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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", Theory of Probability and its applications. 1959, Vol IV, n o 3, 288-299.

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2 Answers 2

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

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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

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