Stochastic Partial Differential Equations Research Articles

From ABC to KPZ

AbstractWe study the equilibrium fluctuations of an interacting particle system evolving on the discrete ring with $$N\in {\mathbb {N}}$$ N ∈ N points, denoted by $${\mathbb {T}}_N$$ T N , and with three species of particles that we name A, B and C, but such that at each site there is only one particle. We prove that proper choices of density fluctuation fields (that match those from nonlinear fluctuating hydrodynamics theory) associated to the (two) conserved quantities converge, in the limit $$N\rightarrow \infty $$ N → ∞ , to a system of stochastic partial differential equations, that can either be the Ornstein–Uhlenbeck equation or the Stochastic Burgers equation. To understand the cross interaction between the two conserved quantities, we derive a general version of the Riemann–Lebesgue lemma which is of independent interest.

Modeling the Causes of Urban Traffic Crashes: Accounting for Spatiotemporal Instability in Cities

Traffic crashes have become one of the key public health issues, triggering significant apprehension among citizens and urban authorities. However, prior studies have often been limited by their inability to fully capture the dynamic and complex nature of spatiotemporal instability in urban traffic crashes, typically focusing on static or purely spatial effects. Addressing this gap, our study employs a novel methodological framework that integrates an Integrated Nested Laplace Approximation (INLA)-based Stochastic Partial Differential Equation (SPDE) model with spatially adaptive graph structures, which enables the effective handling of vast and intricate geospatial data while accounting for spatiotemporal instability. This approach represents a significant advancement over conventional models, which often fail to account for the fluid interplay between time-varying weather conditions, geographical attributes, and crash severity. We applied this methodology to analyze traffic crashes across three major U.S. cities—New York, Los Angeles, and Houston—using comprehensive crash data from 2016 to 2019. Our findings reveal city-specific disparities in the factors influencing severe traffic crashes, which are defined as incidents resulting in at least one person sustaining serious injury or death. Despite some universal trends, such as the risk-enhancing effect of cold weather and pedestrian crossings, we find marked differences across cities in relation to factors like temperature, precipitation, and the presence of certain traffic facilities. Additionally, the adjustment observed in the spatiotemporal standard deviations, with values such as 0.85 for New York and 0.471 for Los Angeles, underscores the varying levels of annual temporal instability across cities, indicating that the fluctuation in crash severity factors over time differs markedly among cities. These results underscore the limitations of traditional modeling approaches, demonstrating the superiority of our spatiotemporal method in capturing the heterogeneity of urban traffic crashes. This work has important policy implications, suggesting a need for tailored, location-specific strategies to improve traffic safety, thereby aiding authorities in better resource allocation and strategic planning.

Pathwise Stochastic Control and a Class of Stochastic Partial Differential Equations

AbstractIn 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.

A remark on randomization of a general function of negative regularity

In the study of partial differential equations (PDEs) with random initial data and singular stochastic PDEs with random forcing, we typically decompose a classically ill-defined solution map into two steps, where, in the first step, we use stochastic analysis to construct various stochastic objects. The simplest kind of such stochastic objects is the Wick powers of a basic stochastic term (namely a random linear solution, a stochastic convolution, or their sum). In the case of randomized initial data of a general function of negative regularity for studying nonlinear wave equations (NLW), we show necessity of imposing additional Fourier–Lebesgue regularity for constructing Wick powers by exhibiting examples of functions slightly outside L 2 ( T d ) L^2(\mathbb {T}^d) such that the associated Wick powers do not exist. This shows that probabilistic well-posedness theory for NLW with general randomized initial data fails in negative Sobolev spaces (even with renormalization). Similar examples also apply to stochastic NLW and stochastic nonlinear heat equations with general white-in-time stochastic forcing, showing necessity of appropriate Fourier–Lebesgue γ \gamma -radonifying regularity in the construction of the Wick powers of the associated stochastic convolution.

An exponential stochastic Runge–Kutta type method of order up to 1.5 for SPDEs of Nemytskii-type

Abstract For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge–Kutta type that allows for convergence with a temporal order of up to $\frac{3}/{2}$ and that can be combined with several spatial discretizations. The developed family of derivative-free schemes is tailored to stochastic partial differential equations of Nemytskii-type, i.e., with pointwise multiplicative noise operators. We prove the strong convergence of these schemes in the root mean-square sense and present some numerical examples that reveal the theoretical results.

Spatial variability and uncertainty associated with soil moisture content using INLA-SPDE combined with PyMC3 probability programming

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.

Instabilities and Pattern Formation in Epidemic Spread Induced by Nonlinear Saturation Effects and Ornstein–Uhlenbeck Noise

Abstract We analytically study the emergence of instabilities and the consequent steady-state pattern formation in a stochastic partial differential equation (PDE) based, compartmental model of spatiotemporal epidemic spread. The model is characterized by: (1) strongly nonlinear forces representing the infection transmission mechanism and (2) random environmental forces represented by the Ornstein–Uhlenbeck (O–U) stochastic process which better approximates real-world uncertainties. Employing second-order perturbation analysis and computing the local Lyapunov exponent, we find the emergence of diffusion-induced instabilities and analyze the effects of O–U noise on these instabilities. We obtain a range of values of the diffusion coefficient and correlation time in parameter space that support the onset of instabilities. Notably, the stability and pattern formation results depend critically on the correlation time of the O–U stochastic process; specifically, we obtain lower values of steady-state infection density for higher correlation times. Also, for lower correlation times the results approach those obtained in the white noise case. The analytical results are valid for lower-order correlation times. In summary, the results provide insights into the onset of noise-induced, and Turing-type instabilities in a stochastic PDE epidemic model in the presence of strongly nonlinear deterministic infection forces and stochastic environmental forces represented by Ornstein–Uhlenbeck noise.

Invariant cones for jump-diffusions in infinite dimensions

In this paper we provide sufficient conditions for stochastic invariance of closed convex cones for stochastic partial differential equations (SPDEs) of jump-diffusion type, and clarify when these conditions are necessary. We emphasize that the diffusion coefficient of the SPDE does not need to be smooth, an assumption which is frequently imposed when dealing with stochastic invariance problems. Our results apply to the positive cone of abstract L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-spaces. Furthermore, we present a series of applications, where we investigate SPDEs arising in natural sciences and economics.

Efficient Modeling of Spatial Extremes over Large Geographical Domains

Various natural phenomena exhibit spatial extremal dependence at short spatial distances. However, existing models proposed in the spatial extremes literature often assume that extremal dependence persists across the entire domain. This is a strong limitation when modeling extremes over large geographical domains, and yet it has been mostly overlooked in the literature. We here develop a more realistic Bayesian framework based on a novel Gaussian scale mixture model, with the Gaussian process component defined by a stochastic partial differential equation yielding a sparse precision matrix, and the random scale component modeled as a low-rank Pareto-tailed or Weibull-tailed spatial process determined by compactly-supported basis functions. We show that our proposed model is approximately tail-stationary and that it can capture a wide range of extremal dependence structures. Its inherently sparse structure allows fast Bayesian computations in high spatial dimensions based on a customized Markov chain Monte Carlo algorithm prioritizing calibration in the tail. We fit our model to analyze heavy monsoon rainfall data in Bangladesh. Our study shows that our model outperforms natural competitors and that it fits precipitation extremes well. We finally use the fitted model to draw inference on long-term return levels for marginal precipitation and spatial aggregates.

A note on invariant manifolds for stochastic partial differential equations in the framework of the variational approach

In this note we provide conditions for local invariance of finite dimensional submanifolds for solutions to stochastic partial differential equations (SPDEs) in the framework of the variational approach. For this purpose, we provide a connection to SPDEs in continuously embedded spaces.

Warning signs for boundary noise and their application to an ocean Boussinesq model

In this paper, we construct and discuss early-warning signs of the approach of a parameter to a deterministic bifurcation on a stochastic partial differential equation (SPDE) model with Gaussian white-noise on the boundary. We specifically focus on the qualitative behaviour of the time-asymptotic autocovariance and autocorrelation of the solutions of the linearized system. We also discuss the reliability of the tools from an analytic perspective and through various examples. Among those, the application of the early-warning signs for an ocean Boussinesq model is explored through numerical simulations. The analytic results obtained expand on previous work and present valuable early-warning signs for various applications.

Stochastic Burgers type equations with two reflecting walls: Existence, uniqueness and exponential ergodicity

In this paper, we study stochastic Burgers type equations with two reflecting walls driven by multiplicative noise. We first establish the existence and uniqueness of the solution by Picard iteration technique, the main trouble is to handle the complicated nonlinear term ∂g(u(x,t))∂x which involves the derivative of the solution u(x, t) and the singularities due to reflection. Furthermore, we prove the exponential ergodicity for the equations driven by possibly degenerate, multiplicative noise through asymptotic coupling. More generally, we also give a simplified criterion for exponential ergodicity of parabolic stochastic partial differential equations with reflections driven by degenerate multiplicative noise.

Dynamics of stochastic differential equations with memory driven by colored noise.

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.

A frequency-independent bound on trigonometric polynomials of Gaussians and applications

We prove a frequency-independent bound on trigonometric functions of a class of singular Gaussian random fields, which arise naturally from weak universality problems for singular stochastic PDEs. This enables us to reduce the regularity assumption on the nonlinearity of the microscopic models (for pathwise convergence) in KPZ and dynamical Φ34 in the previous works of Hairer-Xu and Furlan-Gubinelli to heuristically optimal thresholds required by PDE structures.

Variational methods for some singular stochastic elliptic PDEs

Variational methods for some singular stochastic elliptic PDEs

A new pricing method for integrated energy systems based on geometric Brownian motions under the risk-neutral measure

An integrated energy system (IES) is a promising energy-saving technology. With the advancement of technology and deregulation of the energy market, numerous distributed energy systems (DESs) will be accessed in the IES with various trading modes in the future. This makes it even more challenging to evaluate the fair energy selling price of IES, which is of great concern to market regulators and participants. To address this issue, this study proposes a risk-neutral pricing method for IES based on geometric Brownian motions. First, geometric Brownian motions are introduced to characterize the uncertainty feature of the energy price of the DESs. Then, we propose a stochastic partial differential equation (SPDE) for the energy price of an IES by utilizing the Ito lemma and the properties of the martingale under the risk-neutral measure. Finally, sensitivity analyses of the proposed SPDE on the volatilities and two types of correlation coefficients are conducted by implementing the proposed model in the given scenario.

General quantum-classical dynamics as measurement based feedback

This note derives the stochastic differential equations and partial differential equation of general hybrid quantum–classical dynamics from the theory of continuous measurement and general (non-Markovian) feedback. The advantage of this approach is an explicit parameterization, without additional positivity constraints. The construction also neatly separates the different effects: how the quantum influences the classical and how the classical influences the quantum. This modular presentation gives a better intuition of what to expect from hybrid dynamics, especially when used to construct possibly fundamental theories.

Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps

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 {γ1,γ2,…} 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.

A Microlocal Investigation of Stochastic Partial Differential Equations for Spinors with an Application to the Thirring Model

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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\le 2$$\\end{document}.

Global Existence, Uniqueness, and Non-negativity of the Solutions to Stochastic Partial Differential Equations of Infectious Diseases

Global Existence, Uniqueness, and Non-negativity of the Solutions to Stochastic Partial Differential Equations of Infectious Diseases