I will give some suggestions and you tell me what you think about them. Then i can suggest others.
First of all, some revision:
Topology------ Introduction to metric and topological spaces by Sutherland. Very good book with an excellent motivation for compactness (with an example that appears in analysis)
Real Analysis and Advanced Calculus---David Brannan A first curse in real Analysis The book is very easy tpo read and guide you in the proofs. You learn proofs reading good ones and trying to do some of them by your own. If that book is very easy or when you finish with it, then you can try Bartle and Sherbert Introduction to Real Analysis.
Multivariate Calculus, Advanced Calculus and Analysis all in one---- Mathematical Analysis 1 and 2 by Zorich. I haven't used this book but i have been told that it is very good. And it deals with all the material.
Linear Algebra ---- Algebra Done right Axler.
Then you can jump to PDES or Dynamical Systems.
I'm doing research in the theory of stochastic processes, and in my answer I'll try to give you some idea what people are currently interested in. Note that these are not necessarily the "most imporant" or "most interesting" topics.
In my university I am doing courses like Brownian motion, Empirical Processes, Martingale theory, etc. However I don't find any article on these. Are these too outdated?
Yes and no. There are still many open questions on these topics, but it feels to me that todays probabilits are not too interested in most of them. Partly, I think, it's like with toys: After playing with a toy for a certain time, it's getting boring and you are happy enough to get a new one.
One of these (rather) new toys is fractional Brownian motion which is a generalization of Brownian motion. More generally, this leads to fractional fields and Lévy fields; see for instance
Serge Cohen, Jacques Istas: Fractional Fields and Applications. Springer
for an introduction. Fractional processes are growing more important in applications because they allow to model short- and long range dependence (which is obviously not true for Brownian motion because it has independent increments).
A not-so-new toy are Lévy processes, i.e. stochastic processes with independent and stationary increments (Brownian motion is a very particular case of a Lévy process). There is a lot of interest in heat kernel estimates for the transition densities of Lévy processes. Moreover, there any many open questions on the existence of solutions to Lévy-driven stochastic differential equations, i.e.
$$dX_t = b(X_{t-}) \, dt + \sigma(X_{t-}) \, dL_t \tag{1}$$
where $(L_t)_{t \geq 0}$ is a Lévy process. If $b$ and $\sigma$ are Lipschitz continuous, then everything is nice, but for irregular coefficients $b$ and $\sigma$ (e.g. Hölder continuous) the existence of solutions is not well understood. There is a lot of research going on in this direction. Related topics are backward SDEs (BSDEs), SDEs with delay and stochastic regularization.
During my PhD I studied so-called Feller processes. Roughly speaking they behave locally like a Lévy process, but the Lévy triplet depends on the current position of the process; for instance a solution to the SDE $(1)$ is a Feller process (at least if $b$ and $\sigma$ are bounded). A nice introduction is the first part of the book
Davar Khoshnevisan, René Schilling: From Lévy-Type Processes to Parabolic SDEs. Birkhäuser
and (but this one is for more advanced readers)
Björn Böttcher, René Schilling, Jian Wang: Lévy Matters III. Springer.
Let me list some more active research topics:
- stochastic partial differential equations (very popular and active)
- rough path theory
- stochastic processes on sublinear expectation spaces (e.g. G-Brownian motion)
- large deviations (see e.g. Dembo & Zeitouni for an introduction)
- Markov processes (heat kernel estimates, ergodic theory, functional inequalities (see e.g. Bakry & Gentil), selfsimilar Markov processes, ...)
Best Answer
It's hard to make a remote diagnosis. The mathematical theory behind partial differential equations is a lot of abstract functional analysis, which might be tough for you as a physicist. For example, I suppose you'll deal with compact operators to understand Sturm-Liouville problems. However, as PDEs occur a lot in physics, solving exercises with Fourier analysis/Green's functions might be something you're already a bit acquainted with.
Probability Theory is sometimes approached from a very abstract setting as well, in particular there are deep connections to Measure Theory, which is something that's probably hard to learn on its own. Furthermore, abstract concepts like that of a martingale might be difficult to grasp on your own.
It could go both ways here: If your course on PDEs is somewhat more applied / focused on computations than on theory, it might be easier for you to learn it on its own. If the course on Probability Theory doesn't rely heavily on Measure Theory, then I'd say it might be manageable for you too.
It all depends on your previous knowledge: If you're at ease with functional analysis, I'd take the probability course, if you're at ease with measures, I'd take the PDE course. In doubt, I'd recommend you take the PDE course, because the theory behind existence/uniqueness proofs is astoundingly abstract and highly nontrivial.