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This paper is devoted to analyzing the influence of space-time white noise on the dynamics of biological mathematical models in spatiotemporal scenarios based on stochastic partial differential equations (SPDEs). Here, we propose a stochastic reaction–diffusion three-species food chain model with various functional response functions. The motivation of the SPDEs model construction is mainly twofold: (1) when species migrate in space, space-time white noise is a factor that cannot be ignored; (2) functional response function is an essential indicator for directly measuring the relationship between species in biological mathematical models. To discuss the long-term behaviors of the SPDEs model in spatial heterogeneous environment, we first analyze the well-posedness of mild and strong solutions, respectively. Then, the critical conditions for determining the permanence and extinction of populations are given, which reveal that high-intensity white noise can lead to the extinction of predator populations. The validity of the theoretical results is also performed by numerical simulation. In particular, this new theoretical method may be extended to the study of other stochastic biological mathematical models in spatiotemporal scenarios, such as eco-epidemiological models, high-dimensional population models and epidemic models. [ABSTRACT FROM AUTHOR]

In this paper, we prove a regularization by noise phenomenon for semilinear SPDEs driven by multiplicative cylindrical Brownian motion and singular diffusion coefficient, addressing an open problem in Ref. 8. The analysis combines infinite-dimensional generalizations of arguments from Ref. 5 with a detailed maximal regularity analysis for semilinear SPDEs, utilizing Volterra-sewing techniques from Ref. 23. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of stochastic semilinear parabolic equations subjected to multiplicative noise within singularly perturbed phase spaces. We first establish the existence and smoothness of stable foliations. Then we prove that the long-term behavior of each solution is determined by a solution residing on the pseudo-unstable manifold via a leaf of the stable foliation. Finally, we present the convergence of C 1 invariant foliations as the high dimensional region collapse to low dimensional region. In contrast to the convergence of pseudo-unstable manifolds, we introduce a novel technique to address challenges arising from the singularity of the stable term of hyperbolicity in the proof of convergence of stable manifolds and stable foliations as the space collapses. [ABSTRACT FROM AUTHOR]

We study a continuous-time version of the Hegselmann–Krause model describing the opinion dynamics of interacting agents subject to random perturbations. Mathematically speaking, the opinion of agents is modelled by an interacting particle system with a non-Lipschitz continuous interaction force, perturbed by idiosyncratic and environmental noises. Sending the number of agents to infinity, we derive a McKean–Vlasov stochastic differential equation as the limiting dynamic, by establishing propagation of chaos for regularized versions of the noisy opinion dynamics. To that end, we prove the existence of a unique strong solution to the McKean–Vlasov stochastic differential equation as well as well-posedness of the associated non-local, non-linear stochastic Fokker–Planck equation. [ABSTRACT FROM AUTHOR]

We prove the existence of a B-continuous viscosity solution for a class of infinite dimensional semilinear partial differential equations (PDEs) using probabilistic methods. Our approach also yields a stochastic representation formula for the solution in terms of a scalar-valued backward stochastic differential equation. The uniqueness is proved under additional assumptions using a comparison theorem for viscosity solutions. Our results constitute the first nonlinear Feynman–Kac formula using the notion of B-continuous viscosity solutions and thus introduces a framework allowing for generalizations to the case of fully nonlinear PDEs. [ABSTRACT FROM AUTHOR]

Spatially misaligned data can be fused by using a Bayesian melding model that assumes that underlying all observations there is a spatially continuous Gaussian random field. This model can be employed, for instance, to forecast air pollution levels through the integration of point data from monitoring stations and areal data derived from satellite imagery. However, if the data present preferential sampling, that is, if the observed point locations are not independent of the underlying spatial process, the inference obtained from models that ignore such a dependence structure may not be valid. In this paper, we present a Bayesian spatial model for the fusion of point and areal data that takes into account preferential sampling. Fast Bayesian inference is performed using the integrated nested Laplace approximation and the stochastic partial differential equation approaches. The performance of the model is assessed using simulated data in a range of scenarios and sampling strategies that can appear in real settings. The model is also applied to predict air pollution in the USA. [ABSTRACT FROM AUTHOR]

ABSTRACT This paper deals with homogenization for a class stochastic partial differential equations with nonperiodic coefficients. By means of the stochastic two‐scale convergence approach, the solution of a class stochastic partial differential equations with nonperiodic coefficients (microscopic model) is shown to converge in distribution to the solution of effective equation (macroscopic model). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the almost-sure exponential asymptotic stability of the trivial solution of a parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise near the deterministic Hopf bifurcation point. We show the existence and uniqueness of the invariant measure under appropriate assumptions, and approximate the exponential growth rate via asymptotic expansion, given that the strength of the noise is small. This approximate quantity can readily serve as a robust indicator of the change of almost-sure stability. We apply the results to a simplified stochastic Moore-Greitzer PDE model in detecting the stall instabilities of modern jet-engine under the impact of multiplicative noise. A better understanding of the instability margin will eventually optimize the jet-engine operating range and thus lead to lighter and more efficient jet-engine design. [ABSTRACT FROM AUTHOR]

The validity of the instanton analysis approach is tested numerically in the case of the diffraction–amplification problem ∂ z ψ − i 2 m ∂ x 2 2 ψ = g | S | 2 ψ for ln ⁡ U ≫ 1 , where U = | ψ (0 , L) | 2 . Here, S (x , z) is a complex Gaussian random field, z and x respectively are the axial and transverse coordinates, with 0 ⩽ z ⩽ L , and both m ≠ 0 and g > 0 are real parameters. We consider a class of S, called the 'one-max class', for which we devise a specific biased sampling procedure. As an application, p (U), the probability distribution of U, is obtained down to values less than 10−2270 in the far right tail. We find that the agreement of our numerical results with the instanton analysis predictions in Mounaix (2023 J. Phys. A: Math. Theor. 56 305001) is remarkable. Both the predicted algebraic tail of p (U) and concentration of the realizations of S onto the leading instanton are clearly confirmed, which validates the instanton analysis numerically in the large ln ⁡ U limit for S in the one-max class. [ABSTRACT FROM AUTHOR]

In this paper, we use a stochastic partial differential equation (SPDE) as a model for the density of a population under the influence of random external forces/stimuli given by the environment. We study statistical properties for two crucial parameters of the SPDE that describe the dynamic of the system. To do that, we use the Galerkin projection to transform the problem, passing from the SPDE to a system of independent SDEs; in this manner, we are able to find the maximum likelihood estimator of the parameters. We validate the method by using simulations of the SDEs. We prove consistency and asymptotic normality of the estimators; the latter is showed using the Malliavin–Stein method. We illustrate our results with numerical experiments. [ABSTRACT FROM AUTHOR]

This paper presents an innovative numerical technique for specific classes of stochastic heat equations. Our approach uniquely combines a sixth-order compact finite difference algorithm with fast discrete Fourier transforms. While traditional discrete sine transforms are effective for approximating second-order derivatives, they are inadequate for first-order derivatives. To address this limitation, we introduce an innovative variant based on exponential transforms. This method is rigorously evaluated on two forms of stochastic heat equations, and the solutions are compared with those obtained using the established stochastic ten non-polynomial cubic-spline method. The results confirm the accuracy and applicability of our proposed method, highlighting its potential to enhance the numerical treatment of stochastic heat equations. [ABSTRACT FROM AUTHOR]

We introduce a new class of numerical schemes which allow for low-regularity approximations to the expectation E (| u k (t , v η) | 2) , where u k denotes the k-th Fourier coefficient of the solution u of the dispersive equation and v η (x) the associated random initial data. This quantity plays an important role in physics, in particular in the study of wave turbulence where one needs to adopt a statistical approach in order to obtain deep insight into the generic long-time behaviour of solutions to dispersive equations. Our new class of schemes is based on Wick's theorem and Feynman diagrams together with a resonance-based discretisation (Bruned and Schratz in Forum Math Pi 10:E2, 2022) set in a more general context: we introduce a novel combinatorial structure called paired decorated forests which are two decorated trees whose decorations on the leaves come in pair. The character of the scheme draws its inspiration from the treatment of singular stochastic partial differential equations via regularity structures. In contrast to classical approaches, we do not discretise the PDE itself, but rather its expectation. This allows us to heavily exploit the optimal resonance structure and underlying gain in regularity on the finite dimensional (discrete) level. [ABSTRACT FROM AUTHOR]

Stochastic partial differential equations (SPDEs) driven by Lévy noise are extensively employed across various domains such as physics, finance, and engineering to simulate systems experiencing random fluctuations. In this paper, we focus on a specific type of such SPDEs, namely the nonlinear beam epitaxy equation driven by Lévy noise. The Feynman graph formalism emerges as a potent tool for analyzing these SPDEs, particularly in computing their correlation functions, which are essential for understanding the moments of the solution. In this context, the solution to the SPDE and its truncated moments can be expressed as a sum over particular Feynman graphs. Each graph is evaluated according to a set of established rules, providing a systematic method to derive the properties of the solution. Moreover, the study delves into the behavior of the truncated moments for large times. Truncated moments, which capture the statistical properties of the system up to a certain order, are crucial for characterizing the long-term behavior and stability of the solution. The paper will conclude with a discussion on potential applications, highlighting the broader implications of this approach in various scientific and engineering contexts. [ABSTRACT FROM AUTHOR]

European anchovies and round sardinella play a crucial role, both ecological and commercial, in the Mediterranean Sea. In this paper, we investigate the distribution of their larval stages by analyzing a dataset collected over time (1998–2016) and spaced along the area of the Strait of Sicily. Environmental factors are also integrated. We employ a hierarchical spatio-temporal Bayesian model and approximate the spatial field by a Gaussian Markov Random Field to reduce the computation effort using the Stochastic Partial Differential Equation method. Furthermore, the Integrated Nested Laplace Approximation is used for the posterior distributions of model parameters. Moreover, we propose an index that enables the temporal evaluation of species abundance by using an abundance aggregation within a spatially confined area. This index is derived through Monte Carlo sampling from the approximate posterior distribution of the fitted models. Model results suggest a strong relationship between sea currents' directions and the distribution of larval European anchovies. For round sardinella, the analysis indicates increased sensitivity to warmer ocean conditions. The index suggests no clear overall trend over the years. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the stochastic Landau–Lifshitz–Slonczewski (LLS) equation that describes magnetisation of an infinite nanowire evolving under current-driven spin torque. The current brings into the system a multiplicative gradient noise that appears as a transport term in the equation. We prove the existence, uniqueness and regularity of pathwise solutions to the equation. [ABSTRACT FROM AUTHOR]

In this work we establish weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise, in particular, for the hyperbolic Anderson model. For this class of stochastic partial differential equations the weak convergence rates we obtain are indeed twice the known strong rates. To the best of our knowledge, our findings are the first in the scientific literature which provide essentially sharp weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise. Key ideas of our proof are a sophisticated splitting of the error and applications of the recently introduced mild Itô formula. We complement our analytical findings by means of numerical simulations in Python for the decay of the weak approximation error for SPDEs for four different test functions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an approximative maximum likelihood estimator (MLE) for the drift coefficient of a stochastic partial differential equation in the case where the corresponding Fourier coefficients u k (t) , k = 1 , ... , N over a finite interval of time [ 0 , T ] are observed on a uniform time grid: 0 = t 0 < t 1 < ⋯ < t M = T , with Δ : = t i − t i − 1 = T / M , i ˙ = 1 , ... , M . We provide an explicit Berry–Esseen bound in Kolmogorov distance for this approximative MLE when N , M , T → ∞ , assuming that T 3 N 7 / M 2 → 0 and N 2 / T → 0 . [ABSTRACT FROM AUTHOR]

In recent years, stochastic partial differential equations (SPDEs) have become a wellstudied field in mathematics. With their increase in popularity, it becomes important to efficiently approximate their solutions. Thus, our goal is a contribution towards the development of efficient and practical time-stepping methods for SPDEs. Operator splitting schemes provide powerful, efficient, and flexible numerical methods for deterministic and stochastic differential equations. An example is given by domain decomposition schemes, where one splits the domain into subdomains and constructs the numerical approximation in a divide-and-conquer strategy. Instead of solving one expensive problem on the entire domain, one then deals with cheaper problems on the subdomains. This is particularly useful in modern computer architectures, as the subproblems may often be solved in parallel. While splitting methods have already been used to study domain decomposition methods for deterministic PDEs, this is a new approach for SPDEs. This implies that the existing convergence analysis is not directly applicable, even though the building blocks of the operator splitting domain decomposition method are standard. We provide an abstract convergence analysis of a splitting scheme for stochastic evolution equations and state a domain decomposition scheme as an application of the setting. The theoretical results are verified through numerical experiments. [ABSTRACT FROM AUTHOR]

Natural resource development in the Canadian Arctic—a mostly remote and 'untouched' landscape—is expanding. Raptorial species are key indicators of ecosystem diversity and environmental change; disturbance-mediated changes to Arctic-breeding raptor populations can be assessed to determine impacts from development. From 2012 through 2020, we monitored peregrine falcon (Falco peregrinus) and rough-legged hawk (Buteo lagopus) breeding territories near an iron ore mine on North Baffin Island, Nunavut, Canada. The mine was constructed from 2010 through 2014 and became operational in 2015. Our objective was to evaluate whether proximity to mining disturbance affected occupancy and reproductive success of both species. We quantified occupancy using multi-season occupancy models and reproductive success using stochastic partial differential equations capable of accounting for unexplained spatiotemporal variation. Occupancy of both species was best explained by year effects. Occupancy remained relatively stable across time for peregrine falcons (λ = 0.99 ± 0.04) but fluctuated drastically for rough-legged hawks (λ = 3.41 ± 2.17). For both species, most of the spatiotemporal variation in reproductive success was unexplained (presumably from underlying abiotic and biotic factors), which led to the differential presence and count of nestlings across the study area and time. Neither distance to disturbance nor primary production explained variation in occupancy and reproductive success. [ABSTRACT FROM AUTHOR]

In this article, we study a stochastic optimal control problem in the pathwise sense, as initially proposed by Lions and Souganidis in [C. R. Acad. Sci. Paris Ser. I Math., 327 (1998), pp. 735-741]. The corresponding Hamilton-Jacobi-Bellman (HJB) equation, which turns out to be a non-adapted stochastic partial differential equation, is analyzed. Making use of the viscosity solution framework, we show that the value function of the optimal control problem is the unique solution of the HJB equation. When the optimal drift is defined, we provide its characterization. Finally, we describe the associated conserved quantities, namely the space-time transformations leaving our pathwise action invariant. [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]

In this article, we derive the convergence rate for the Wong–Zakai equation of some approximation of stochastic evolution equations with multiplicative noise. To be more precise, the diffusion coefficient in front of the noise is the multiplication operator, and, is therefore not bounded, a situation not treated in the literature. Since our motivation comes from problems in numerical ling, we consider a finite, high-dimensional problem approximating a stochastic evolution equation on a random time grid. By imposing suitable stability conditions on the drift term and the time grid, we achieve a convergence rate in the mean square of order $ \min \{1-\delta,2-2\gamma \} $ min { 1 − δ , 2 − 2 γ } , for some $ 0 \lt \delta \lt 1 $ 0 < δ < 1 and $ 0 \lt \gamma \lt 1/2 $ 0 < γ < 1 / 2. [ABSTRACT FROM AUTHOR]

The article presents a methodology for constructing a generator matrix for dynamical systems using a data-driven approach, focusing on the Lorenz equations as an example. It discusses partitioning strategies, Bayesian methods for quantifying uncertainties, and the use of random matrices to represent uncertainty. The text explores convergence properties, symmetries, matrix decomposition, and connections to dynamic mode decomposition, with applications to a simple harmonic oscillator case. It also includes references to scientific papers on topics such as spectral analysis, Markov models, and data-driven approximations of the Koopman operator, making it a valuable resource for researchers in fluid mechanics, molecular dynamics, and computational methods in physics and chemistry. [Extracted from the article]

The article presents a methodology for discretizing the Liouville equation of high-dimensional dynamical systems, focusing on the compressible Euler equations with gravity and rotation. By constructing a data-driven finite-volume discretization, researchers can recover statistical properties like steady-state distributions and autocorrelations. The study applies this method to the Held-Suarez atmospheric test case, demonstrating its effectiveness across various discretization strategies and showcasing its application in analyzing complex dynamical systems. The text explores global Koopman modes, temporal autocorrelations, and two partitioning strategies for computing statistical quantities, highlighting the importance of partitioning strategy selection in statistical representation. [Extracted from the article]

Spatial variability and uncertainty associated with soil volumetric moisture content (SVMC) is crucial in moisture prediction accuracy, this paper sets out to address this point of SVMC by developing data-driven model. Grid samples of SVMC covered approximately a 3-ha field during the jointing growth stage of winter wheat, and SVMC were measured by Time Domain Reflectometry (TDR), located in North China Plain, China. Bayesian inference was performed to explore spatial heterogeneity, robustness, transparency, interpretability and uncertainty related to SVMC using python-based PyMC3 combined with Integrated Nested Laplace Approximation with the Stochastic Partial Differential Equation (INLA-SPDE) model. The results showed that the prediction surface of SVMC, the lower and upper limits of 95% credible intervals quantified uncertainty associated with SVMC, cauchy prior of the flexibility and adaptability to obtain state-of-the-art predictive performance is more robust than gaussian prior for SVMC prediction, the transparency and interpretability of SVMC prediction model were revealed by MCMC (Markov-Chain Monte-Carlo) trace plots, KDE (Kernel density estimates), and rank plots. The uncertainty associated with SVMC can explicitly be described using the highest-posterior density interval, the prediction lower and upper limits. [ABSTRACT FROM AUTHOR]

The article delves into the linear stability theory and molecular simulations of nanofilm dewetting, focusing on factors like disjoining pressure, liquid-solid slip, and thermal fluctuations. Molecular dynamics simulations reveal distinct behaviors in nanofilm instability based on slip strength, leading to the derivation of a new stochastic lubrication equation for strong-slip dewetting. The study aims to predict experimental outcomes and offers insights into the behavior of thin liquid films on substrates with varying slip lengths. Additionally, the document provides a list of scientific research articles exploring various aspects of fluid mechanics, including topics like stick-slip motion, thin film rupture, and nanoscale flows, which may be valuable for researchers interested in fluid dynamics and surface phenomena. [Extracted from the article]

In this paper, we will show two approaches to analyze the dynamics of a stochastic partial differential equation (PDE) with long time memory, which does not generate a random dynamical system and, consequently, the general theory of random attractors is not applicable. On the one hand, we first approximate the stochastic PDEs by a random one via replacing the white noise by a colored one. The resulting random equation does generate a random dynamical system which possesses a random attractor depending on the covariance parameter of the colored noise. On the other hand, we define a mean random dynamical system via the solution operator and prove the existence and uniqueness of weak pullback mean random attractors when the problem is driven by a more general white noise. [ABSTRACT FROM AUTHOR]

We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances in several numerical experiments. [ABSTRACT FROM AUTHOR]

We consider a class of linear Vlasov partial differential equations driven by Wiener noise. Different types of stochastic perturbations are treated: additive noise, multiplicative Itô and Stratonovich noise, and transport noise. We propose to employ splitting integrators for the temporal discretization of these stochastic partial differential equations. These integrators are designed in order to preserve qualitative properties of the exact solutions depending on the stochastic perturbation, such as preservation of norms or positivity of the solutions. We provide numerical experiments in order to illustrate the properties of the proposed integrators and investigate mean-square rates of convergence. [ABSTRACT FROM AUTHOR]

We investigated a novel stochastic fractional partial differential equation (FPDE) characterized by a mixed operator that integrated the standard Laplacian, the fractional Laplacian, and the gradient operator. The equation was driven by a random noise, which admitted a covariance measure structure with respect to the time variable and behaved as a Wiener process in space. Our analysis included establishing the existence of a solution in the general case and deriving an explicit form for its covariance function. Additionally, we delved into a specific case where the noise was modeled as a generalized fractional Brownian motion (gfBm) in time, with a particular emphasis on examining the regularity of the solution's sample paths. [ABSTRACT FROM AUTHOR]

In this paper, approximate solutions for stochastic Fitzhugh–Nagumo partial differential equations are obtained using two‐dimensional shifted Legendre polynomial (2DSLP) approximation. The problem's suitability and solvability are confirmed. The convergence analysis for the proposed methodology and the error analysis in the L2$$ {L}&#x0005E;2 $$ norm are carried out. Using Maple software, an algorithm is created and implemented to arrive at the numerical solution. The solution thus obtained is compared with the exact solution and the solution obtained using the explicit order RK1.5 method. [ABSTRACT FROM AUTHOR]

Accurate forecasting of high-resolution particulate matter 2.5 (PM2.5) levels is essential for the development of public health policy. However, datasets used for this purpose often contain missing observations. This study presents a two-stage approach to handle this problem. The first stage is a multivariate spatial time series (MSTS) model, used to generate forecasts for the sampled spatial units and to impute missing observations. The MSTS model utilizes the similarities between the temporal patterns of the time series of the spatial units to impute the missing data across space. The second stage is the high-resolution prediction model, which generates predictions that cover the entire study domain. The second stage faces the big N problem giving rise to complex memory and computational problems. As a solution to the big N problem, we propose a Gaussian Markov random field (GMRF) for innovations with the Matérn covariance matrix obtained from the corresponding Gaussian field (GF) matrix by means of the stochastic partial differential equation (SPDE) method and the finite element method (FEM). For inference, we propose Bayesian statistics and integrated nested Laplace approximation (INLA) in the R-INLA package. The above approach is demonstrated using daily data collected from 13 PM2.5 monitoring stations in Jakarta Province, Indonesia, for 1 January–31 December 2022. The first stage of the model generates PM2.5 forecasts for the 13 monitoring stations for the period 1–31 January 2023, imputing missing data by means of the MSTS model. To capture temporal trends in the PM2.5 concentrations, the model applies a first-order autoregressive process and a seasonal process. The second stage involves creating a high-resolution map for the period 1–31 January 2023, for sampled and non-sampled spatiotemporal units. It uses the MSTS-generated PM2.5 predictions for the sampled spatiotemporal units and observations of the covariate's altitude, population density, and rainfall for sampled and non-samples spatiotemporal units. For the spatially correlated random effects, we apply a first-order random walk process. The validation of out-of-sample forecasts indicates a strong model fit with low mean squared error (0.001), mean absolute error (0.037), and mean absolute percentage error (0.041), and a high R² value (0.855). The analysis reveals that altitude and precipitation negatively impact PM2.5 concentrations, while population density has a positive effect. Specifically, a one-meter increase in altitude is linked to a 7.8% decrease in PM2.5, while a one-person increase in population density leads to a 7.0% rise in PM2.5. Additionally, a one-millimeter increase in rainfall corresponds to a 3.9% decrease in PM2.5. The paper makes a valuable contribution to the field of forecasting high-resolution PM2.5 levels, which is essential for providing detailed, accurate information for public health policy. The approach presents a new and innovative method for addressing the problem of missing data and high-resolution forecasting. [ABSTRACT FROM AUTHOR]

We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have q players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form Ito-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents { γ 1 , γ 2 , ... } under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI. [ABSTRACT FROM AUTHOR]

For temperatures below and beyond the Curie temperature, the stochastic Landau-Lifshitz-Bloch equation describes the evolution of spins in ferromagnetic materials. In this work, we consider the stochastic Landau-Lifshitz-Bloch equation driven by a real valued Wiener process and show Wong-Zakai type approximations for the same. We consider non-zero contribution from the helicity term in the energy. First, using a Doss-Sussmann type transform, we convert the stochastic partial differential equation into a deterministic equation with random coefficients. We then show that the solution of the transformed equation depends continuously on the driving Wiener process. We then use this result, along with the properties of the said transform to show that the solution of the originally considered equation depends continuously on the driving Wiener process. [ABSTRACT FROM AUTHOR]

The two-dimensional stochastic Euler equations (EEs) perturbed by a linear multiplicative noise of Itô type on the bounded domain O have been considered in this work. Our first aim is to prove the existence of global weak (analytic) solutions for stochastic EEs when the divergence-free initial data u* ∈ H1(O), and the external forcing f ∈ L²loc(R;H¹(O)). In order to prove the existence of weak solutions, a vanishing viscosity technique has been adopted. In addition, if curl ... and curl ..., we establish that the global weak (analytic) solution is unique. This work appears to be the first one to discuss the existence and uniqueness of global weak (analytic) solutions for stochastic EEs driven by linear multiplicative noise. Second, we prove the existence of a pullback stochastic weak attractor for stochastic nonautonomous EEs using the abstract theory available in the literature. Finally, we propose an abstract theory for weak asymptotic autonomy of pullback stochastic weak attractors. Then we consider the 2D stochastic EEs perturbed by a linear multiplicative noise as an example to discuss how to prove the weak asymptotic autonomy for concrete stochastic partial differential equations. As EEs do not contain any dissipative term, the results on attractors (deterministic and stochastic) are available in the literature for dissipative (or damped) EEs only. Since we are considering stochastic EEs without dissipation, all the results of this work for 2D stochastic EEs perturbed by a linear multiplicative noise are totally new. [ABSTRACT FROM AUTHOR]

We present a neural network approach for approximating the value function of high-dimensional stochastic control problems. Our training process simultaneously updates our value function estimate and identifies the part of the state space likely to be visited by optimal trajectories. Our approach leverages insights from optimal control theory and the fundamental relation between semilinear parabolic partial differential equations and forward-backward stochastic differential equations. To focus the sampling on relevant states during neural network training, we use the stochastic Pontryagin maximum principle (PMP) to obtain the optimal controls for the current value function estimate. By design, our approach coincides with the method of characteristics for the non-viscous Hamilton--Jacobi--Bellman equation arising in deterministic control problems. Our training loss consists of a weighted sum of the objective functional of the control problem and penalty terms that enforce the HJB equations along the sampled trajectories. Importantly, training is unsupervised in that it does not require solutions of the control problem. Our numerical experiments highlight our scheme's ability to identify the relevant parts of the state space and produce meaningful value estimates. Using a two-dimensional model problem, we demonstrate the importance of the stochastic PMP to inform the sampling and compare it to a finite element approach. With a nonlinear control affine quadcopter example, we illustrate that our approach can handle complicated dynamics. For a 100-dimensional benchmark problem, we demonstrate that our approach improves accuracy and time-to-solution, and, via a modification, we show the wider applicability of our scheme. [ABSTRACT FROM AUTHOR]

This paper implements the stochastic filters for state and parameter estimation of nonuniform transmission lines (NTL). In general, transmission line (TL) problem is a continuous time and space problem. By taking the line loading noise into account, the TL equations become a stochastic partial differential equation (PDE) rather than a simple set of coupled finite stochastic differential equations (SDE). By transforming the spatial variables into the Fourier domain, the stochastic PDE can be transformed into an infinite sequence of SDE. After truncation to a finite set of Fourier series coefficients, it becomes a finite set of coupled linear SDE, which is the required domain in which extended Kalman filter (EKF) and unscented Kalman filter (UKF) can be applied. For state space equation of EKF and UKF, the voltage and current of periodic NTL are expanded into an infinite set of spatial harmonics. In this way, the voltage and current measurement of NTL become an eigenvalue problem. The NTL is considered as cascade of small infinite NTL and the four distributed primary parameters of the periodic NTL are expressed using Fourier series expansion. Finally, the Kalman filter (KF)-based state estimation and the EKF- and UKF-based parameter estimation have been compared with recursive least squares (RLS) method. The simulation results present the superiority of the method. [ABSTRACT FROM AUTHOR]

In this article, we explore Freidlin-Wentzell's large deviation principle for a stochastic partial differential equation with the nonlinear diffusion-convection operator in divergence form satisfying p-type growth, involving coercivity assumptions, perturbed by small multiplicative Brownian noise. We use the weak convergence method to prove the Laplace principle, which is equivalent to the large deviation principle in our framework. [ABSTRACT FROM AUTHOR]

Forward–backward doubly stochastic differential equations (FBDSDEs) serve as a probabilistic interpretation of stochastic partial differential equations (SPDEs) with diverse applications. Coupled FBDSDEs encounter numerous challenges in numerical approximation compared to forward–backward stochastic differential equations (FBSDEs) and decoupled FBDSDEs, including ensuring the measurability of the numerical solutions, accounting for the mutual influences between forward and backward processes, and considering the relationship with respect to SPDEs rather than PDEs. This paper introduces, for the first time, a numerical method for solving multi-dimensional coupled FBDSDEs. By integrating an optimal control-based approach with deep neural networks, it effectively addresses the coupling-related challenges between forward and backward equations. Computational examples of coupled FBDSDEs with explicit solutions demonstrate that the proposed deep learning-based numerical algorithm achieves commendable performance in terms of both accuracy and efficiency. [ABSTRACT FROM AUTHOR]

We study nonlinear stochastic partial differential equations with Wick-analytic type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fisher–KPP equations, stochastic Allen–Cahn, stochastic Newell–Whitehead–Segel, and stochastic Fujita–Gelfand equations. By implementing the theory of $ C_0- $ C 0 − semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove the existence and uniqueness of solutions for this class of stochastic partial differential equations. [ABSTRACT FROM AUTHOR]

Climate change significantly impacts marine ecosystems worldwide, leading to alterations in the composition and structure of marine communities. In this study, we aim to explore the effects of temperature on demersal fish communities in the Central Mediterranean Sea, using data collected from a standardized monitoring program over 23 years. Computationally efficient Bayesian inference is performed using the integrated nested Laplace approximation and the stochastic partial differential equation approach to model the spatial and temporal dynamics of the fish communities. We focused on the mean temperature of the catch (MTC) as an indicator of the response of fish communities to changes in temperature. Our results showed that MTC decreased significantly with increasing depth, indicating that deeper fish communities may be composed of colder affinity species, more vulnerable to future warming. We also found that MTC had a step-wise rather than linear increase with increasing water temperature, suggesting that fish communities may be able to adapt to gradual changes in temperature up to a certain threshold before undergoing abrupt changes. Our findings highlight the importance of considering the non-linear dynamics of fish communities when assessing the impacts of temperature on marine ecosystems and provide important insights into the potential impacts of climate change on demersal fish communities in the Central Mediterranean Sea. [ABSTRACT FROM AUTHOR]

Stochastic optimal control and games have a wide range of applications, from finance and economics to social sciences, robotics, and energy management. Many real-world applications involve complex models that have driven the development of sophisticated numerical methods. Recently, computational methods based on machine learning have been developed for solving stochastic control problems and games. In this review, we focus on deep learning methods that have unlocked the possibility of solving such problems, even in high dimensions or when the structure is very complex, beyond what traditional numerical methods can achieve. We consider mostly the continuous time and continuous space setting. Many of the new approaches build on recent neural-network-based methods for solving high-dimensional partial differential equations or backward stochastic differential equations, or on model-free reinforcement learning for Markov decision processes that have led to breakthrough results. This paper provides an introduction to these methods and summarizes the state-of-the-art works at the crossroad of machine learning and stochastic control and games. [ABSTRACT FROM AUTHOR]

On a d-dimensional Riemannian, spin manifold (M, g) we consider non-linear, stochastic partial differential equations for spinor fields, driven by a Dirac operator and coupled to an additive Gaussian, vector-valued white noise. We extend to the case in hand a procedure, introduced in Dappiaggi et al (Commun Contemp Math 27(07):2150075, 2022), for the scalar counterpart, which allows to compute at a perturbative level the expectation value of the solutions as well as the associated correlation functions accounting intrinsically for the underlying renormalization freedoms. This framework relies strongly on tools proper of microlocal analysis and it is inspired by the algebraic approach to quantum field theory. As a concrete example we apply it to a stochastic version of the Thirring model proving in particular that it lies in the subcritical regime if d ≤ 2 . [ABSTRACT FROM AUTHOR]