Stability analysis of probabilistic soil moisture dynamics

We study the dynamics of soil moisture in the presence of precipitation, evapotranspiration and leakage. This study utilizes a probabilistic framework, and considers the evolution of the probability density function of soil moisture. Our main objective is to assess when the stationary solution is a physically realizable state of the system. Mathematically, this is equivalent to analyzing when the stationary solutions are stable. We show the following main results: For a general rainfall and evapotranspiration model, in a multiple plant framework, the stationary solution is a stable attractor in L₁. However, the stationary solution is at best conditionally stable in L_∞. This has important implications in terms of (1) the validity of moment analysis of the transient problem, (2) the physical importance of transient solutions, and (3) the numerical solution of the stationary problem as a late time solution of the transient problem.