# Is it Possible to Construct all Proofs in Complex Analysis using Brownian Motion?

(First, I am very aware of the fact that Brownian motion is actually probably more difficult to understand than at least basic complex analysis, so the pedagogical merits of such an approach would be questionable for anyone besides a probabilist wanting to refresh or reshape their already existing complex analysis knowledge.)

During my stochastic processes lecture, my professor said something to the effect that:

"Every statement in complex analysis can be proven using Brownian motion, in particular using the fact that the image of a Brownian path under a conformal map is Brownian motion up to a time change."

To what extent is this true?

Even after he gave a proof of Liouville's Theorem using Brownian motion and promised to give a proof of the Riemann Mapping Theorem during the next lecture, I was still unconvinced.

However, now the more that I think about it, it becomes more plausible -- Brownian motion is very closely related to the theory of harmonic functions, and analytic functions are just 2-dimensional harmonic functions satisfying the Cauchy-Riemann equations (is this correct?).

Brownian motion has already been shown to have numerous applications to potential theory and PDEs (to the best of my knowledge), so is a similar formulation of complex analysis in terms of Brownian motion also theoretically possible? (even if not desirable?)

• If he really could have given a one-hour proof of RMT via Brownian motion, then the method must be very powerful indeed. – Lubin May 15 '16 at 0:46
• I think your professor was making an overstatement. Did he really give a proof of the RMT in the next lecture? – Moishe Kohan May 16 '16 at 14:23
• I haven't had the lecture yet so we'll see – Chill2Macht May 16 '16 at 14:24

See §10.6. Intermission: Harmonic Functions and Brownian Motion of Ulrich's Complex Made Simple.

The following links appear to be relevant -- also I need to ask my professor again where he found the proof of the Riemann mapping theorem using Brownian motion, because I don't remember the author's last name.

http://www.stat.berkeley.edu/~peres/bmbook.pdf

https://www.math.washington.edu/~morrow/336_13/papers/yuxuan.pdf

http://math.mit.edu/~sswatson/pdfs/partiiiessay.pdf

http://people.maths.ox.ac.uk/belyaev/papers/BM-CA09.pdf

I think the following proves the maximum principle using Brownian motion and the optional stopping theorem: https://www.scribd.com/doc/312821091/Rogers-Complex-Analysis-and-Brownian-Motion

EDIT: Here are the notes my professor was referring to:

http://www.math.uchicago.edu/~lawler/harmonic.pdf

The middle third of the proof allows one to use the optional stopping theorem in place of the maximum principle.

EDIT 2: Per the recommendation of @Ihf:

https://www.scribd.com/doc/312845466/Ullrich-Brownian-Motion