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 -
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...
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.
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...