I am reading Cedric Villani’s book “Optimal Transport: old and new” and I am stuck on one paragraph (see page 26/27 in this book). He speaks about random solutions to an ODE and I simply cannot figure out what he means with a random solution to an ODE defined by a vector field. I understand the part about weak solutions to ODEs (I am familiar with that) but I struggle to understand what a random solution to an ODE should be. Could you tell me how a random solution is defined and what he means with that?
Let me specify: let $\xi:\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}^d$ be a vector field. As we don't assume that $\xi$ is Lipschitz-continuous, the ODE $$ \frac{d}{dt}x(t)=\xi(x(t),t) $$ does not necessarily have a unique solution. Cellani nows talks about $T_t(x)$ as a random solution to the above ODE with starting condition $x$. Could somebody rigorously define to me what a random solution is? Does it mean that the solution satisfies the ODE in a weak sense? In particular, I am curious: How can one simulate a solution in this case?
