A note on stochastic elliptic models

There exist two types of commonly studied stochastic elliptic models in literature: (I) −∇ · (a(x, ω)∇u(x, ω)) = f(x) and (II) -∇·(a(x,ω)♢∇u(x,ω))=f(x), where ω indicates randomness, ♢ the Wick product, and a(x, ω) is a positive random process. Model (I) is widely used in engineering and physical applications while model (II) is usually studied from the mathematical point of view. The difference between the above two stochastic elliptic models has not been fully clarified. In this work, we discuss the difference between models (I) and (II) when a(x, ω) is a log-normal random process. We show that the difference between models (I) and (II) is mainly characterized by a scaling factor, which is an exponential function of the degree of perturbation of a(x, ω). We then construct a new stochastic elliptic model (III): -∇·((a-1)♢(-1)♢∇u(x,ω))=f(x), which has the same scaling factor as model (I). After removing the divergence from the scaling factor, models (I) and (III) can be highly comparable for many cases. We demonstrate this by a numerical study for a one-dimensional problem.