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1 [made Community Wiki]

Some other probability PDE techniques:

1) Percolation: The Aizenman Barsky proof of exponential decay in subcritical percolation hinged on establishing a number of differential inequalities.

2) Conformal Invariance and SLE: Many conformal invariance proofs reduce to showing that the discrete stochastic process in question satisfies a Riemann Hilbert boundary value problem along with defining a flow on the state space which is divergence and curl free. This makes it clear how Cardy's formula arises as the hypergeometric function which solves the appropriate differential equation.