We establish well-posedness results for systems of a finite number of stochastic particles driven by independent Brownian motions and subject to a strongly singular drift induced by a Lennard-Jones interaction. In addition to the pairwise force, the dynamics includes a nonlocal drift mediated by an environmental field, whose evolution is coupled to the particle configuration through a regularized empirical density. We then extend the analysis to a reaction model in which the switching (or killing) rate also depends on the field. An interlacing technique is considered for establishing the well-posedness of the full system. The model is motivated by the challenge to provide a stochastic microscopic description of the sulphation phenomenon in cultural heritage materials.

In this paper, we discuss and compare two probabilistic approaches for associating a stochastic differential equation with a McKean-type partial differential equation featuring a reaction term and path-dependent coefficients. The non-conservative nature of the macroscopic dynamics leads to two possible interpretations of the sub-probability measure and of the associated SDE equation at the microscale: on the one hand, as a measure-valued solution of a Feynman-Kac-type equation; on the other hand, as the sub-probability associated with an SDE defined up to a survival time with a reaction-dependent rate. These different interpretations give rise to two different microscopic stochastic models and therefore to two different techniques of probabilistic analysis. Finally, by considering the interacting particle systems associated with both models, we discuss how their empirical densities provide two different kernel estimators for the PDE solution.

We present a hybrid stochastic-continuum model to study the sulphation of calcium carbonate and the consequent formation of gypsum, a key phenomenon driving marble deterioration. While calcium carbonate and gypsum are continuous random fields evolving according to random ordinary differential equations, the dynamics of sulfuric acid particles follow Itô-type stochastic differential equations. The particle evolution incorporates both strong repulsion between particles via the Lennard‒Jones potential and non-local interactions with the continuum environment. The particle–continuum coupling is also achieved through a chemical reaction, which is modelled as a Poisson counting process. We simulate the spatiotemporal evolution of this corrosion process using the Euler‒Maruyama algorithm with varying initial data combined with finite elements to address spatial discretization. Despite symmetric initial data, our simulations highlight an uneven progression of corrosion due to stochastic influences in the model.

A McKean–Vlasov stochastic differential equation subject to killing associated to a regularised non-conservative and path-dependent nonlinear parabolic partial differential equation is studied. The existence and pathwise uniqueness of a strong solution and the regularity properties of its sub-probability law are proved. The density of such a law may be seen as a weak solution of the considered PDE. The well-posedness of the associated particle system is also discussed.

We study the well-posedness of a nonlinear reaction diffusion partial differential equation system on the half-line coupled with a stochastic dynamical boundary condition, a random system arising in the description of the evolution of the chemical reaction of sulphur dioxide with the surface of calcium carbonate stones. The boundary condition is given by a Jacobi process, solution to a Brownian motion-driven stochastic differential equation with a mean reverting drift and a bounded diffusion coefficient. The main result is the global existence and the pathwise uniqueness of mild solutions. The proof relies on a splitting strategy, which allows to deal with the low regularity of the dynamical boundary condition.

We study the well-posedness of a nonlinear reaction diffusion partial differential equation system on the half-line coupled with a stochastic dynamical boundary condition, a random system arising from the description of the chemical reaction of sulphur dioxide with calcium carbonate stones. The boundary condition is given by a Jacobi process, solution to a stochastic differential equation with a mean-reverting drift and a bounded diffusion coefficient. The main result is the global existence and the pathwise uniqueness of mild solutions. The proof relies on a splitting strategy, which allows to deal with the low regularity of the dynamical boundary condition.

The marble sulphation process, the phenomenon of the formation of a patina of gypsum on the surface of calcium carbonate monuments, is characterized by different scales: from the nanoscale where molecules interact and react, to the macroscale where averaged quantities are measured. We propose a first stochastic model at the nanoscale. It is given by a first order dynamics for the molecules given by a system of stochastic differential equations driven by Wiener processes, coupled with point processes properly describing the chemical reactions. In particular we suggest a new model for the activating energy related to the reaction. By an appropriate rescaling procedure, we heuristically derive both a measure-valued equation for the empirical measure and a natural mean-field approximation, and compare it with the closest deterministic model in literature. Possible developments of the proposed stochastic model are discussed.

A space discrete approximation to a highly nonlinear reaction-diffusion system endowed with a stochastic dynamical boundary condition is analyzed and the convergence of the discrete scheme to the solution to the corresponding continuum random system is established. A splitting strategy allows us to decompose the random system into a space-discrete heat equation with a stochastic boundary condition, and a nonlinear and nonlocal space-discrete differential system coupled with the first one and with deterministic initial and boundary conditions. The convergence result is obtained by first establishing some a priori estimates for both space-discrete splitted variables and then exploiting compact embedding theorems for time-space Besov spaces on the positive lattice. The convergence of a fully discrete approximation of the random system is also discussed.

We prove existence and uniqueness of strong solutions to a large class of autonomous stochastic differential equations on an open domain, where the drift exhibits a singular behaviour at the boundary. The main result involves a drift composed of the gradient of a singular potential and an additional possibly singular force. In order to achieve the well-posedness of the model, we employ a probabilistic regularization approach. Under suitable conditions, it is shown that the explosion time of the solution process is infinite. The result is finally applied to the case of an interacting particle system subject to a Lennard-Jones potential, which is singular at the origin.

In the framework of the dynamical evolution of the chemical reactions of the sulphur dioxide with the surface of calcium carbonate stones in the process of the degradation of the cultural heritage, starting from a well known deterministic mathematical model, in order to better describe the high variability of the external sulphur dioxide concentration we introduce a suitable stochastic dynamical boundary condition. As boundary condition we take a Jacobi process, solution to a Brownian motion driven stochastic differential equation. We discuss both the mathematical problems arising from considering a lower regular boundary condition and in particular the global existence and (pathwise) uniqueness of the reaction diffusion system coupled with this stochastic boundary condition. The proof relies on a splitting strategy, which allows to deal with the low regularity of the boundary condition. A discretization scheme based on the same splitting is proposed and some numerical samples are shown.

We investigate the qualitative behaviour of the solutions of a stochastic boundary value problem on the half-line for a nonlinear system of parabolic reaction-diffusion equations, from a numerical point of view. The model describes the chemical aggression of calcium carbonate stones under the attack of sulphur dioxide. The dynamical boundary condition is given by a Pearson diffusion, which is original in the context of the degradation of cultural heritage. We first discuss a scheme based on the Lamperti transformation for the stochastic differential equation to preserve the boundary and a splitting strategy for the partial differential equation based on recent theoretical results. Positiveness, boundedness, and stability are stated. The impact of boundary noise on the solution and its qualitative behaviour both in the slow and fast regimes is discussed in several numerical experiments.

Research conducted on the degradation of cultural and architectural heritage caused by the deposition of air pollutants indicates that the deterioration of carbonate materials is due primarily to the interaction of the substrate with sulphur dioxide (SO2) and particulate matter (PM) deriving from the combustion of fossil fuels. The main chemical degradation process is the sulphation of the substrate, which consists in the initial conversion of SO2 into sulphuric acid (H2SO4) and the subsequent reaction of this product with calcium carbonate (CaCO3). This leads to the formation of black crusts composed mainly of gypsum (CaSO4 & sdot;2 H2O), inside which PM is embedded. These carbonaceous particles also contain heavy metals that can act as catalysts in both stages of the sulphation process, favouring the degradation of the substrate. However, the degree to which each specific metal is able to affect sulphation is still unclear. This research aims to evaluate which heavy metals activate the sulphation process by carrying out targeted exposure mock-up tests in special climatic chambers. The selected cations were the following: Fe3+, Cu2+, Mn2+, Pb2+, Cr3+, V5+ (deposited concentrations were calculated based on data from urban PM). In addition, mixtures involving three or more metal cations were also used to evaluate possible synergistic effects. Finally, PM2.5 extracted from quartz-fibre filters sampled in the city of Milan was also included in the experimentation. The physicochemical characterization of the different mock-up samples was performed both in the pre-exposure and post-exposure phases using different analytical techniques such as: colorimetric analysis, stereomicroscopic observations, SEM-EDX (Scanning Electron Microscopy coupled to Energy Dispersive X-Ray Spectroscopy), IC (Ion Chromatography), and XRPD (X-ray powder diffraction). Results show that some metal cations (Pb, Cu, Cr) are able to activate the catalytic process faster than others (Fe, Mn). Also, samples treated with metal mixtures and PM2.5 exhibited the greatest catalytic action, highlighting a synergistic effect of more heavy metals acting together.

We provide a probabilistic interpretation of a weakly parabolic PDE–ODE system with a reaction term, which makes the dynamics non-conservative. As a consequence, the solution is represented as the density of a sub-probability measure solving a Feynman–Kac-type equation, where the time-marginal law of the underlying process is weighted by a survival probability induced by the reaction. This leads to a coupled stochastic formulation consisting of a non-Markovian stochastic differential equation with path-dependent coefficients and the associated Feynman–Kac-type equation. We prove well-posedness of the resulting stochastic system. Finally, we introduce the corresponding interacting particle system and show that its empirical measure, suitably weighted by the survival probability associated with the reaction rate, converges to the limiting sub-probability.

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.

Within the rough path framework we prove the continuity of the solution to random differential equations driven by fractional Brownian motion with respect to the Hurst parameter $H$ when $H \in (1/3, 1/2]$.

We propose a model for the description and the forecast of the gross prices of electricity in the liberalized Italian energy market via an additive two-factor model driven by both a Hawkes and a fractional Brownian processes. We discuss the seasonality, the identification of spikes and the estimates of the Hurst coefficient. After the calibration and the validation of the model, we discuss its forecasting performance via a class of adequate evaluation metrics.

Dual Phase steel (DP steel) has shown high potential for automotive and other applications, due to its remarkable combined properties of high strength and good formability. The mechanical properties of the material are strictly related to the spatial distribution of the two steel phases, ferrite and martensite, and with their stochastic geometry. Unfortunately the experimental costs to obtain images of sections of steel samples are very high, so that one important industrial problem is to reduce the required number of 2D sections in order to either reconstruct the 3D geometry of the material, or to simulate realistic ones. In this work we will present a germ-grain statistical model which can be used for a best fitting of the main geometric characteristics of the martensite phase. The parameters of the model are estimated on the basis of morphological characteristics of the images of about 150 tomographic sections taken from a real sample. After optimization or tuning of the relevant parameters, the statistical model can then be used to identify the minimum number of sections of the sample which are needed to estimate the parameters in a reliable way.

Purpose: To investigate myopic choroidal neovascularization (mCNV) by fluorescein angiography (FA), spectral-domain optical coherence tomography (SD-OCT), near-infrared (NIR) reflectance, and autofluorescence (AF).Methods: This retrospective study included 65 eyes of 62 Caucasian patients with a mean age of 66.72 years (95% confidence interval [CI] 63-70 years) and a mean refraction of -9.72 diopters (95% CI -8.74 to -10.70 diopters).Results: Most of the mCNV cases were foveal-juxtafoveal (60/65, 92.3%), with thickening of the corresponding retina (62/65, 95.3%) and leakage on FA (44/65, 67.6%). No retinal fluid was detectable in 32 (49.2%) eyes and there was no hemorrhage in 25 (38.4%) eyes. Papillary chorio-retinal atrophy was evident in 58 (89.2%), a shadowing effect in 48 (73.8%), and an epiretinal membrane in 38 (58.4%) eyes. If an area of macular chorioretinal atrophy was present, mCNV frequently developed adjacent to it and was hyperfluorescent rather than with leakage (P<0.001). In eyes with edema or hemorrhage, hyper-reflective foci were more frequent (P<0.005). NIR and AF features were indeterminable in 19 (29.2%) and 27 (41.5%) eyes, respectively. The predominant feature was black or grayish on NIR (34/65, 52.3%) and patchy (hypo-and hyperfluorescence was observed) on AF (25/65, 38.4%). FA and SD-OCT correctly detected mCNV in 49 (75.3%) and 48 (73.8%) eyes, respectively, whereas NIR and AF exhibited limited diagnostic sensitivity. Doubtful diagnosis was associated with hyperfluorescent mCNV (P<0.001), absence of retinal fluid and epiretinal membrane (P<0.05), and presence of macular chorioretinal atrophy (P<0.01).Conclusion: Tomographic, angiographic, AF, and NIR features of mCNV are described in this study. Combination of SD-OCT and FA is recommendable for diagnosis.

Ion channels are of major interest, and form an area of intensive research in the fields of biophysics and medicine since they control many vital physiological functions. The aim of this work is on the one hand to propose a fully stochastic and discrete model describing the main characteristics of a multiple channel system. The movement of the ions is coupled, as usual, with a Poisson equation for the electrical field; we have considered, in addition, the influence of exclusion forces. On the other hand, we have discussed the nondimensionalization of the stochastic system by using real physical parameters, all supported by numerical simulations. The specific features of both cases of micro- and nanochannels have been taken into consideration with particular attention to the latter case in order to show that it is necessary to consider a discrete and stochastic model for ions movement inside the channels.

The mechanical properties of dual Phase steels (DP steels) are strictly related to the spatial distribution and the geometry of the two phases composing the steel, ferrite and martensite. Due to the high costs to obtain images of sections of steel samples, one important industrial problem is the reduction of the number of 2D sections needed to build and simulate a geometric model which may reproduce in a realistic way the 3D geometry of the material. In this context, the availability of suitable techniques of parameter estimation or identification is fundamental to solve the problem.