This will turn into a standard equation of the type dv/dt=kv+ noise after a change of variable. For some general methods for solving SDEs, I hope the following link will be of much help -
http://math.berkeley.edu/~evans/SDE.course.pdf
Have you come across Korner's ' A Companion to analysis'? I find it a good second course, even after having studied Rudin's & Apostol's books.
Separate textbooks on complex analysis & on differential geometry would help , as these have much special attributes. An overview of books...
It won't be simply the sum of the two. It will consist of alternate pieces of the graphs of the two. We must consider the time intervals in which x>0 & x<0 separately.
Have you tried 'Differential Equations' by Simmons? It's a good book & contains many problems in physics.
For a course in PDEs, you must be familiar with calculus of several variables & a little functional analysis.
The curvature of potential energy is definitely related to distribution function, but the dependence is weird.I can't give any physical meaning to curvature which might suit the dependence.
I can't think of a definition of the lie derivative with respect to a covector off my head.
However, we may talk about the lie derivative with respect to a totally contravariant tensor.We could define it as the tensor product of component-wise lie derivatives. Such a quantity could be another...