Nonlinear Stochastic Equations Research Articles

Direct computation of nth-order correlations of the solution of a non-linear stochastic equation

SummaryA system of nonlinear stochastic equations excited by a Gaussian random term leading to a statistically stationary solution is considered. The Carleman linearization is used to handle the nonlinearity and the statistical characterization of the solution is formulated in terms of a sequence of correlations of increasing order. Truncated in a finite but arbitrary order, the problem leads to a linear system of equations yielding directly the correlations. It is demonstrated in two examples that, if the excitation is not too strong, the solution converges as a function of the truncation order and provides an alternative to the Monte Carlo approach consisting in ensemble averaging a large number of time-dependent random solutions. For a low-dimensional system, it is shown that replacing the tensor indexing by a numbering accounting for redundancies makes it possible to keep the total problem dimension within reasonable limits even if relatively high-order correlations are accounted for. A model reduction based on the singular value decomposition of the second-order correlation matrix is tested with success for a case of a partial differential equation (Burgers’ equation), showing that the method can be potentially applied even to high-dimensional systems originating in partial differential equations.

Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two

We study the Gaussian fluctuations of a nonlinear stochastic heat equation in spatial dimension two. The equation is driven by a Gaussian multiplicative noise. The noise is white in time, smoothed in space at scale \(\varepsilon \), and tuned logarithmically by a factor \(\frac{1}{\sqrt{\log \varepsilon ^{-1}}}\) in its strength. We prove that, after centering and rescaling, the solution random field converges in distribution to an Edwards-Wilkinson limit as \(\varepsilon \downarrow 0\). The tool we used here is the Malliavin-Stein’s method. We also give a functional version of this result.

Controllability of Stochastic Delay Systems Driven by the Rosenblatt Process

In this work, we consider dynamical systems of linear and nonlinear stochastic delay-differential equations driven by the Rosenblatt process. With the aid of the delayed matrix functions of these systems, we derive the controllability results as an application. By using a delay Gramian matrix, we provide sufficient and necessary criteria for the controllability of linear stochastic delay systems. In addition, by employing Krasnoselskii’s fixed point theorem, we present some necessary criteria for the controllability of nonlinear stochastic delay systems. Our results improve and extend some existing ones. Finally, an example is given to illustrate the main results.

Nonlinear fractional stochastic heat equation driven by Gaussian noise rough in space

In this paper we consider a class of nonlinear fractional stochastic heat equation∂∂tu(t,x)=Dδαu(t,x)+σ(u(t,x))W˙(t,x),(t,x)∈[0,T]×R, in (1+1)-dimension with T>0, where Dδα is a nonlocal fractional differential operator with α∈(1,2) and |δ|≤2−α. The diffusion coefficient σ(⋅) is a measurable function. W˙ is a Gaussian noise which is white in time and behaves as a fractional Brownian motion with Hurst index H satisfying 3−α4<H<12 in the space variable. Under some mild assumptions, we prove the existence and uniqueness of the mild solution in some function spaces. Along the way, we study the moment bounds for the solution and show that the solution is weakly full intermittent.

General decay stability of backward Euler–Maruyama method for nonlinear stochastic integro-differential equations

The general decay stability of backward Euler–Maruyama(EM) method is discussed for stochastic integro-differential equations(SIDEs) in this paper. Under polynomial growth condition, both continuous and discrete solutions admit high nonlinearity of the underlying equations. Besides, the considered stability namely the general decay stability is a generalized stability including almost sure polynomially stability and almost sure exponentially stability. Making using of the nonnegative semi-martingale convergence theorem, for SIDEs we propose sufficient conditions to obtain the general decay stability of backward EM method.

Generalized Space-Time Fractional Stochastic Kinetic Equation

In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in Rd with Gaussian noise which is white in time and homogeneous in space. This type of equation constitutes an extension of the nonlinear stochastic heat equation involving fractional derivatives in time and fractional Laplacian in space. We firstly give a necessary condition on the spatial covariance for the existence and uniqueness of the solution. Furthermore, we also study various properties of the solution, such as Hölder regularity, the upper bound of second moment, and the stationarity with respect to the spatial variable in the case of linear additive noise.

On the Impact of Noise on Hyperbolic-Type Traveling Wave Solutions of Some Stochastic Evolution Equations

In this paper, traveling wave solutions of some nonlinear stochastic evolution equations emerging in miscellaneous fields such as modeling of flame propagation, magneto-acoustic waves in plasma and small-amplitude water waves with surface tension are investigated. By means of Galilean transformation and tanh method, we obtain some exact solutions such as kink wave solution, solitary wave solution and periodic solution. To illustrate the impact of noise on the solutions, we assigned different noise functions for the external noise. The results showed that the waveform deforms as the noise intensity increases.

Wellposedness and stability of fractional stochastic nonlinear heat equation in Hilbert space

In this work, we interest in the study of the wellposedness and the stability of fractional stochastic nonlinear heat equation in the Hilbert space $$ L^{2}(0,1) $$ ; perturbed by a trace-class noise and driven by the fractional Laplacian. Precisely, we use the fixed point theorem to prove the wellposedness of the problem. Moreover, we prove the $$ p^{th} $$ -moment exponential stability and the almost surely exponential stability by imposing an additional assumption. Some examples are considered in order to confirm and support the validity of our theoretical results.

Sampling constrained stochastic trajectories using Brownian bridges.

We present a new method to sample conditioned trajectories of a system evolving under Langevin dynamics based on Brownian bridges. The trajectories are conditioned to end at a certain point (or in a certain region) in space. The bridge equations can be recast exactly in the form of a non-linear stochastic integro-differential equation. This equation can be very well approximated when the trajectories are closely bundled together in space, i.e., at low temperature, or for transition paths. The approximate equation can be solved iteratively using a fixed point method. We discuss how to choose the initial trajectories and show some examples of the performance of this method on some simple problems. This method allows us to generate conditioned trajectories with a high accuracy.

Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence* *The second author is supported by the VIDI subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO).

In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an L p -setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete problems and their quasilinear variants are given. This includes Burgers’ equation, the Allen–Cahn equation, the Cahn–Hilliard equation, reaction–diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. The proofs are based on recent developments in maximal regularity theory for the linearized problem for deterministic and stochastic evolution equations. In particular, our theory can be seen as a stochastic version of the theory of critical spaces due to Prüss–Simonett–Wilke (2018). Sharp weighted time-regularity allow us to deal with rough initial values and obtain instantaneous regularization results. The abstract well-posedness results are obtained by a combination of several sophisticated splitting and truncation arguments.

Three kinds of novel multi-symplectic methods for stochastic Hamiltonian partial differential equations

Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, the splitting technique and the partitioned Runge–Kutta method. Specific numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schrödinger equations, stochastic Korteweg-de Vries equations and stochastic Maxwell equations. We take stochastic wave equations as examples to perform numerical experiments, which indicate the validity of the proposed methods.

Stochastic response determination of U-OWC energy harvesters: a statistical linearization solution treatment accounting for intermittent wave excitation

A novel statistical linearization technique is developed for determining approximately the response statistics and the power output of U-Oscillating Water Column (U-OWC) energy harvesting systems. In this regard, first, the governing equations are derived by employing the unsteady Bernoulli equation. Note that the intermittent, i.e., non-stationary, nature of the wave excitation, occurring in severe sea states due to uncovering of the U-OWC inlet, is explicitly accounted for in the herein proposed model. This is done by multiplying the excitation process with a Heaviside function dependent on the instantaneous free surface displacement. Next, the resulting coupled system of nonlinear integro-differential stochastic equations is solved approximately by relying on a statistical linearization technique. Specifically, the original system of nonlinear equations is replaced by an equivalent linear one, whose parameters and response first- and second-order statistics are obtained by minimizing the mean square error between the two systems. A significant novel aspect of the technique relates to the fact that the Heaviside function is replaced in the equivalent linear system by an “equivalent excitation” coefficient to be determined as part of the statistical linearization solution scheme. Further, compared with other relevant solution schemes adopted in earlier research efforts in the literature, it is shown that the developed technique can be construed as a direct generalization that exhibits an enhanced accuracy degree. The U-OWC installed in the Civitavecchia harbor (Rome, Italy) is considered as an illustrative numerical example, where the reliability of the approximate technique is demonstrated by comparisons with pertinent Monte Carlo simulation data.

Nonlinear stochastic wave equation driven by rough noise

In this paper, we obtain the existence and uniqueness of the strong solution to one (spatial) dimensional stochastic wave equation ∂2u(t,x)∂t2=∂2u(t,x)∂x2+σ(t,x,u(t,x))W˙(t,x) assuming σ(t,x,0)=0, where W˙ is a mean zero Gaussian noise which is white in time and fractional in space with Hurst parameter H∈(1/4,1/2).

Galerkin Approximation for Stochastic Volterra Integral Equations with Doubly Singular Kernels

This paper is concerned with the more general nonlinear stochastic Volterra integral equations with doubly singular kernels, whose singular points include both s=t and s=0. We propose a Galerkin approximate scheme to solve the equation numerically, and we obtain the strong convergence rate for the Galerkin method in the mean square sense. The rate is min{2−2(α1+β1),1−2(α2+β2)} (where α1,α2,β1,β2 are positive numbers satisfying 0&lt;α1+β1&lt;1, 0&lt;α2+β2&lt;12), which improves the results of some numerical schemes for the stochastic Volterra integral equations with regular or weakly singular kernels. Moreover, numerical examples are given to support the theoretical result and explain the priority of the Galerkin method.

On ill-posedness of nonlinear stochastic wave equations driven by rough noise

We highlight a fundamental ill-posedness issue for nonlinear stochastic wave equations driven by a fractional noise. Namely, if the noise becomes too rough (i.e., the sum of its Hurst indexes becomes too small), then there is essentially no hope to provide a systematic interpretation of the model, whether directly or through a Wick-type renormalization procedure. This phenomenon can be compared with the situation of a general SDE driven by a two-dimensional fractional noise of index H≤14.Our results clarify and extend previous similar properties exhibited in Deya (2020) or in Oh and Okamoto (2021).

A forward-backward SDE from the 2D nonlinear stochastic heat equation

We consider a nonlinear stochastic heat equation in spatial dimension $d=2$, forced by a white-in-time multiplicative Gaussian noise with spatial correlation length $\varepsilon>0$ but divided by a factor of $\sqrt{\log\varepsilon^{-1}}$. We impose a condition on the Lipschitz constant of the nonlinearity so that the problem is in the "weak noise" regime. We show that, as $\varepsilon\downarrow0$, the one-point distribution of the solution converges, with the limit characterized in terms of the solution to a forward-backward stochastic differential equation (FBSDE). We also characterize the limiting multipoint statistics of the solution, when the points are chosen on appropriate scales, in similar terms. Our approach is new even for the linear case, in which the FBSDE can be solved explicitly and we recover results of Caravenna, Sun, and Zygouras (Ann. Appl. Probab. 27(5):3050--3112, 2017).

A nonlinear stochastic heat equation with variable thermal conductivity and multiplicative noise

A nonlinear stochastic heat equation with variable thermal conductivity and multiplicative noise

THIRD ORDER ITERATIVE METHOD FOR SOLVING NON-LINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATION IN FINANCIAL APPLICATION

In this paper, third order iterative scheme is presented for working the solution the non-linear stochastic parabolic equation in one dimensional space. First, the given result sphere is discretized by using invariant discretization grid point. Next, by using Taylor series expansion we gain the discretization of the model problem. From this, we gain the system of nonlinear ordinary difference equations. By rearranging this scheme, we gain iterative schemes which is called gauss Jacobean iterative scheme. To validate the convergences of the proposed system, three model illustrations are considered and answered it at each specific grid point on its result sphere. The coincident (convergent) analysis of the present techniques is worked by supported the theoretical and fine statements and the delicacy of the result is attained. The delicacy of the present techniques has been shown in the sense of average absolute error (AAE), root mean square error norm and point-wise maximum absolute error norm and comparing gets crimes in the result attained in literature and these results are also presented in tables and graphs. The physical gets of results between numerical versus are also been presented in terms of graphs. As we can see from the table and graphs, the present system approach are approximates the exact result veritably well and it's relatively effective and virtually well suited for working the solution for non-linear parabolic equation.

On stochastic response of fractional-order generalized birhythmic van der Pol oscillator subjected to delayed feedback displacement and Gaussian white noise excitation

This paper addresses the occurrence of P-bifurcation in a fractional-delay modified birhythmic van der Pol (BVDP) oscillator, for enzyme-substrate reactions in brain waves, under Gaussian white noise excitation. The minimum mean-square error is used to reduce the system to its equivalent integer-order nonlinear stochastic equation. An averaged Itô equation is obtained via the stochastic averaging method, with the amplitude being the solution of the Fokker-Planck-Kolmogorov equation. From the latter, the stationary density functions are found analytically. This helps to predict the appearance of birhythmicity theoretically and shows itself to respond to parameter changes, namely, fractional-orders, fractional coefficient, and noise intensity. There is an agreement between the theoretical solutions and the numerical solution, which confirms the accuracy of our predictions. In general, bifurcation scenarios are dominated by the changes in fractional orders, as strongly supported by the behaviors of the calculated potential barriers.

A game-theoretic perspective to study a nonlinear stochastic parabolic model of population competition

In this paper, a game of competition between two populations is designed. Populations are considered as two players competing against each other to increase their sizes. Players have been provided with costly two-component strategies. Population densities are described by two coupled nonlinear stochastic second-order parabolic equations. We have added stochastic terms to the deterministic model to consider random perturbations. First, we have considered the stochastic payoff functions and proved the game has a unique stochastic Nash equilibrium. Then, we have shown the stochastic Nash equilibrium is also a Nash equilibrium for the game with deterministic payoff functions defined by expectation. Cooperative game is also studied and stochastic Pareto efficient strategies are obtained. Then, the existence and uniqueness analyses of the deterministic Nash equilibrium and deterministic Pareto efficient strategies for the game with deterministic payoff functions are provided. The explicit forms of Nash equilibriums and Pareto efficient strategies are presented. Results about the losses the players experience when they don't cooperate are given. Finally, some numerical experiments are included to illustrate what we have claimed in the statements of theorems and show the effects of the Nash equilibrium and Pareto efficient strategy on the payoffs of the players.