How should one prepare for graduate studies in probability?

Question

I have an undergraduate degree in Probability and Statistics and am contemplating joining a graduate program for PhD. I believe I am reasonably strong in probability theory and would like to do research in the area.

However, I am a bit confused about where the current research in probability is going. I went through some reputed journals in Probability like the Annals of Probability and Journal of Theoretical Probability. I was a bit disappointed to see that currently research in probability constitutes either combinatorics or differential equations. There are lots of models, primarily from Statistical Physics literature.

I like probability and mathematical statistics. I do not really want to be in combinatorics -- never really liked the field. But from what probabilists are currently interested in, it seems that finding expected cluster sizes in random graphs and doing triangulations and stuff are all that are happening.

Where did the research similar to that of Stein go? 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 Are these too outdated? Has the market been completely occupied by triangles and graphs? Is probability too immersed in "models"? One of my professors said, "Everything is a model!" Then where is the identity of a subject? Everybody who seems to do probability and is sufficiently famous seems to be doing these!

I think it is time to really understand where the research in probability is going at present. I have been long disappointed to see no big program in probability, like Perfectoid spaces or Langland's program in Number Theory. There is no big theory which people try to understand.

How does a prospective graduate student understand what is the aim of probability? What are the non-combinatorial areas of probability? How active are they? What are some resources or influential papers on these?

What should one learn in order to attack modern problems in probability? As I said, in other subjects there are things to learn. There are interconnections between Algebraic Geometry and Algebraic K Theory. I understand such theories are not there in probability since it is too young. Eve if they exist, are they really active areas?

Answer 1

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, ...)