  Semi-linear elliptic Partial Differential Equations (PDEs) such as the
non-linear Poisson Boltzmann Equation (nPBE) is highly relevant for non-linear
electrostatics in computational biology and chemistry. It is of particular
importance for modeling potential fields from molecules in solvents or plasmas
with stochastic fluctuations. The extensive applications include ones in
condensed matter and solid state physics, chemical physics, electrochemistry,
biochemistry, thermodynamics, statistical mechanics, and materials science,
among others. In this paper we study the complex analytic properties of
semi-linear elliptic Partial Differential Equations with respect to random
fluctuations on the domain. We first prove the existence and uniqueness of the
nPBE on a bounded domain in $\mathbb{R}^3$. This proof relies on the
application of a contraction mapping reasoning, as the standard convex
optimization argument for the deterministic nPBE no longer applies. Using the
existence and uniqueness result we subsequently show that solution to the nPBE
admits an analytic extension onto a well defined region in the complex
hyperplane with respect to the number of stochastic variables. Due to the
analytic extension, stochastic collocation theory for sparse grids predict
algebraic to sub-exponential convergence rates with respect to the number of
knots. A series of numerical experiments with sparse grids is consistent with
this prediction and the analyticity result. Finally, this approach readily
extends to a wide class of semi-linear elliptic PDEs.
