scholarly journals Limit theorems for time-dependent averages of nonlinear stochastic heat equations

Bernoulli ◽  
2022 ◽  
Vol 28 (1) ◽  
Author(s):  
Kunwoo Kim ◽  
Jaeyun Yi
Keyword(s):  
Limit Theorems ◽  
Time Dependent ◽  
Heat Equations ◽  
Stochastic Heat Equations

scholarly journals Asymptotic Distributions for Power Variations of the Solution to the Spatially Colored Stochastic Heat Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Wensheng Wang ◽  
Xiaoying Chang ◽  
Wang Liao
Keyword(s):  
Heat Equation ◽  
White Noise ◽  
Harmonic Analysis ◽  
Central Limit ◽  
Gaussian Processes ◽  
Limit Theorems ◽  
Colored Noise ◽  
Asymptotic Distributions ◽  
Heat Equations ◽  
Stochastic Heat Equations

Let u α , d = u α , d t , x ,   t ∈ 0 , T , x ∈ ℝ d be the solution to the stochastic heat equations (SHEs) with spatially colored noise. We study the realized power variations for the process u α , d , in time, having infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. We use the underlying explicit kernels and spectral/harmonic analysis, yielding temporal central limit theorems for SHEs with spatially colored noise. This work builds on the recent works on delicate analysis of variations of general Gaussian processes and SHEs driven by space-time white noise.

scholarly journals Some properties of non-linear fractional stochastic heat equations on bounded domains

2017 ◽  
Vol 102 ◽  
pp. 86-93 ◽  
Author(s):  
Mohammud Foondun ◽  
Ngartelbaye Guerngar ◽  
Erkan Nane
Keyword(s):  
Heat Equations ◽  
Bounded Domains ◽  
Non Linear ◽  
Stochastic Heat Equations

scholarly journals A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 904 ◽  
Author(s):  
Afshin Babaei ◽  
Hossein Jafari ◽  
S. Banihashemi
Keyword(s):  
Order Of Convergence ◽  
Additive Noise ◽  
Numerical Solutions ◽  
Numerical Test ◽  
Test Problems ◽  
Heat Equations ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Stochastic Heat Equations ◽  
Collocation Approach

A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding the exact solution of such equations is difficult in many cases. Thus, a collocation method based on sixth-kind Chebyshev polynomials (SKCPs) is introduced to assess their numerical solutions. This collocation approach reduces the considered problem to a system of linear algebraic equations. The convergence and error analysis of the suggested scheme are investigated. In the end, numerical results and the order of convergence are evaluated for some numerical test problems to illustrate the efficiency and robustness of the presented method.

Harmonic analysis tools for stochastic magnetohydrodynamics equations in Besov spaces

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640025 ◽  
Author(s):  
Mamadou Sango ◽  
Tesfalem Abate Tegegn
Keyword(s):  
Harmonic Analysis ◽  
Existence Result ◽  
Three Dimensional ◽  
Regularity Result ◽  
Heat Equations ◽  
Analysis Tools ◽  
Large Numbers ◽  
Magnetohydrodynamics Equation ◽  
Stochastic Heat Equations ◽  
Time Existence

We establish a regularity result for stochastic heat equations in probabilistic evolution spaces of Besov type and we use it to prove a global in time existence and uniqueness of solution to a stochastic magnetohydrodynamics equation. The existence result holds with a positive probability which can be made arbitrarily close to one. The work is carried out by blending harmonic analysis tools such as Littlewood–Paley decomposition, Jean–Micheal Bony paradifferential calculus and stochastic calculus. The law of large numbers is a key tool in our investigation. Our global existence result is new in three-dimensional spaces.

scholarly journals pth Moment Exponential Stability of Nonlinear Hybrid Stochastic Heat Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Xuetao Yang ◽  
Quanxin Zhu ◽  
Zhangsong Yao
Keyword(s):  
Fixed Point ◽  
Exponential Stability ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Heat Equations ◽  
Stochastic Heat Equations ◽  
Pth Moment Exponential Stability ◽  
Infinite State ◽  
Moment Exponential Stability ◽  
Nonlinear Hybrid

We are concerned with the exponential stability problem of a class of nonlinear hybrid stochastic heat equations (known as stochastic heat equations with Markovian switching) in an infinite state space. The fixed point theory is utilized to discuss the existence, uniqueness, andpth moment exponential stability of the mild solution. Moreover, we also acquire the Lyapunov exponents by combining the fixed point theory and the Gronwall inequality. At last, two examples are provided to verify the effectiveness of our obtained results.

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