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We prove and implement stochastic solution (or Feynman-Kac) formulas forboundary value problems involving the spectral fractional Laplacian withnonzero Dirichlet boundary condition. The main tools used in the proofs are theabstract Cauchy problem for Feller semigroups together with Balakrishnan'stheory of fractional powers. We show the the spectral fractional Laplacian withnonzero Dirichlet boundary conditions is the generator of an appropriate Fellersemigroup for subordinate stopped Brownian motion, and obtain a stochasticsolution formula for the fractional heat equation in a bounded domain. We thenobtain precise regularity and steady-state convergence properties of theparabolic problem using the eigenfunction expansion of the classical solution,which leads to estimates for the survival probability of subordinate stoppedBrownian motion. These results allow us to take the long-time limit in theparabolic formula to establish a stochastic solution formula for the Dirichletboundary value problem. These stochastic solution formulas for the spectralfractional Laplacian with nonzero boundary conditions are novel, and allow forefficient, embarrassingly parallel local solution of the above boundary valueproblems. We discuss the discretization of these formulas, and verify them indimensions two and three with benchmark examples. We study the effect of thenumber of path samples and the time step size for path discretization on theaccuracy of the solution.

Stochastic Solution of Elliptic and Parabolic Boundary Value Problems for the Spectral Fractional Laplacian