The expected value of e^{-λτ}φ(B_τ) over Brownian exit times solves Δh = λh with boundary value φ on negatively curved manifolds.
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On Sullivan's construction of eigenfunctions via exit times of Brownian motion
The expected value of e^{-λτ}φ(B_τ) over Brownian exit times solves Δh = λh with boundary value φ on negatively curved manifolds.