The following three items are from Chapters 14-16 of my book Measure Theory and Integration. They provide background for the study of random processes and diffusion.

1. Ergodic theory

2. Probability spaces and random variables

3. Wiener measure and Brownian motion

The next item is from Chapter 11 of my PDE text (vol.2).

4. Brownian motion and potential theory

Section 6 of this chapter discusses a Brownian motion approach to Newtonian capacity. Click here to see notes on a general class of capacities.

The next items discuss various topics regarding diffusion and random processes.

5. Levy processes

6. Remarks on fractional diffusion equations  Click here for figures for this paper.

7. Multidimensional random fields

8. Remarks on M. Pinsky’s derivation of the modulus of continuity for Brownian motion

9. Stochastic operators and an infinite dimensional version of the Perron-Frobenius theorem

Here is a set of notes on the Central Limit Theorem.

10. Varieties of Central Limit Theorems

The following monograph gathers together material in items 5–10 above, and adds further material on stochastic integrals.

11. Topics in Probability and Random Processes.