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 MULTIFRACTAL FLUCTUATIONS IN FINANCE

2000 ◽  
Vol 03 (03) ◽  
pp. 361-364 ◽  
Author(s):  
FRANCOIS SCHMITT ◽  
DANIEL SCHERTZER ◽  
SHAUN LOVEJOY
Keyword(s):  
Brownian Motion ◽  
Exchange Rate ◽  
Foreign Exchange ◽  
Stochastic Models ◽  
Evolution Equations ◽  
Foreign Exchange Rate ◽  
Scaling Exponent ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Multiplicative Processes

We consider the structure functions S(q)(τ), i.e. the moments of order q of the increments X(t + τ)-X(t) of the Foreign Exchange rate X(t) which give clear evidence of scaling (S(q)(τ)∝τζ(q)). We demonstrate that the nonlinearity of the observed scaling exponent ζ(q) is incompatible with monofractal additive stochastic models usually introduced in finance: Brownian motion, Lévy processes and their truncated versions. This nonlinearity correspond to multifractal intermittency yielded by multiplicative processes. The non-analyticity of ζ(q) corresponds to universal multifractals, which are furthermore able to produce "hyperbolic" pdf tails with an exponent qD > 2. We argue that it is necessary to introduce stochastic evolution equations which are compatible with this multifractal behaviour.

Attractors with higher-order regularity of stochastic reaction–diffusion equations on time-varying domains

2020 ◽  
Vol 20 (02) ◽  
pp. 2050041
Author(s):  
Lu Yang ◽  
Meihua Yang ◽  
Peter Kloeden
Keyword(s):  
Reaction Diffusion ◽  
Higher Order ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Initial Time ◽  
Regular Space ◽  
Time Varying ◽  
Regularity Properties ◽  
Random Attractors ◽  
Stochastic Reaction

Random attractors and their higher-order regularity properties are studied for stochastic reaction–diffusion equations on time-varying domains. Some new a priori estimates for the difference of solutions near the initial time and the continuous dependence in initial data in [Formula: see text] are proved. Then attraction of the random attractors in the higher integrability space [Formula: see text] for any [Formula: see text] and the regular space [Formula: see text] is established.

A THEOREM DUAL TO YAMADA–WATANABE THEOREM FOR STOCHASTIC EVOLUTION EQUATIONS

2010 ◽  
Vol 10 (03) ◽  
pp. 367-374 ◽  
Author(s):  
HUIJIE QIAO
Keyword(s):  
Evolution Equation ◽  
Strong Solution ◽  
Existence And Uniqueness ◽  
Variational Approach ◽  
Evolution Equations ◽  
Stochastic Evolution Equation ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Uniqueness In Law ◽  
Strong Existence

In this paper, we prove that uniqueness in law and strong existence for a stochastic evolution equation [Formula: see text] imply existence and uniqueness of a strong solution in the framework of the variational approach. This result seems to be dual to Yamada–Watanabe theorem in [7].

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