What are the applications of stochastic differential equations to number theory?

Question

This semester i'm taking a course about stochastic differential equations. This made me wonder what applications does this topic have to areas like number theory and algebraic geometry, specially arithmetic geometry. Unfortunately i wasn't able to find anything online, all i found was about applications to usual differential equations to number theory in this question.

Answer 1

First, let me mention a nice established connection between probability and zeta function

• random matrices and the Riemann zeta function
• "Notes on L-functions and Random Matrix Theory

it says the the eigenvalues for certain random matrices behave like the zeroes of zeta function along the imaginary line.

So then one can further study this problem using Dyson-Brownian motion (which is an evolution on the space of matrices) eg. as done here

"Relaxation to equilibrium and the spacing distribution of zeros of the Riemann ζ function".

The DBM is described by an SDE for its eigenvalues

$$\lambda_i=d B_i+\sum_{1 \leq j \leq n: j \neq i} \frac{d t}{\lambda_i-\lambda_j}$$

where $$B_1, ..., B_n$$ are different and independent Wiener process|Wiener processes. Start with a Hermitian matrix with eigenvalues $\lambda_1(0), \lambda_2(0), ..., \lambda_n(0)$, then let it perform Brownian motion in the space of Hermitian matrices. Its eigenvalues constitute a Dyson Brownian motion.