scholarly journals Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities

2019 ◽  
Vol 40 (2) ◽  
pp. 1005-1050 ◽  
Author(s):  
Arnulf Jentzen ◽  
Primož Pušnik
Keyword(s):  
Strong Convergence ◽  
Approximation Method ◽  
Evolution Equations ◽  
Convergence Rates ◽  
Reaction Diffusion ◽  
Convergence Result ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Lipschitz Continuous ◽  
Strong Convergence Rates

Abstract In this article we propose a new, explicit and easily implementable numerical method for approximating a class of semilinear stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. We establish strong convergence rates for this approximation method in the case of semilinear stochastic evolution equations with globally monotone coefficients. Our strong convergence result, in particular, applies to a class of stochastic reaction–diffusion partial differential equations.

scholarly journals Weak pullback mean random attractors for stochastic evolution equations and applications

2021 ◽  
pp. 2240001
Author(s):  
Anhui Gu
Keyword(s):  
Stochastic Models ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Random Attractors ◽  
Porous Media Equations ◽  
Stochastic Reaction

In this paper, we investigate the existence and uniqueness of weak pullback mean random attractors for abstract stochastic evolution equations with general diffusion terms in Bochner spaces. As applications, the existence and uniqueness of weak pullback mean random attractors for some stochastic models such as stochastic reaction–diffusion equations, the stochastic [Formula: see text]-Laplace equation and stochastic porous media equations are established.

scholarly journals ON STOCHASTIC EVOLUTION EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS

2009 ◽  
Vol 09 (04) ◽  
pp. 549-595 ◽  
Author(s):  
XICHENG ZHANG
Keyword(s):  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Reaction Diffusion ◽  
Stochastic Equations ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Uniqueness Of Solutions ◽  
Volterra Type ◽  
Porous Medium Equations ◽  
Stochastic Functional

In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with non-Lipschitz coefficients, that contains backward stochastic evolution equations, stochastic Volterra type evolution equations and stochastic functional evolution equations. In particular, the results can be used to treat a large class of quasi-linear stochastic equations, which includes the reaction diffusion and porous medium equations.

scholarly journals Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise

2021 ◽  
Author(s):  
Carlo Marinelli ◽  
Lluís Quer-Sardanyons
Keyword(s):  
Absolute Continuity ◽  
Multiplicative Noise ◽  
Evolution Equations ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Stochastic Pde ◽  
Stochastic Evolution Equations ◽  
Lipschitz Continuous ◽  
Locally Lipschitz

AbstractWe prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L2(G), where G is an open bounded domain in $\mathbb {R}^{d}$ ℝ d with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

scholarly journals Smoothness of solutions of stochastic evolution equations and the existence of a filtering transition density

1981 ◽  
Vol 84 ◽  
pp. 195-208 ◽  
Author(s):  
B. L. Rozovskii ◽  
A. Shimizu
Keyword(s):  
Evolution Equations ◽  
Transition Density ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Smoothness Of Solutions

In this paper, we shall discuss the smoothness of solutions of stochastic evolution equations, which has been investigated in N. V. Krylov and B. L. Rozovskii [2] [3], to establish the existence of a filtering transition density.

scholarly journals On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Jing Cui ◽  
Litan Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Banach Fixed Point Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Composition Theorem

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.

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