Semilinear Stochastic Partial Differential Equations Research Articles

Lower and upper bounds for the explosion times of a system of semilinear SPDEs

In this paper, we obtain lower and upper bounds for the blow-up times for a system of semilinear stochastic partial differential equations. Under suitable assumptions, lower and upper bounds of the explosive times are obtained by using explicit solutions of an associated system of random partial differential equations and a formula due to Yor. We provide lower and upper bounds for the probability of finite-time blow-up solution as well. The above obtained results are also extended for semilinear SPDEs forced by two-dimensional Brownian motions.

Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise

In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits a unique invariant probability measure. Under the condition that the nonlinearity is once differentiable, the optimal convergence orders of the numerical invariant measures are obtained based on the time-independent weak error, but not relying on the associated Kolmogorov equation. More precisely, the obtained convergence orders are O(λN−γ) in space and O(τγ) in time, where γ∈(0,1] from the assumption ∥Aγ−12Q12∥L2 is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings.

Higher order time discretization method for a class of semilinear stochastic partial differential equations with multiplicative noise

In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly 1 for the approximate solution. The proof is based on new Hölder continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain H1 stability, higher moment H1 stability, L2 stability, and higher moment L2 stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results.

Schauder regularity results in separable Hilbert spaces

We prove Schauder type estimates for solutions of stationary and evolution equations driven by weak generators of transition semigroups associated to a semilinear stochastic partial differential equations with values in a separable Hilbert space.

Propagation of Chaos for Weakly Interacting Mild Solutions to Stochastic Partial Differential Equations

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

The Large Deviation of Semi-linear Stochastic Partial Differential Equation Driven by Brownian Sheet

We prove the the large deviation principle(LDP) for the law of the one-dimensional semilinear stochastic partial differential equations driven by nonlinear multiplicative noise. Firstly, combining the energy estimate and approximation procedure, we obtain the existence of global solution. Then the large deviation principle is obtained via weak convergence method.

Averaging principle for semilinear stochastic partial differential equations involving space–time white noise

We study the averaging principle for a class of semilinear stochastic partial differential equations perturbed by space–time white noise. Using the factorization method and Burkholder’s inequality, the estimation of stochastic integral involving the heat kernel is obtained. Under suitable assumptions, we show that the original stochastic systems can be approximated by the averaged equations.

Large time behaviour of semilinear stochastic partial differential equations perturbed by a mixture of Brownian and fractional Brownian motions

We study the trajectorywise blowup behaviour of a semilinear partial differential equation that is driven by a mixture of multiplicative Brownian and fractional Brownian motion, modelling different types of random perturbations. The linear operator is supposed to have an eigenfunction of constant sign, and we show its influence, as well as the influence of its eigenvalue and of the other parameters of the equation, on the occurrence of a blowup in finite time of the solution. We give estimates for the probability of finite time blowup and of blowup before a given fixed time. Essential tools are the mild and weak form of an associated random partial differential equation.

Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach

Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic processes with arbitrary finitely regular paths. These results are proved by using the notion of \(\Delta \)-power variations, introduced herein, along with the Hölder-Zygmund norms. Consequently, we prove a new central limit theorem for \(\Delta \)-power variations of the iterated integrals of a fractional Brownian motion. These abstract results, besides being of independent interest, in the second part of the paper are applied to estimation of the drift and volatility coefficients of semilinear stochastic partial differential equations in dimension one, driven by an additive Gaussian noise white in time and possibly colored in space. In particular, we solve the earlier conjecture from Cialenco et al. (Stat. Inference Stoch. Process. 23:83-103, 2020) about existence of a nontrivial bias in the estimators derived by naive approximations of derivatives by finite differences. We give an explicit formula for the bias and derive the convergence rates of the corresponding estimators. Theoretical results are illustrated by numerical examples.

On approximations for reflected SDEs and SPDEs with Neumann boundary conditions

In this paper we study an approximation for reflected stochastic differential equations in non-smooth domains by replacing the driving Brownian motion with its smooth approximation and prove that the solutions of approximating equations converge in the mean square to the solution of the Stratonovich reflected stochastic differential equation. We also study the smooth approximation for Neumann-type semilinear stochastic partial differential equations, which is completed through establishing the convergence in $ L^2 $ of backward doubly stochastic differential equations coupled with reflected SDEs.

Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)

This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11-1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of n-th (n=2, 3,...) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).

Uniform large deviations for a class of semilinear stochastic partial differential equations driven by a Brownian sheet

Uniform large deviations for a class of semilinear stochastic partial differential equations driven by a Brownian sheet

An addendum to “Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients”

In this addendum we provide an existence and uniqueness result for mild solutions to semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures in the framework of the semigroup approach with locally monotone coefficients, where the semigroup is allowed to be pseudo-contractive. This improves an earlier paper of the author, where the equation was only driven by Wiener processes, and where the semigroup was only allowed to be a semigroup of contractions.

Lp-regularity theory for semilinear stochastic partial differential equations with multiplicative white noise

We establish the Lp-regularity theory for a semilinear stochastic partial differential equation with multiplicative white noise:du=(aijuxixj+biuxi+cu+b¯i|u|λuxi)dt+σk(u)dwtk,(ω,t,x)∈Ω×(0,∞)×Rd;u(0,⋅)=u0, where λ>0, the set {wtk,k=1,2,…} is a set of one-dimensional independent Wiener processes, and the function u0=u0(ω,x) is nonnegative random initial data. The coefficients aij,bi,c depend on (ω,t,x), and b¯i depends on (ω,t,x1,…,xi−1,xi+1,…,xd). The coefficients aij,bi,c,b¯i are uniformly bounded and twice continuously differentiable. The leading coefficient a satisfies the ellipticity condition. Depending on the diffusion coefficient σk(u)=σk(ω,t,x,u), we consider two cases:(i)λ∈(0,∞) and σk(u) has a Lipschitz continuity and linear growth in u.(ii)λ,λ0∈(0,1/d) and σk(ω,t,x,u)=μk(ω,t,x)|u|1+λ0 with |∑k|μk(ω,t,⋅)|2|C(Rd)<∞ for all (ω,t)∈Ω×[0,∞). We obtain the existence, uniqueness, Lp regularity, and Hölder regularity of the solution. It should be noted that each case has different regularity results. For example, in the case of (i), for ε>0u∈Ct,x1/2−ε,1−ε([0,T]×Rd)∀T<∞, almost surely. On the other hand, in the case of (ii), if λ,λ0∈(0,1/d), for ε>0u∈Ct,x1−(λd)∨(λ0d)2−ε,1−(λd)∨(λ0d)−ε([0,T]×Rd)∀T<∞ almost surely. It should be noted that λ can be any positive number and that the solution regularity is independent of the nonlinear terms in case (i). In case (ii), however, λ and λ0 should satisfy λ,λ0∈(0,1/d), and the regularities of the solution are affected by λ,λ0 and d. This difference is due to the need to use a different proof method for the super-linear diffusion coefficient σk(u).

Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme

We consider the long-time behavior of an explicit tamed exponential Euler scheme applied to a class of parabolic semilinear stochastic partial differential equations driven by additive noise, under a one-sided Lipschitz continuity condition. The setting encompasses nonlinearities with polynomial growth. First, we prove that moment bounds for the numerical scheme hold, with at most polynomial dependence with respect to the time horizon. Second, we apply this result to obtain error estimates, in the weak sense, in terms of the time-step size and of the time horizon, to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process. We justify the efficiency of using the explicit tamed exponential Euler scheme to approximate the invariant distribution, since the computational cost does not suffer from the at most polynomial growth of the moment bounds. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for SPDEs with non-globally Lipschitz coefficients using an explicit tamed scheme.

Exponential Stability Analysis of Stochastic Semi-Linear Systems With Lèvy Noise

The exponential stability of semi-linear stochastic partial differential equations (SPDEs) involving L&#x00E8;vy type noise is investigated in this paper. By constructing an appropriate Lyapunov function, a new set of sufficient conditions are established in terms of linear matrix inequalities (LMIs) which ensures the mean-square exponentially stability (MSES) of given system with Neumann boundary conditions. Then the <i>H</i>_&#x221E; performance index is introduced to eliminate the disturbance which occurs in the considered system. The boundary control gain is obtained by solving the LMI conditions using the standard MATLAB software. Finally, a numerical example is provided to demonstrate the usefulness of the proposed methods.

The dual Yamada–Watanabe theorem for mild solutions to stochastic partial differential equations

We provide the dual result of the Yamada–Watanabe theorem for mild solutions to semilinear stochastic partial differential equations with path-dependent coefficients. An essential tool is the so-called “method of the moving frame”, which allows us to reduce the proof to infinite dimensional stochastic differential equations.

On generators of transition semigroups associated to semilinear stochastic partial differential equations

Let X be a real separable Hilbert space. Let Q be a linear, bounded, positive and compact operator on X and let A:Dom(A)⊆X→X be a linear, self-adjoint operator generating a strongly continuous semigroup on X. Let F:X→X be a (smooth enough) function and let {W(t)}t≥0 be a X-valued cylindrical Wiener process. For any α≥0, we are interested in the mild solution X(t,x) of the semilinear stochastic partial differential equation{dX(t,x)=(AX(t,x)+F(X(t,x)))dt+QαdW(t),t>0;X(0,x)=x∈X, and in its associated transition semigroup(0.1)P(t)φ(x):=E[φ(X(t,x))],φ∈Bb(X),t≥0,x∈X; where Bb(X) denotes the space of the real-valued, bounded and Borel measurable functions on X. In this paper we study the behavior of the semigroup P(t) in the space L2(X,ν), where ν is the unique invariant probability measure of (0.1), when F is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincaré inequalities and we study the maximal Sobolev regularity for the stationary equationλu−N2u=f,λ>0,f∈L2(X,ν); where N2 is the infinitesimal generator of P(t) in L2(X,ν).

Strong solutions of semilinear SPDEs with unbounded diffusion

We prove a modification to the classical maximal inequality for stochastic convolutions in 2-smooth Banach spaces using the factorization method. This permits to study semilinear stochastic partial differential equations with unbounded diffusion operators driven by cylindrical Brownian motion via the mild solution approach. In the case of finite dimensional driving noise, provided sufficient regularity on the coefficients, we establish existence and uniqueness of strong solutions. In the case of “critical unboundedness”, we show how to use the stochastic compactness method to obtain a martingale solution.

Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients

We provide an existence and uniqueness result for mild solutions to semilinear stochastic partial differential equations in the framework of the semigroup approach with locally monotone coefficients. An important component of the proof is an application of the dilation theorem of Nagy, which allows us to reduce the problem to infinite dimensional stochastic differential equations on a larger Hilbert space. Properties of the solutions like the Markov property are discussed as well.