"the paths of the integrator are too rough." Does this mean that you view the integral as a path integral over the path of the stochastic process? I'm new to this, so please excuse the basic questions. And what does it mean for Ito integrals to be defined in $L^2$ rather than pathwise?

@leomonsaingeon looking these terms up, thank you for sharing! Is displacement convexity a concept specific to Wasserstein space? And does it mean "$F$ is displacement convex if it is convex along optimal transport plans"?

@Steve interpolation wrt $W_2$ is closer to what I was thinking of! I'm interested in convexity of a set in the ambient space $(P,W_2)$ which is why this definition makes more sense. So far, I'm not convinced that {product densities} is not convex wrt $W_2$...

@DieterKadelka I figured that the distance metric would affect the definition of the path between two points, but I guess that doesn't have to be the case. Thanks for pointing that out

Thanks! So if you don't mind me asking, what are some of those applications of Ricci flow to computer vision, etc? I'm now wondering why mean curvature flow wasn't used in those cases.