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Stochastic Dynamical Systems with Non-Gaussian and Singular Noises.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Stochastic Dynamical Systems with Non-Gaussian and Singular Noises./
作者:	Zhang, Qi.
面頁冊數:	1 online resource (144 pages)
附註:	Source: Dissertations Abstracts International, Volume: 84-04, Section: B.
Contained By:	Dissertations Abstracts International84-04B.
標題:	Applied mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29321224click for full text (PQDT)
ISBN:	9798845408815

Stochastic Dynamical Systems with Non-Gaussian and Singular Noises.
Zhang, Qi.

Stochastic Dynamical Systems with Non-Gaussian and Singular Noises. - 1 online resource (144 pages)

Source: Dissertations Abstracts International, Volume: 84-04, Section: B.

Thesis (Ph.D.)--Illinois Institute of Technology, 2022.

Includes bibliographical references

In order to describe stochastic fluctuations or random potentials arising from science and engineering, non-Gaussian or singular noises are introduced in stochastic dynamical systems. In this thesis we investigate stochastic differential equations with non-Gaussian Levy noise, and the singular two-dimensional Anderson model equation with spatial white noise potential. This thesis consists of the following three main parts. In the first part, we establish a linear response theory for stochastic differential equations driven by an α-stable Levy noise (1<α<2). We first prove the ergodic property of the stochastic differential equation and the regularity of the corresponding stationary Fokker-Planck equation. Then we establish the linear response theory. This result is a general fluctuation-dissipation relation between the response of the system to the external perturbations and the Levy type fluctuations at a steady state.In the second part, we study the global well-posedness of the singular nonlinear parabolic Anderson model equation on a two-dimensional torus. This equation can be viewed as the nonlinear heat equation with a random potential. The method is based on paracontrolled distribution and renormalization. After splitting the original nonlinear parabolic Anderson model equation into two simpler equations, we prove the global existence by some a priori estimates and smooth approximations. Furthermore, we prove the uniqueness of the solution by classical energy estimates. This work improves the local well-posedness results in earlier works.In the third part, we investigate the variation problem associated with the elliptic Anderson model equation in a two-dimensional torus in the paracontrolled distribution framework. The energy functional in this variation problem is arising from the Anderson localization. We obtain the existence of minimizers by a direct method in the calculus of variations, and show that the Euler-Lagrange equation of the energy functional is an elliptic singular stochastic partial differential equation with the Anderson Hamiltonian. We further establish the L2 estimates and Schauder estimates for the minimizer as weak solution of the elliptic singular stochastic partial differential equation. Therefore, we uncover the natural connection between the variation problem and the singular stochastic partial differential equation in the paracontrolled distribution framework.Finally, we summarize our results and outline some research topics for future investigation.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2023

Mode of access: World Wide Web

ISBN: 9798845408815Subjects--Topical Terms:

2122814
Applied mathematics.
Subjects--Index Terms:

Anderson modelIndex Terms--Genre/Form:

542853
Electronic books.

Stochastic Dynamical Systems with Non-Gaussian and Singular Noises.
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504 $a Includes bibliographical references
520 $a In order to describe stochastic fluctuations or random potentials arising from science and engineering, non-Gaussian or singular noises are introduced in stochastic dynamical systems. In this thesis we investigate stochastic differential equations with non-Gaussian Levy noise, and the singular two-dimensional Anderson model equation with spatial white noise potential. This thesis consists of the following three main parts. In the first part, we establish a linear response theory for stochastic differential equations driven by an α-stable Levy noise (1<α<2). We first prove the ergodic property of the stochastic differential equation and the regularity of the corresponding stationary Fokker-Planck equation. Then we establish the linear response theory. This result is a general fluctuation-dissipation relation between the response of the system to the external perturbations and the Levy type fluctuations at a steady state.In the second part, we study the global well-posedness of the singular nonlinear parabolic Anderson model equation on a two-dimensional torus. This equation can be viewed as the nonlinear heat equation with a random potential. The method is based on paracontrolled distribution and renormalization. After splitting the original nonlinear parabolic Anderson model equation into two simpler equations, we prove the global existence by some a priori estimates and smooth approximations. Furthermore, we prove the uniqueness of the solution by classical energy estimates. This work improves the local well-posedness results in earlier works.In the third part, we investigate the variation problem associated with the elliptic Anderson model equation in a two-dimensional torus in the paracontrolled distribution framework. The energy functional in this variation problem is arising from the Anderson localization. We obtain the existence of minimizers by a direct method in the calculus of variations, and show that the Euler-Lagrange equation of the energy functional is an elliptic singular stochastic partial differential equation with the Anderson Hamiltonian. We further establish the L2 estimates and Schauder estimates for the minimizer as weak solution of the elliptic singular stochastic partial differential equation. Therefore, we uncover the natural connection between the variation problem and the singular stochastic partial differential equation in the paracontrolled distribution framework.Finally, we summarize our results and outline some research topics for future investigation.
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653 $a Paracontrolled distribution
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