# Weak solutions of linear parabolic PDEs and corresponding SDEs

It is well known that for an Stochastic differential equation (on the real line) of the form: $dX_t = \mu(X_t)dt + \sigma(X_t)dW$ where $W$ is the standard Wiener process, the transition probability densities of the process can be related to a PDE of the form $\frac{\partial u}{\partial t} = \frac{\partial^2}{\partial x^2}(a^2(x)u)-\frac{\partial}{\partial x}(v(x)u)$. Using standard PDE theory it can be proved that weak solutions of the PDE exist in suitably weighted Lesbesgue spaces for essentially bounded coefficients $a^2(x)$ and $v(x)$ with suitable growth conditions (and ellipticity). However almost all books/papers on SDEs I see that $\mu$ and $\sigma$ are taken to be at least continuous. Are there any counterexamples where solutions for PDEs exist but not for the corresponding SDE? Or even if solutions for the SDE exist is there a counterexample such that the sample paths are not continuous any more?

• The phrase "existence for the solution of the SDE" is a bit ambiguous. There are several notions of solution. Strong solutions (i.e. where the process X is a measurable function of W) may fail to exist when $\mu$ is discontinuous. See Tsirelson's example: tau.ac.il/~tsirel/Research/mydrift/main.html – ofer zeitouni Dec 4 '15 at 15:11
• Under boundedness of $\mu$ (and continuity of $\sigma$), weak solutions to the SDE exist by Girsanov's theorem (and time change). – ofer zeitouni Dec 4 '15 at 15:13