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Some SPDEs/SDEs driven by fractional noises and their related properties
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SPDEs/SDEs driven by fractional noises
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- Author / Creator
- Wang, Xiong
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In this dissertation, various problems related to stochastic (partial) differential equations are investigated. These problems include well-posedness, H\"older continuity of the solution, moments of the solution and their asymptotics. This thesis is divided into three parts. The first part studies the existence and uniqueness problems of nonlinear stochastic differential equations, including stochastic heat equation and stochastic wave equation driven by multiplicative Gaussian noises. The main feature of this part is that the Gaussian noise has the covariance of a fractional Brownian motion with Hurst parameter $H\in (1/4,1/2)$ in the spatial variable. Our contributions are to remove an artificial assumption on diffusion coefficient in the nonlinear stochastic heat equation and to surmount the barrier caused by the absence of semi-group property of wave kernel. The second part of the dissertation explores intermittency properties for various stochastic PDEs with varieties of space-time Gaussian noises via matching upper and lower moment bounds. This part introduces the Feynman diagram formula for the moments of the solution and the small ball nondegeneracy for the Green's function to obtain the sharp lower bounds for all moments for various interesting equations, including stochastic heat equations, stochastic wave equations, stochastic heat equations with fractional Laplacians, and stochastic diffusions which are both fractional in time and in space. The third part of this thesis considers stability problems in the mean square sense for stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (1/2,1)$. Both the mean-square stability of the solution and its stochastic theta scheme for linear and nonlinear equations are investigated by introducing a set of analytic and probabilistic tools. Numerical examples are carried out to illustrate our theoretical results.
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- Subjects / Keywords
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- stochastic partial differential equations
- stochastic differential equations
- fractional noise
- Gaussian noise
- Gaussian processes
- rough noise
- Hurst parameters
- existence and uniqueness problem
- stochastic heat equation
- stochastic wave equation
- intermittency
- Feynman diagram
- upper and lower moments
- fractional Brownian motions
- mean-square stability
- Gaussian correlation inequality
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- Graduation date
- Fall 2022
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- Type of Item
- Thesis
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- Degree
- Doctor of Philosophy
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- License
- This thesis is made available by the University of Alberta Library with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.