Abstract
In this note, we shall obtain an explicit formula on the probability of the level line of the Gaussian Free Field (GFF) with mixed boundary condition terminating at the free boundary, which generalizes the results of GFF with Dirichlet boundary condition. Our proof of the main result relies on both the martingale methods, and finding out the explicit form of the solution of the partial differential equation derived from the Ito’s calculus with extra efforts.
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