We study Laplace equation and heat equation with the Gaussian white noise as the boundary/initial condition. Using the stochastic pre-orthogonal adaptive Fourier decomposition on the Bergman space, we give sparse expansions of the solutions of the boundary/initial value problems. The proposed methodology breaks away from the \({\mathcal {H}}\) - \(H_K\) structure. The key point to the solution is to use the Wiener isometry property of the Gaussian white noise integral and the corresponding Green’s function to represent the covariance of the solution. For the heat equation with the Gaussian white noise initial value, we define a class of parabolic Bergman space following E. Stein, prove existence of Bergman kernel, and obtain its expression, and further verify the boundary vanishing property needed for generating a sparse expansion. In order to make the heat flow satisfy a long-time decay property, we induce a subspace of the parabolic Bergman space, which corresponds to a class of homogeneous Sobolev spaces.
