions to SDEs, even with discontinuous drift, and many examples of nonexistence or nonuniqueness of strong solutions. Are there examples of nonexistence, or nonuniqueness in probability law, of weak solutions to a one-dimensional SDE with a uniformly positive diffusion coefficient? I would like a reference to such an example. I checked the books Karatzas and Shreve 1988, Oksendal 2010 and did not find such examples. The reason to require a positive diffusion coefficient is to rule out ODEs as examples.
A strong solution is a weak solution, but the nonuniqueness of a strong solution does not imply nonuniqueness of a weak solution, because the uniqueness concept is different. Strong solutions may have the same probability law, but be different pathwise.
Intuitively, it seems that even with the drift and diffusion coefficients discontinuous, an SDE will still have a weak solution, because the solutions on intervals of continuity can be "patched together".
