scholarly journals Stochastic heat equations for infinite strings with values in a manifold

2020 ◽  
Vol 374 (1) ◽  
pp. 407-452
Author(s):  
Xin Chen ◽  
Bo Wu ◽  
Rongchan Zhu ◽  
Xiangchan Zhu
Keyword(s):  
Heat Equations ◽  
Stochastic Heat Equations ◽  
Infinite Strings

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.

scholarly journals A note on parameter estimation for discretely sampled SPDEs

2019 ◽  
Vol 20 (03) ◽  
pp. 2050016 ◽  
Author(s):  
Igor Cialenco ◽  
Yicong Huang
Keyword(s):  
Parameter Estimation ◽  
Bounded Domain ◽  
Asymptotic Properties ◽  
Mesh Size ◽  
Fixed Time ◽  
Estimation Problem ◽  
Joint Estimation ◽  
Heat Equations ◽  
Whole Space ◽  
Stochastic Heat Equations

We consider a parameter estimation problem for one-dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We prove that, the real valued parameter next to the Laplacian (the drift), and the constant parameter in front of the noise (the volatility) can be consistently estimated under somewhat surprisingly minimal information. Namely, it is enough to observe the solution at a fixed time and on a discrete spatial grid, or at a fixed space point and at discrete time instances of a finite interval, assuming that the mesh-size goes to zero. The proposed estimators have the same form and asymptotic properties regardless of the nature of the domain –bounded domain or whole space. The derivation of the estimators and the proofs of their asymptotic properties are based on computations of power variations of some relevant stochastic processes. We use elements of Malliavin calculus to establish the asymptotic normality properties in the case of bounded domain. We also discuss the joint estimation problem of the drift and volatility coefficient. We conclude with some numerical experiments that illustrate the obtained theoretical results.

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