Stochastic Reaction-diffusion Equations Research Articles

Approximation of optimal feedback controls for stochastic reaction-diffusion equations

In this paper we present a method to approximate optimal feedback controls for stochastic reaction-diffusion equations. We derive two approximation results providing the theoretical foundation of our approach and allowing for explicit error estimates. The approximation of optimal feedback controls by neural networks is discussed as an explicit application of our method. We illustrate our findings in the case of a linear quadratic control problem with a numerical example.

Pullback Measure Attractors for Non-autonomous Fractional Stochastic Reaction-Diffusion Equations on Unbounded Domains

Pullback Measure Attractors for Non-autonomous Fractional Stochastic Reaction-Diffusion Equations on Unbounded Domains

A non-autonomous framework for climate change and extreme weather events increase in a stochastic energy balance model

We develop a three-timescale framework for modeling climate change and introduce a space-heterogeneous one-dimensional energy balance model. This model, addressing temperature fluctuations from rising carbon dioxide levels and the super-greenhouse effect in tropical regions, fits within the setting of stochastic reaction–diffusion equations. Our results show how both mean and variance of temperature increase, without the system going through a bifurcation point. This study aims to advance the conceptual understanding of the extreme weather events frequency increase due to climate change.

Novel solitary wave solutions for stochastic nonlinear reaction–diffusion equation with multiplicative noise

The primary objective of this paper is to explore solitary wave solutions within the context of the stochastic Reaction–Diffusion equation, a problem that has not been previously investigated with respect to multiplicative noise. This marks a novel aspect of the current study. To accomplish this goal, we utilize two distinct integration methods: the extended Kudryashov’s method and the G′G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( \\frac{G'}{G}\\right) $$\\end{document}-expansion method. These methods are chosen for their efficacy in handling nonlinear differential equations and are applied in tandem to uncover new trigonometric and hyperbolic stochastic exact solutions. The employment of these methods is crucial in generating solutions that have not been previously identified in this particular context. Subsequently, to assess the validity and usefulness of these newly discovered solutions, we conduct numerical simulations. These simulations involve implementing the derived solutions across a variety of scenarios or samples, allowing for an evaluation of their performance and applicability. This paper represents a significant contribution to the field of stochastic Reaction–Diffusion equations by introducing a pioneering investigation into the effects of multiplicative noise on solitary wave solutions. Furthermore, it offers a fresh perspective on obtaining exact solutions in the presence of such noise, showcasing the potential of the integration methods in tackling complex nonlinear stochastic systems.

Large Deviations of Invariant Measures of Stochastic Reaction–Diffusion Equations on Unbounded Domains

This paper is concerned with the large deviation principle of invariant measures of the stochastic reaction–diffusion equation with polynomial drift driven by additive noise defined on the entire space Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document}. Since the standard Sobolev embeddings on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} are not compact and the spectrum of the Laplace operator on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} are not discrete, there are many issues for proving the large deviations of invariant measures in the case of unbounded domains, including the difficulties for proving the compactness of the level sets of rate functions, the uniform Dembo–Zeitouni large deviations on compact sets as well as the exponential tightness on compact sets. Currently, there is no result available in the literature on the large deviations of invariant measures for stochastic PDEs on unbounded domains, and this paper is the first one to deal with this issue. The non-compactness of the standard Sobolev embeddings on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} is circumvented by the idea of uniform tail-ends estimates together with the arguments of weighted spaces.

High-order stability for a stochastic reaction-diffusion equation under random fluctuation on ℝ N

In this article, we systematically study the high-order stability for a stochastic reaction-diffusion equation under random fluctuation. We obtain the convergence of solutions and upper semi-continuity of random attractors for the fluctuation equations in the initial space L 2(ℝ N ), without an additional assumption on the nonlinearity. By means of a splitting method, we technically prove that the L p ∩H 1-difference of solutions is controlled by the L 2-difference of initial values near the initial time, where p , p > 2 , is the dissipation exponent of the nonlinearity. This along with the L 2-stability is sufficient for us to establish the L p ∩H 1-stability of random attractors. By utilizing some iteration arguments, we show that the solution is uniformly bounded on a bounded interval near the initial time in L δ (ℝ N ) for arbitrary δ > p, in which space the convergence of solutions and the upper semi-continuity of random attractors are also established.

Large Deviation Principle for Stochastic Reaction–Diffusion Equations with Superlinear Drift on $$\mathbb {R}$$ Driven by Space–Time White Noise

Large Deviation Principle for Stochastic Reaction–Diffusion Equations with Superlinear Drift on $$\mathbb {R}$$ Driven by Space–Time White Noise

Asymptotic Behavior of Stochastic Reaction–Diffusion Equations

In this paper, we concentrate on the propagation dynamics of stochastic reaction–diffusion equations, including the existence of travelling wave solution and asymptotic wave speed. Based on the stochastic Feynman–Kac formula and comparison principle, the boundedness of the solution of stochastic reaction–diffusion equations can be obtained so that we can construct a sup-solution and a sub-solution to estimate the upper bound and the lower bound of wave speed.

Pullback measure attractors for non-autonomous stochastic reaction-diffusion equations on thin domains

This paper is concerned with pullback measure attractors of the non-autonomous stochastic reaction-diffusion equations defined in thin domains. We first prove the existence and uniqueness of pullback measure attractors for the inhomogeneous Markov process associated with the stochastic equations. We also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback measure attractors. Then we establish the upper semi-continuity of these attractors as a family of thin domains collapses into a lower-dimensional domain.

Effective splitting of invariant measures for a stochastic reaction diffusion equation with multiplicative noise

This work is concerned with the effective dynamics for the stochastic reaction diffusion equations with cubic nonlinearity driven by a multiplicative noise. By splitting the solution into the finite dimension kernel space and its complement space with some appropriate multi-scale, it derives the dominant solution and the effective invariant measure in the sense of the Wasserstein distance, which capture the complex dynamical behaviors of the original system as a singular parameter is enough small. Furthermore, the effective invariant measure is decomposed to a product of two invariant measures.

Reaction-diffusion equations on metric graphs with edge noise

We investigate stochastic reaction-diffusion equations on finite metric graphs. On each edge in the graph a multiplicative cylindrical Gaussian noise driven reaction-diffusion equation is given. The vertex conditions are the standard continuity and generalized, non-local Neumann-Kirchhoff-type law in each vertex. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. The model is a generalization of the problem in [14] where polynomials with much more restrictive assumptions are considered and no first order differential operator is involved. We utilize the semigroup approach from [15] to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph.

Model reduction for stochastic systems with nonlinear drift

In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such nonlinearities in high dimensional settings occur, e.g., when stochastic reaction diffusion equations are discretized in space. We provide a brief discussion around existence, uniqueness and stability of solutions. (Almost) stability then is the basis for new concepts of Gramians that we introduce and study in this work. With the help of these Gramians, dominant subspace is identified leading to a balancing related highly accurate reduced order SDE. We provide an algebraic error criterion and an error analysis of the propose model reduction schemes. The paper is concluded by applying our method to spatially discretized reaction diffusion equations.

Schauder estimates for stationary and evolution equations associated to stochastic reaction-diffusion equations driven by colored noise

We consider stochastic reaction-diffusion equations with colored noise on the space of real-valued and continuous functions on a compact subset of ℝ d for d = 1 , 2 , 3 . We prove Schauder-type estimates, which will depend on the color of the noise, for the stationary and evolution problems associated with the corresponding transition semigroup.

Random exponential attractor for a stochastic reaction-diffusion equation in $L^{2p}(D)$

AbstractIn this paper, we establish some sufficient conditions for the existence of a random exponential attractor for a random dynamical system in a Banach space. As an application, we consider a stochastic reaction-diffusion equation with multiplicative noise. We show that the random dynamical system $\phi(t,\omega)$ ϕ ( t , ω ) generated by this stochastic reaction-diffusion equation is uniformly Fréchet differentiable on a positively invariant random set in $L^{2p}(D)$ L 2 p ( D ) and satisfies the conditions of the abstract result, then we obtain the existence of a random exponential attractor in $L^{2p}(D)$ L 2 p ( D ) , where p is the growth of the nonlinearity satisfying $1< p\leq 3$ 1 < p ≤ 3 .

Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional

<abstract><p>In this paper, the asymptotic behavior of solutions to a fractional stochastic nonlocal reaction-diffusion equation with polynomial drift terms of arbitrary order in an unbounded domain was analysed. First, the stochastic equation was transformed into a random one by using a stationary change of variable. Then, we proved the existence and uniqueness of solutions for the random problem based on pathwise uniform estimates as well as the energy method. Finally, the existence of a unique pullback attractor for the random dynamical system generated by the transformed equation is shown.</p></abstract>

Quasi-Ergodicity of transient patterns in stochastic reaction-diffusion equations

Quasi-Ergodicity of transient patterns in stochastic reaction-diffusion equations

Topological dimensions of random attractors for a stochastic reaction-diffusion equation with delay

The aim of this paper is to obtain an estimation of Hausdorff as well as fractal dimensions of random attractors for a stochastic reaction-diffusion equation with delay. The stochastic equation is firstly transformed into a delayed random partial differential equation by means of a random conjugation, which is then recast into an auxiliary Hilbert space. For the obtained equation, it is firstly proved that it generates a random dynamical system (RDS) in the auxiliary Hilbert space. Then it is shown that the equation possesses random attractors by a uniform estimate of the solution and the asymptotic compactness of the generated RDS. After establishing the variational equation in the auxiliary Hilbert space and the almost surely differentiable properties of the RDS, upper estimates of both Hausdorff and fractal dimensions of the random attractors are obtained.

Importance sampling for stochastic reaction–diffusion equations in the moderate deviation regime

We develop a provably efficient importance sampling scheme that estimates exit probabilities of solutions to small-noise stochastic reaction–diffusion equations from scaled neighborhoods of a stable equilibrium. The moderate deviation scaling allows for a local approximation of the nonlinear dynamics by their linearized version. In addition, we identify a finite-dimensional subspace where exits take place with high probability. Using stochastic control and variational methods we show that our scheme performs well both in the zero noise limit and pre-asymptotically. Simulation studies for stochastically perturbed bistable dynamics illustrate the theoretical results.

Limiting dynamics for fractional stochastic reaction-diffusion equations driven by a Wong–Zakai approximation process on Rn

Limiting dynamics for fractional stochastic reaction-diffusion equations driven by a Wong–Zakai approximation process on Rn

Poisson stable solutions for stochastic PDEs driven by Lévy noise

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 L2-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.