There is a well known connection between parabolic and elliptic partial differential equations and Brownian motion. By now it very well explored formally (e.g. the probabilistic proof of Hörmander's theorem due to Malliavin) but it used to be the case that people would get there their intuition from Brownian motion and then prove a theorem by completely different means.

One example is the following quote from Nash's 1958 paper "Continuity of solutions of parabolic and elliptic equations":

The methods here were inspired by physical intuition, but the ritual of mathematical exposition tends to hide this natural basis. For parabolic equations, diffusion, Brownian movement, and flow of heat or electrical charge all provide helpful interpretations.

As a side note. One of the people who contributed the most to establishing the formal connection between Brownian motion and parabolic/elliptic equations was Joseph Doob. He had done his Phd. thesis on harmonic analysis, but couldn't find a job anywhere (this was a couple of years after the Great Depression) until he got offered a post at a probability department. He started working on formal (i.e. Kolmogorov) foundations of probability and ended up establishing the connection between harmonic functions and Martingales. He's one of my favorite mathematicians and I think his contributions are underrated.

There is a well known connection between parabolic and elliptic partial differential equations and Brownian motion. By now it very well explored formally (e.g. the probabilistic proof of Hörmander's theorem due to Malliavin) but it used to be the case that people would get there intuition from Brownian motion and then prove a theorem by completely different means.

One example is the following quote from Nash's 1958 paper "Continuity of solutions of parabolic and elliptic equations":

The methods here were inspired by physical intuition, but the ritual of mathematical exposition tends to hide this natural basis. For parabolic equations, diffusion, Brownian movement, and flow of heat or electrical charge all provide helpful interpretations.

As a side note. One of the people who contributed the most to establishing the formal connection between Brownian motion and parabolic/elliptic equations was Joseph Doob. He had done his Phd. thesis on harmonic analysis, but couldn't find a job anywhere (this was a couple of years after the Great Depression) until he got offered a post at a probability department. He started working on formal (i.e. Kolmogorov) foundations of probability and ended up establishing the connection between harmonic functions and Martingales. He's one of my favorite mathematicians and I think his contributions are underrated.