In this talk, we will first briefly mention the recent research on CLT results of random field solutions to stochastic heat equations and stochastic wave equations

with various Gaussian noises. Then, we will talk about the spatial ergodicity (first order result) and CLT (second order fluctuation)

for a stochastic linear wave equation driven by Lévy noise, which is based on a joint work with R. Balan (Ottawa)

[arXiv:2302.14178]. Finally, we will talk about the associated almost sure central limit theorem, based on a joint work with R. Balan (Ottawa) and P. Xia (Auburn).

Given a (finitely supported, symmetric) random walk X_n on a countable groups G, one may consider several quantities associated to the random walk, such as the expected distance E|X_n|, its Shannon entropy H(X_n), its return probabilities P(X_n=id) and more.  For example, on polynomial growth groups (e.g. Z^d) it is well known the walk is always diffusive, has logarithmic entropy and polynomial decaying return probabilities, while on the free group the speed and entropy of a random walk increase linearly with n and the return probabilities decay exponentially.

A natural problem is to understand what are the possible range of behaviours of such quantities for random walks on groups and how they relate to other group properties. 

In this lecture I will survey some old and new results on the subject, including some more recent works on speed and entropy of random walks on intermediate growth groups and on possible behaviours of the Law of Iterated Logarithm.

The talk is based on joint works with B. Virag, G. Blachar, C. Saroussi and T. Zheng,  though we will also touch upon works by A. Erschler, and by J. Brieussel and T. Zheng.

All notions will be defined in the talk, and in particular no knowledge of groups is required for the talk.