Randomness in a Nonlinear Model of Sulphation Phenomena

The problem of the degradation of sandstones, limestones, and marble stones with different porosity used as building materials for thousands of years is a very important issue that has been observed in the last century. The first cause is due to the atmospheric pollutants, in particular the reaction of sulphur dioxide with calcareous surfaces which forms gypsum and black crusts. Recently, some mathematical models have been used to study the evolution of degradation phenomena, either pure statistical or deterministic partial differential equations (PDE) models. Here we present a first attempt of introduction of randomness in the modelling starting from a deterministic PDE model, existing in literature. Randomness is introduced via stochastic dynamical boundary conditions. We motivate our choice via an analysis of the sulphur dioxide time series in the area of Milano, Italy, through a filtration procedure for the identification of the deterministic and stochastic components of the process. We discuss the possible choices of the dynamical boundary conditions and the consequences for the solution to the PDE model. In particular, we take a mean reverting process with bounded noise as dynamical boundary condition. Then, we perform a comparison study of a system of PDE describing the evolution of the sulphur dioxide and the calcite both with deterministic and stochastic boundary condition, via numerical experiments.