Proofs of main probability results from other fields

Question

Making connections between different areas is very exciting and probability has already made connections with other fields (BM used in proving complex analysis and PDE results).

To keep it short, I will greatly appreciate any references to multidisciplinary proofs for any of the following results:

1. Central limit theorem (simple version, Lindeberg's or Donsker's theorem)

2. Law of large numbers

3. Recurrence in $d=1,2$ and transience in $d>2$ of random walk.

4. Conformal and time change: if $f$ is conformal then $f(B_{t})\stackrel{d}{=}\widetilde{B}_{\int_{0}^{t}|f'(B_{s})|^{2}ds}$ where $B,\widetilde{B}$ are planar Brownian motions.

5. Feynman-Kac formula of Brownian motion

The more far-fetched the proofs the better. For example, I will be surprised (and glad) to see proofs using mainly topology, number theory or algebra.

For example CLT has a very interesting approach from information theory:

"An information-theoretic proof of the central limit theorem with the Lindeberg condition". Ju. V. Linnik (Transl. by R.A. Silverman). Theory of Probability & Its Applications. 1959. Vol IV, No 3. 288-299.

Answer 1

A well-known example is the heat-equation proof of the central limit theorem due to [Petrovsky and Kolmogorov].

Answer 2

As for (3) (recurrence in $d\leq 2$ and transience in $d\geq 3$ of simple random walk), there are "electric networks"-proofs of these facts. See the classical book of Doyle and Snell "Random walks and electric networks", http://arxiv.org/abs/math/0001057

Answer 3

Maybe this is a stretch, but laws of large numbers, and more precise concentration of measure bounds, can be proven by the "isoperimetric" approach which can be called geometric and was pioneered by Talagrand.

For example, consider $n$ independent fair coin flips, i.e. the set of binary sequences of length $n$; viewed as points in $\mathbb{R}^n$, we can see that a law of large numbers is roughly equivalent to the statement that almost all of these sequences fall in a ball of small radius as $n \to \infty$.

More generally, Talagrand states that in a product measure space, if $A$ has large measure, then the measure of points within distance $\epsilon$ of $A$ (i.e. $A$ plus its perimeter) is very large.

References: E.g. "Notes on Talagrand's Isoperimetric Inequality" by Nick Cook.