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The connection between Markov processes and Potential Theory is well known, as is conformal invariance of Brownian motion which allows probabilistic proofs of statements in Complex Analysis, like Picard's theorem. What are some results in these areas (Potential Theory, Complex Analysis, maybe Complex or Differential Geometry) that admit probabilistic proofs, and possibly for which no other proof is known? To what extent are probabilistic methods in these areas actually crucial in proving new theory and to what extent are they more of an interpretation of facts that are already known? It would be great if there were some survey about this.

Thanks in advance for your answers.

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  • $\begingroup$ Perhaps one place to look is the (relatively recent) development of the Schramm-Loewner equation in complex analysis: terrytao.wordpress.com/2018/05/29/…. Probably most of these results have proofs heavily relying on probability, and for which there are probably no other proofs known. $\endgroup$
    – D.R.
    Commented Jun 9, 2023 at 6:20

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