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This paper is devoted to proving the small noise asymptotic behavior, particularly large deviation principle, for multi-scale stochastic dynamical systems with fully local monotone coefficients driven by multiplicative noise. The main techniques rely on the weak convergence approach, the theory of pseudo-monotone operators and the time discretization scheme. The main results derived in this paper have broad applications to various multi-scale models, where the slow component could be such as stochastic porous medium equations, stochastic Cahn-Hilliard equations and stochastic 2D Liquid crystal equations. [ABSTRACT FROM AUTHOR]

This paper introduces a new p-dependent coercivity condition through which $ L^p $ L p -moments for solutions can be obtained for a large class of SPDEs in the variational framework. If p = 2, our condition reduces to the classical coercivity condition, which only yields second moments for the solution. The abstract result is shown to be optimal. Moreover, the results are applied to obtain $ L^p $ L p -moments of solutions for several classical SPDEs such as stochastic heat equations with Dirichlet and Neumann boundary conditions, Burgers' equation and the Navier–Stokes equations in two spatial dimensions. Furthermore, we can recover recent results for systems of SPDEs and higher-order SPDEs using our unifying coercivity condition. [ABSTRACT FROM AUTHOR]

In this paper, we propose a low rank approximation method for efficiently solving stochastic partial differential equations. Specifically, our method utilizes a novel low rank approximation of the stiffness matrices, which can significantly reduce the computational load and storage requirements associated with matrix inversion without losing accuracy. To demonstrate the versatility and applicability of our method, we apply it to address two crucial uncertainty quantification problems: stochastic elliptic equations and optimal control problems governed by stochastic elliptic PDE constraints. Based on varying dimension reduction ratios, our algorithm exhibits the capability to yield a high-precision numerical solution for stochastic partial differential equations, or provides a rough representation of the exact solutions as a pre-processing phase. Meanwhile, our algorithm for solving stochastic optimal control problems allows a diverse range of gradient-based unconstrained optimization methods, rendering it particularly appealing for computationally intensive large-scale problems. Numerical experiments are conducted and the results provide strong validation of the feasibility and effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]

Abstract Background Gabon still bears significant malaria burden despite numerous efforts. To reduce this burden, policy-makers need strategies to design effective interventions. Besides, malaria distribution is well known to be related to the meteorological conditions. In Gabon, there is limited knowledge of the spatio-temporal effect or the environmental factors on this distribution. This study aimed to investigate on the spatio-temporal effects and environmental factors on the distribution of malaria prevalence among children 2–10 years of age in Gabon. Methods The study used cross-sectional data from the Demographic Health Survey (DHS) carried out in 2000, 2005, 2010, and 2015. The malaria prevalence was obtained by considering the weighting scheme and using the space–time smoothing model. Spatial autocorrelation was inferred using the Moran’s I index, and hotspots were identified with the local statistic Getis-Ord General Gi. For the effect of covariates on the prevalence, several spatial methods implemented in the Integrated Nested Laplace Approximation (INLA) approach using Stochastic Partial Differential Equations (SPDE) were compared. Results The study considered 336 clusters, with 153 (46%) in rural and 183 (54%) in urban areas. The prevalence was highest in the Estuaire province in 2000, reaching 46%. It decreased until 2010, exhibiting strong spatial correlation (P

This paper is mainly concerned with the existence, uniqueness and Poisson stability (including stationarity, periodicity, almost periodicity and almost automorphy) of solutions for a class of stochastic partial differential equations driven by Lévy noise, where the involved coefficients are assumed to be strictly monotone. Based on the variational method, we establish the well-posedness of L 2 -bounded solution and then prove that it has the same characters of periodicity, almost periodicity and almost automorphy as the coefficients of the equation. Moreover, we also investigate the additive noise case under strong monotone condition. In particular, we illustrate our results by applying to concrete models such as stochastic reaction-diffusion equations, stochastic porous media equations and stochastic p -Laplace equations. [ABSTRACT FROM AUTHOR]

Background: Gabon still bears significant malaria burden despite numerous efforts. To reduce this burden, policy-makers need strategies to design effective interventions. Besides, malaria distribution is well known to be related to the meteorological conditions. In Gabon, there is limited knowledge of the spatio-temporal effect or the environmental factors on this distribution. This study aimed to investigate on the spatio-temporal effects and environmental factors on the distribution of malaria prevalence among children 2–10 years of age in Gabon. Methods: The study used cross-sectional data from the Demographic Health Survey (DHS) carried out in 2000, 2005, 2010, and 2015. The malaria prevalence was obtained by considering the weighting scheme and using the space–time smoothing model. Spatial autocorrelation was inferred using the Moran's I index, and hotspots were identified with the local statistic Getis-Ord General Gi. For the effect of covariates on the prevalence, several spatial methods implemented in the Integrated Nested Laplace Approximation (INLA) approach using Stochastic Partial Differential Equations (SPDE) were compared. Results: The study considered 336 clusters, with 153 (46%) in rural and 183 (54%) in urban areas. The prevalence was highest in the Estuaire province in 2000, reaching 46%. It decreased until 2010, exhibiting strong spatial correlation (P < 0.001), decreasing slowly with distance. Hotspots were identified in north-western and western Gabon. Using the Spatial Durbin Error Model (SDEM), the relationship between the prevalence and insecticide-treated bed nets (ITNs) coverage was decreasing after 20% of coverage. The prevalence in a cluster decreased significantly with the increase per percentage of ITNs coverage in the nearby clusters, and per degree Celsius of day land surface temperature in the same cluster. It slightly increased with the number of wet days and mean temperature per month in neighbouring clusters. Conclusions: In summary, this study showed evidence of strong spatial effect influencing malaria prevalence in household clusters. Increasing ITN coverage by 20% and prioritizing hotspots are essential policy recommendations. The effects of environmental factors should be considered, and collaboration with the national meteorological department (DGM) for early warning systems is needed. [ABSTRACT FROM AUTHOR]

Land subsidence in intertidal environments may change the flooding regime and sediment composition, two drivers of the macrozoobenthic community. In the Dutch Wadden Sea, a UNESCO world heritage site, gas extraction has resulted in an average subsidence of intertidal mudflats of 2 mm year−1. These mudflats support an abundant macrozoobenthic community that offers important resources for birds and fish. The area is managed through the 'hand on the tap' principle, meaning that human activities should be halted if they affect the natural values. To what extent land subsidence affects sediment and macrozoobenthos remains unknown and is increasingly important given sea level rise.Taking advantage of a large‐scale monitoring program, we evaluated the effect of anthropogenically caused subsidence on sediment composition and macrozoobenthos. Nearly 4600 points were sampled yearly (2008–2020) across the Dutch Wadden Sea, allowing us to compare sediment composition and macrozoobenthos biomass within and outside the subsidence area while controlling for the main drivers of these variables. We also compared population trends within and outside the subsidence area for 31 species with different habitat use.Mud fraction was 3% higher within the subsided area and median grain size decreased at 1 μm year−1 while remaining constant in other mudflats. This had no effect on the total biomass of macrozoobenthos. Within the subsidence area, however, the biomass of species that use deeper areas increased compared to outside, and the opposite was true for species using shallower habitat.Policy implications: Land subsidence is related to changes in median grain size and macrozoobenthic community composition. However, because thresholds have not been defined, it is not clear if this requires management actions. For a successful implementation of the 'hand on the tap' principle in the Wadden Sea, it is necessary to define beforehand the relevant variables that represent the natural values, implement proper monitoring and define thresholds above which effects are not acceptable. We propose median grain size, mud fraction and macrozoobenthic composition as good measures of the natural values of the Wadden Sea, and the methods used here as a way for identifying anthropogenic effects on them. [ABSTRACT FROM AUTHOR]

The stochastic partial differential equation (SPDE) approach is widely used for modeling large spatial datasets. It is based on representing a Gaussian random field u on R d as the solution of an elliptic SPDE L β u = W where L is a second-order differential operator, 2 β ∈ N is a positive parameter that controls the smoothness of u and W is Gaussian white noise. A few approaches have been suggested in the literature to extend the approach to allow for any smoothness parameter satisfying β > d / 4 . Even though those approaches work well for simulating SPDEs with general smoothness, they are less suitable for Bayesian inference since they do not provide approximations which are Gaussian Markov random fields (GMRFs) as in the original SPDE approach. We address this issue by proposing a new method based on approximating the covariance operator L − 2 β of the Gaussian field u by a finite element method combined with a rational approximation of the fractional power. This results in a numerically stable GMRF approximation which can be combined with the integrated nested Laplace approximation (INLA) method for fast Bayesian inference. A rigorous convergence analysis of the method is performed and the accuracy of the method is investigated with simulated data. Finally, we illustrate the approach and corresponding implementation in the R package rSPDE via an application to precipitation data which is analyzed by combining the rSPDE package with the R-INLA software for full Bayesian inference. for this article are available online. [ABSTRACT FROM AUTHOR]

Traffic deaths and injuries are one of the major global public health concerns. The present study considers accident records in an urban environment to explore and analyze spatial and temporal in the incidence of road traffic accidents. We propose a spatio-temporal model to provide predictions of the number of traffic collisions on any given road segment, to further generate a risk map of the entire road network. A Bayesian methodology using Integrated nested Laplace approximations with stochastic partial differential equations (SPDE) has been applied in the modeling process. As a novelty, we have introduced SPDE network triangulation to estimate the spatial autocorrelation restricted to the linear network. The resulting risk maps provide information to identify safe routes between source and destination points, and can be useful for accident prevention and multi-disciplinary road safety measures. [ABSTRACT FROM AUTHOR]

The normal inverse Gaussian (NIG) and generalized asymmetric Laplace (GAL) distributions can be seen as skewed and semi-heavy-tailed extensions of the Gaussian distribution. Models driven by these more flexible noise distributions are then regarded as flexible extensions of simpler Gaussian models. Inferential procedures tend to overestimate the degree of non-Gaussianity in the data and therefore we propose controlling the flexibility of these non-Gaussian models by adding sensible priors in the inferential framework that contract the model towards Gaussianity. In our venture to derive sensible priors, we also propose a new intuitive parameterization of the non-Gaussian models and discuss how to implement them efficiently in Stan. The methods are derived for a generic class of non-Gaussian models that include spatial Mat´ern fields, autoregressive models for time series, and simultaneous autoregressive models for aerial data. The results are illustrated with a simulation study and geostatistics application, where priors that penalize model complexity were shown to lead to more robust estimation and give preference to the Gaussian model, while at the same time allowing for non-Gaussianity if there is sufficient evidence in the data. [ABSTRACT FROM AUTHOR]

Gridded observational products of the main climate parameters are essential in climate science. Current interpolation approaches, implemented to derive such products, often lack of a proper uncertainty propagation and representation. In this study, we introduce a Bayesian spatiotemporal approach based on the integrated nested Laplace approximation (INLA) and the stochastic partial differential equation (SPDE). The method is described and discussed by using a real case study based on high‐resolution monthly 2‐m maximum (Tmax) and minimum (Tmin) air temperature over Italy in 1961–2020. The INLA‐SPDE based approach is able to properly take into account uncertainties in the final gridded products and offers interesting promising advantages to deal with nonstationary and non‐Gaussian multisource data. [ABSTRACT FROM AUTHOR]

Natural hazards like floods, cyclones, earthquakes, or, tsunamis have deep impacts on the environment and society causing damage to both life and property. These events can cause widespread destruction and can lead to long-term socio-economic disruption often affecting the most vulnerable populations in society. Computational modeling provides an essential tool to estimate the damage by incorporating spatial uncertainties and examining global risk assessments. Classical stationary models in spatial statistics often assume isotropy and stationarity. It causes inappropriate smoothing over features having boundaries, holes, or physical barriers. Despite this, nonstationary models like barrier model have been little explored in the context of natural disasters in complex land structures. The principal objective of the current study is to evaluate the influence of barrier models compared to classical stationary models by analysing the incidence of natural disasters in complex spatial regions like islands and coastal areas. In the current study, we have used tsunami records from the island nation of Maldives. For seven atoll groups considered in our study, we have implemented three distinct categories of stochastic partial differential equation meshes, two for stationary models and one that corresponds to the barrier model concept. The results show that when assessing the spatial variance of tsunami incidence at the atoll scale, the barrier model outperforms the other two models while maintaining the same computational cost as the stationary models. In the broader picture, this research work contributes to the relatively new field of nonstationary barrier models and intends to establish a robust modeling framework to explore spatial phenomena, particularly natural hazards, in complex spatial regions having physical barriers. [ABSTRACT FROM AUTHOR]

The main aim of this work is to investigate small noise limiting behavior of multi-scale McKean-Vlasov stochastic dynamical systems, where we allow the coefficients depend on the distributions of both slow and fast components. Firstly, the strong convergence in the functional law of large numbers is established by the time discretization scheme. Secondly, in order to characterize the probability of deviations away from the averaged limit, we prove the large deviation principle by the weak convergence approach for McKean-Vlasov equations. [ABSTRACT FROM AUTHOR]

This paper is devoted to proving the strong averaging principle for slow–fast stochastic partial differential equations with locally monotone coefficients, where the slow component is a stochastic partial differential equations with locally monotone coefficients and the fast component is a stochastic partial differential equations with strongly monotone coefficients. The result is applicable to a large class of examples, such as the stochastic porous medium equation, the stochastic p-Laplace equation, the stochastic Burgers type equation and the stochastic 2D Navier–Stokes equation, which are the nonlinear stochastic partial differential equations. The main techniques are based on time discretization and the variational approach to stochastic partial differential equations. [ABSTRACT FROM AUTHOR]

This article investigates the propagation of chaos property for weakly interacting mild solutions to semilinear stochastic partial differential equations whose coefficients might not satisfy Lipschitz conditions. Furthermore, we establish existence and uniqueness results for mild solutions to SPDEs with distribution dependent coefficients, so-called McKean–Vlasov SPDEs. [ABSTRACT FROM AUTHOR]

In this paper we present a framework in which one can rigorously study the effect of spatio-temporal noise on traveling waves, stationary patterns, and oscillations that are invariant under the action of a finite-dimensional set of continuous isometries (such as translation or rotation). This formalism can accommodate patterns, waves, and oscillations in reaction-diffusion systems and neural field equations. To do this, we define the phase by precisely projecting the infinite-dimensional system onto the manifold of isometries. We outline a precise stochastic differential equation for the phase. The phase is then used to show that the probability of the system leaving the attracting basin of the manifold after an exponentially long period of time (in ε -2, the magnitude of the noise) is exponentially unlikely. [ABSTRACT FROM AUTHOR]

In this work the existence and uniqueness of strong solutions are established for a class of stochastic integral evolution equations with locally monotone and non-Lipschitz coefficients. In particular, our results can be applied to a large class of stochastic partial differential equation models with hereditary or memory effects such as quasilinear SPDEs like stochastic porous medium equations and semilinear SPDEs like stochastic 2D Navier—Stokes equations with hereditary or memory terms. [ABSTRACT FROM AUTHOR]

A fully discrete finite difference scheme for stochastic reaction-diffusion equations driven by a 1+1 -dimensional white noise is studied. The optimal strong rate of convergence is proved without posing any regularity assumption on the nonlinear reaction term. The proof relies on stochastic sewing techniques. [ABSTRACT FROM AUTHOR]

In recent years, the growing frequency and severity of wildfires, influenced by both human activities and climate change, have posed significant challenges worldwide. Among the regions most affected by wildfires in Mexico is the state of Jalisco, which has the largest accumulated burned area in the last five decades. In this paper, we present an in-depth analysis of the spatio-temporal patterns of wildfire occurrence and size in the state of Jalisco, spanning the period from 2001 to 2020. Our approach included modeling the spatial distribution of the area burned by wildfires, employing Bayesian methodology with Integrated Nested Laplace Approximation (INLA) and Stochastic Partial Differential Equations (SPDE). Our findings highlight the critical roles of vegetation, temperature, and human activities in shaping wildfire behavior. Additionally, our model suggests four distinct wildfire-prone regions within the state. The insights gained from this study can serve as a foundation for future research and localized studies, aiding in the development of more targeted and effective wildfire management strategies in Jalisco. • The use of INLA-SPDE methodology is an innovation to wildfire research in Mexico. • Environmental factors exhibit varying impacts on wildfire occurrence and size. • Wildfires in Jalisco are associated with vegetation cover and human activities. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of the solution of a linear SPDE driven by a multiplicative multifractional Brownian motion. Moreover, we study an optimal control problem with a linear quadratic objective functional involving the solution of the studied SPDE. We prove the existence and uniqueness of the optimal control. [ABSTRACT FROM AUTHOR]

The main aim of this work is to investigate the large deviation principle for a class of stochastic integral evolution equations. As applications, our results can be applied to a large class of stochastic models with hereditary or memory effects such as stochastic integral porous medium equations, stochastic integral $ p $-Laplace equations and stochastic integral 2D Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

The impact of invasive species is not limited to the loss of biodiversity; it also represents significant threats to agriculture on a global scale. The Japanese beetle Popillia japonica (native to Japan but an invasive agricultural pest in North America) recently occurred in the Po plain (Italy), one of the most cultivated areas in southern Europe. Thus, our aims were to identify (i) the main landscape predictors related to the occurrence of the Japanese beetle and (ii) the areas of potential invasion of the Japanese beetle in the two Northern Italian regions in which this invasive species currently occurs, Piedmont and Lombardy. Specifically, we combined Japanese beetle occurrences available in the citizen science online platform iNaturalist with high-resolution landscape predictors in an ensemble approach and averaged the results of Bayesian generalized linear and additive models developed with the integrated nested Laplace approximation (with stochastic partial differential equation). We found that the occurrence of the Japanese beetle was negatively related to the percentage of broadleaf forests and pastures, while it was positively related to sparse and dense human settlements as well as intensive crops. Moreover, the occurrence of the Japanese beetle increased in relation to the percentage of rice fields until a peak at around 50%. The Japanese beetle was likely to occur in 32.49% of our study area, corresponding to 16,000.02 km2, mainly located in the Po plain, low hills, and mountain valleys. We stress that the Japanese beetle is a high-risk invasive species in human-dominated landscapes. Thus, we strongly recommend that local administrations quickly enact pest management in order to reduce further spread. [ABSTRACT FROM AUTHOR]

In this paper, we aim to study the asymptotic behavior for a class of McKean-Vlasov stochastic partial differential equations with slow and fast time-scales. Using the variational approach and classical Khasminskii time discretization, we show that the slow component strongly converges to the solution of the associated averaged equation. In particular, the corresponding convergence rates are also obtained. The main results can be applied to demonstrate the averaging principle for various McKean-Vlasov nonlinear SPDEs such as stochastic porous media type equation, stochastic p -Laplace type equation and also some McKean-Vlasov stochastic differential equations. [ABSTRACT FROM AUTHOR]

The spread of invasive species is a threat to global biodiversity. The Japanese beetle is native to Japan, but alien populations of this insect occur in North America, and recently, also in southern Europe. This beetle was recently included on the list of priority species of European concern, as it is a highly invasive agricultural pest. Thus, in this study, we aimed at (i) assessing its current distribution range, and identifying areas of potential invasion, and (ii) predicting its distribution using future climatic and land-use change scenarios for 2050. We collected species occurrences available on the citizen science platform iNaturalist, and we combined species data with climatic and land-use predictors using a Bayesian framework, specifically the integrated nested Laplace approximation, with a stochastic partial differential equation. We found that the current distribution of the Japanese beetle was mainly, and positively, driven by the percentage of croplands, the annual range of temperature, habitat diversity, percentage of human settlements, and human population density; it was negatively related to the distance to airports, elevation, mean temperature diurnal range, wetlands, and waters. As a result, based on current conditions, the Japanese beetle is likely to occur in 47,970,200 km2, while its distribution will range from between 53,418,200 and 59,126,825 km2, according to the 2050 climatic and land-use change scenarios. We concluded that the Japanese beetle is a high-risk invasive species, able to find suitable conditions for its colonization in several regions around the globe, especially in light of ongoing climatic change. Thus, we strongly recommend strict biosecurity checks and quarantines, as well as regular pest management surveys, in order to reduce its spread. [ABSTRACT FROM AUTHOR]

MultiLevel Monte Carlo (MLMC) attracts great interest for numerical simulations of Stochastic Partial Differential Equations (SPDEs), due to its superiority over the standard Monte Carlo (MC) approach. MLMC combines in a proper manner many cheap fast simulations with few slow and expensive ones, the variance is reduced, and a significant speed up is achieved. Simulations with MC/MLMC consist of three main components: generating random fields, solving deterministic problem and reduction of the variance. Each part is subject to a different degree of parallelism. Compared to the classical MC, MLMC introduces “levels” on which the sampling is done. These levels have different computational cost, thus, efficiently utilizing the parallel resources becomes a non-trivial problem. The main focus of this paper is the parallelization of the MLMC Algorithm.

This work is concerned with a Freidlin--Wentzell type large deviation principle for a family of multiscale quasilinear and semilinear stochastic partial differential equations. Employing the weak convergence method and Khasminskii's time discretization approach, the Laplace principle (equivalently, large deviation principle) for a general class of multiscale SPDEs is derived. In particular, we succeed in dropping the compactness assumption of embedding in the Gelfand triple in order to deal with the case of bounded and unbounded domains in applications. Our main results are applicable to various multiscale SPDE models such as stochastic porous media equations, stochastic p-Laplace equations, stochastic fast-diffusion equations, stochastic two-dimensional hydrodynamical type models, stochastic power law fluid equations, and stochastic Ladyzhenskaya models. [ABSTRACT FROM AUTHOR]

Aims: Caesarean section (CS) is a medical intervention performed in Norway when a surgical delivery is considered more beneficial than a vaginal. Because deliveries with higher risk are centralized to larger hospitals, use of CS varies considerably between hospitals. We describe how the use of CS varies geographically by municipality. Since indications for CS should have little variation across the relatively homogenous population of Norway, we expect fair use of CS to be evenly distributed across the municipalities. Methods: Data from the Medical Birth Registry of Norway were used in our analyses (810,914 total deliveries, 133,746 CSs, 440 municipalities). We propose a spatial correlation model that takes the location into account to describe the variation in use of CS across the municipalities. The R packages R-INLA and TMB are used to estimate the yearly municipal CS rate and the spatial correlation between the municipalities. We also apply stratified models for different categories of delivering women (Robson groups). Estimated rates are displayed in maps and model parameters are shown in tables. Results: The CS rate varies substantially between the different municipalities. As expected, there was strong correlation between neighbouring municipalities. Similar results were found for different Robson groups. Conclusions: The substantial difference in CS use across municipalities in Norway is not likely to be due to specific medical reasons, but rather to hospitals' different policies towards the use of CS. The policy to be either more or less restrictive to CS was not specific to any category of deliveries. [ABSTRACT FROM AUTHOR]

Wildfires play an important role in shaping landscapes and as a source of CO2 and particulate matter, and are a typical spatial point process studied in many papers. Modeling the spatial variability of a wildfire could be performed in different ways and an important issue is the computational facilities that the new techniques afford us. The most common approaches have been through point pattern analysis or by Markov random fields. These methods have made it possible to build risk maps, but for many forest managers it is very useful to know the size of the fire as well as its location. In this work, we use Stochastic Partial Differential Equation (SPDE) with Integrated Nested Laplace Approximation (INLA) to model the size of the forest fires observed in the Valencian Community, Spain. But the most important element in this paper is the process that needs to be carried out prior to simulating and analyzing the different point patterns, namely, the choice of the most suitable mesh for the database. We describe and take advantage of the Bayesian methodology by including INLA and SPDE in the modeling process in all the scenarios. [ABSTRACT FROM AUTHOR]

In spatial data, especially in geostatistics data where measurements are often provided by satellite scanning, some parts of data may get missed. Due to spatial dependence in the data, these missing values probably are caused by some latent spatial random fields. In this case, ignoring missingness is not logical and may lead to invalid inferences. Thus incorporating the missingness process model into the inferences could improve the results. There are several approaches to take into account the non-ignorable missingness, one of them is the shared parameter model method. In this paper, we extend it for spatial data so that we will have a joint spatial Bayesian shared parameter model. Then the missingness process will be jointly modeled with the measurement process and one or more latent spatial random fields as shared parameters would describe their association. Bayesian inference is implemented by Integrated nested Laplace approximation. A computationally effective approach is applied via a stochastic partial differential equation for approximating latent Gaussian random field. In a simulation study, the proposed spatial joint model is compared with a model that assumes data are missing at random. Based on these two models, the lake surface water temperature data for lake Vänern in Sweden are analyzed. The results of estimation and prediction confirm the efficiency of the spatial joint model. [ABSTRACT FROM AUTHOR]

Strong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have, loosely speaking, been investigated since 2003 and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions. In the recent article (Conus et al. in Ann Appl Probab 29(2):653–716, 2019) this weak convergence problem has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome this weak convergence problem in the case of a class of time-discrete Euler-type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp weak convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are applications of a mild Itô-type formula and the use of suitable semilinear integrated counterparts of the time-discrete numerical approximation processes. [ABSTRACT FROM AUTHOR]

We develop a method for probabilistic prediction of extreme value hot-spots in a spatio-temporal framework, tailored to big datasets containing important gaps. In this setting, direct calculation of summaries from data, such as the minimum over a space-time domain, is not possible. To obtain predictive distributions for such cluster summaries, we propose a two-step approach. We first model marginal distributions with a focus on accurate modeling of the right tail and then, after transforming the data to a standard Gaussian scale, we estimate a Gaussian space-time dependence model defined locally in the time domain for the space-time subregions where we want to predict. In the first step, we detrend the mean and standard deviation of the data and fit a spatially resolved generalized Pareto distribution to apply a correction of the upper tail. To ensure spatial smoothness of the estimated trends, we either pool data using nearest-neighbor techniques, or apply generalized additive regression modeling. To cope with high space-time resolution of data, the local Gaussian models use a Markov representation of the Matérn correlation function based on the stochastic partial differential equations (SPDE) approach. In the second step, they are fitted in a Bayesian framework through the integrated nested Laplace approximation implemented in R-INLA. Finally, posterior samples are generated to provide statistical inferences through Monte-Carlo estimation. Motivated by the 2019 Extreme Value Analysis data challenge, we illustrate our approach to predict the distribution of local space-time minima in anomalies of Red Sea surface temperatures, using a gridded dataset (11315 days, 16703 pixels) with artificially generated gaps. In particular, we show the improved performance of our two-step approach over a purely Gaussian model without tail transformations. [ABSTRACT FROM AUTHOR]

Irregularly spaced time series are commonly encountered in the analysis of time series. A particular case is that in which the collection procedure over time depends also on the observed values. In such situations, there is stochastic dependence between the process being modeled and the times at which the observations are made. Ignoring this dependence can lead to biased estimates and misleading inferences. In this paper, we introduce the concept of preferential sampling in the temporal dimension and we propose a model to make inference and prediction. The methodology is illustrated using artificial data as well a real data set.

Aim: Fine‐scale topography and canopy cover can play an important role in mediating effects of regional‐scale climate change on the below‐canopy environment in mountain forests. The aim of this study was to determine how below‐canopy temperatures in a high‐elevation Rocky Mountain forest have been affected by canopy change resulting from severe wildfire and spruce beetle outbreak within the last 10–15 years. Location: Eastern San Juan Mountains, Colorado, USA. Taxon: Picea engelmannii, Abies lasiocarpa, Dendroctonus rufipennis. Methods: We used a network of sensors to record temperatures for a full year in burned and beetle‐impacted areas. Using a Bayesian model that accounted for spatial structure in temperatures, we derived covariate parameters to determine the relative influence of topographic variables (elevation, aspect, slope, topographic position and solar radiation), live tree basal area and burned/unburned status on average daily maximum and minimum temperatures (Tmax, Tmin) in summer and winter. Results: Model parameters indicated that burned areas were warmer than unburned forest, with three of four average temperature variables (summer Tmax, winter Tmax, and winter Tmin) having >95% likelihood of a positive temperature difference in burned versus unburned locations. Mean temperature changes for these variables ranged from 0.41 to 0.74°C. Conversely, canopy loss in unburned, beetle‐killed forests did not meaningfully affect Tmax but resulted in slight cooling of Tmin. Modelled temperature changes resulting from 100% overstorey mortality were −1.29°C for summer Tmin (95% credible interval: −0.027 to −2.56°C) and −1.31°C for winter Tmin (95% credible interval: 0.17 to −2.79°C). Main Conclusions: Our results indicate that severe wildfire may exacerbate effects of climate change and increase the probability of ecosystem transitions. However, the effects of bark beetle outbreaks are more complex. Cooling of overnight minimum temperatures may counteract warming trends, but an increase in diurnal temperature ranges may have uncertain ecological consequences. [ABSTRACT FROM AUTHOR]

We introduce Lévy-driven causal CARMA random fields on R d , extending the class of CARMA processes. The definition is based on a system of stochastic partial differential equations which generalize the classical state-space representation of CARMA processes. The resulting CARMA model differs fundamentally from the CARMA random field of Brockwell and Matsuda. We show existence of the model under mild assumptions and examine some of its features including the second-order structure and path properties. In particular, we investigate the sampling behavior and formulate conditions for the causal CARMA random field to be an ARMA random field when sampled on an equidistant lattice. [ABSTRACT FROM AUTHOR]

In air pollution studies, dispersion models provide estimates of concentration at grid level covering the entire spatial domain and are then calibrated against measurements from monitoring stations. However, these different data sources are misaligned in space and time. If misalignment is not considered, it can bias the predictions. We aim at demonstrating how the combination of multiple data sources, such as dispersion model outputs, ground observations, and covariates, leads to more accurate predictions of air pollution at grid level. We consider nitrogen dioxide (NO2) concentration in Greater London and surroundings for the years 2007–2011 and combine two different dispersion models. Different sets of spatial and temporal effects are included in order to obtain the best predictive capability. Our proposed model is framed in between calibration and Bayesian melding techniques for data fusion. Unlike other examples, we jointly model the response (concentration level at monitoring stations) and the dispersion model outputs on different scales, accounting for the different sources of uncertainty. Our spatiotemporal model allows us to reconstruct the latent fields of each model component, and to predict daily pollution concentrations. We compare the predictive capability of our proposed model with other established methods to account for misalignment (e.g., bilinear interpolation), showing that in our case study the joint model is a better alternative. [ABSTRACT FROM AUTHOR]

In this article, we close a gap in the literature by proving existence of invariant measures for reflected stochastic partial differential equations with only one reflecting barrier. This is done by arguing that the sequence (u (t , ·)) t ≥ 0 is tight in the space of probability measures on continuous functions and invoking the Krylov–Bogolyubov theorem. As we no longer have an a priori bound on our solution as in the two-barrier case, a key aspect of the proof is the derivation of a suitable L p bound which is uniform in time. [ABSTRACT FROM AUTHOR]

We consider an asymptotic SPDE description of a large portfolio model where the underlying asset prices evolve according to certain stochastic volatility models with default upon hitting a lower barrier. The asset prices and their volatilities are correlated through systemic Brownian motions, and the SPDE is obtained on the positive half-space along with a Dirichlet boundary condition. We study the convergence of the loss from the system, which is given in terms of the total mass of a solution to our stochastic initial-boundary value problem, under fast mean-reversion of the volatility. We consider two cases. In the first case, the volatilities are sped up towards a limiting distribution and the system converges only in a weak sense. On the other hand, when only the mean-reversion coefficients of the volatilities are allowed to grow large, we see a stronger form of convergence of the system to its limit. Our results show that in a fast mean-reverting volatility environment, we can accurately estimate the distribution of the loss from a large portfolio by using an approximate constant volatility model which is easier to handle. [ABSTRACT FROM AUTHOR]

This work proposes and analyzes a family of spatially inhomogeneous epidemic models. This is our first effort to use stochastic partial differential equations (SPDEs) to model epidemic dynamics with spatial variations and environmental noise. After setting up the problem, the existence and uniqueness of solutions of the underlying SPDEs are examined. Then, definitions of permanence and extinction are given, and certain sufficient conditions are provided for permanence and extinction. Our hope is that this paper will open up windows for investigation of epidemic models from a new angle. [ABSTRACT FROM AUTHOR]

We study Kardar–Parisi–Zhang equation in spatial dimension 3 or larger driven by a Gaussian space–time white noise with a small convolution in space. When the noise intensity is small, it is known that the solutions converge to a random limit as the smoothing parameter is turned off. We identify this limit, in the case of general initial conditions ranging from flat to droplet. We provide strong approximations of the solution which obey exactly the limit law. We prove that this limit has sub-Gaussian lower tails, implying existence of all negative (and positive) moments. [ABSTRACT FROM AUTHOR]

In this paperwe show the existence and uniqueness of strong solutions for a large class of backward SPDEs, where the coefficients satisfy a specific type Lyapunov condition instead of the classical coercivity condition. Moreover, based on the generalized variational framework, we also use the local monotonicity condition to replace the standard monotonicity condition, which is applicable to various quasilinear and semilinear BSPDE models. [ABSTRACT FROM AUTHOR]

The spatio-temporal study of wildfires has two complex elements that are the computational efficiency and longtime processing. Modelling the spatial variability of a wildfire could be performed in different ways, and an important issue is the computational facilities that the new methodological techniques afford us. The Markov random fields methods have made possible to build risk maps, but for many forest managers, it is more advantageous to know the size of the fire and its location. In the first part of this work, Stochastic Partial Differential Equation with Integrated Nested Laplace Approximation is utilised to model the size of the forest fires observed in the Valencian Community (Spain) and so it does the inclusion of the time effect, and the study of the emergency calls. The most crucial element in this paper is the inclusion of the improved meshes for the spatial effect and the time, these are, 2d (locations) and 1d (time) respectively. The advantage of the use of spatio-temporal meshes is described with the inclusion of Bayesian methodology in all the scenarios. [ABSTRACT FROM AUTHOR]

Spatio-temporal disease mapping models give a great worth in epidemiology, especially in describing the pattern of disease incidence across geographical space and time. This paper analyses the spatial and temporal variability of dengue disease rates based on generalized linear mixed models. For spatio-temporal study, the models incorporate spatially correlated random effects as well as temporal effects. In this study, two different spatial random effects are applied and compared. The first model is based on Leroux spatial model, while the second model is based on the stochastic partial differential equation approach. For the temporal effects, both models follow an autoregressive model of first-order model. The models are fitted within a hierarchical Bayesian framework with integrated nested Laplace approximation methodology. The main objective of this study is to compare both spatio-temporal models in terms of their ability in representing the disease phenomenon. The models are applied to weekly dengue fever data in Peninsular Malaysia reported to the Ministry of Health Malaysia in the year 2017 according to the district level. [ABSTRACT FROM AUTHOR]