Some contribution to analysis and stochastic analysis

Liu, X

Abstract:: The dissertation consists of two parts. The first part (Chapter 1 to 4) is on some contributions to the development of a non-linear analysis on the quintessential fractal set Sierpinski gasket and its probabilistic interpretation. The second part (Chapter 5) is on the asymptotic tail decays for suprema of stochastic processes satisfying certain conditional increment controls.

Chapters 1, 2 and 3 are devoted to the establishment of a theory of backward problems for non-linear st...
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APA Style

Liu, X. (2018). Some contribution to analysis and stochastic analysis [PhD thesis]. University of Oxford.

MLA Style

Liu, X. Some Contribution to Analysis and Stochastic Analysis. University of Oxford, 2018.

Chicago Style

Liu, X. 2018. “Some Contribution to Analysis and Stochastic Analysis.” PhD thesis, University of Oxford.
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Files:: DPhil Thesis_XuanLiu.pdf

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+ Liu, X More by this author

Division:: MPLS
Department:: Mathematical Institute
Role:: Author

Contributors

+ Qian, Z

Role:: Supervisor

DOI:: 10.5287/ora-nzabg9jdn
Type of award:: DPhil
Level of award:: Doctoral
Awarding institution:: University of Oxford

Language:: English
Keywords:: Sobolev inequalities

Singular measures

Semi-linear partial differential equations

Backward stochastic differential equations

Sierpinski gasket
Subjects:: Stochastic Analysis

Mathematics
UUID:: uuid:485474c0-2501-4ef0-a0bc-492e5c6c9d62
Deposit date:: 2018-08-13

Related item:: Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket
Description:: This is a joint work by the author and Z. Qian. In this article, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along fractals (which can be considered as models of simplified rough porous media). Unlike the regular space case, such parabolic type equations involving non-linear convection terms must take a different form, due to the fact that convection terms must be singular to the “linear part” determined the heat semigroup. In order to study these parabolic type equations, a new kind of Sobolev inequalities for the Dirichlet form on the gasket will be established. These Sobolev inequalities, which are interesting on their own and in contrast to the case of Euclidean spaces, involve two $L^p$ norms with respect to two mutually singular measures. By examining properties of singular convolutions of the associated heat semigroup, we derive the space-time regularity of solutions to these parabolic equations under a few technical conditions. The Burgers equations on the Sierpinski gasket are also studied, for which a maximum principle for solutions is derived using techniques from backward stochastic differential equations, and the existence, uniqueness, and regularity of its solutions are obtained.
Related item:: Brownian motion on the Sierpinski gasket and a stochastic maximum principle
Description:: This is an article by the author. In this item, we consider stochastic control problems on the Sierpinski gasket. An order comparison lemma is derived using heat kernel estimate for Brownian motion on the gasket. Using the order comparison lemma and techniques of BSDEs, we establish a Pontryagin stochastic maximum principle for these control problems. It turns out that the stochastic maximum principle on the Sierpinski gasket involves two necessity equations in contrast to its counterpart on Euclidean spaces. This effect is due to singularity between the Hausdorff measure and the energy dominant measure on the gasket, which is a common feature shared by many fractal spaces. The linear regulator problems on the gasket is also considered as an example.
Related item:: Backward problems for stochastic differential equations on the Sierpinski gasket
Description:: This is a joint work by the author and Z. Qian. In this work, we study the non-linear backward problems (with deterministic or stochastic durations) of stochastic differential equations on the Sierpinski gasket. We prove the existence and uniqueness of solutions of backward stochastic differential equations driven by Brownian martingale on the Sierpinski gasket. The exponential integrability of quadratic processes for martingale additive functionals is obtained, and as an application, a Feynman-Kac representation formula for weak solutions to semi-linear parabolic PDEs on the gasket is also established.
Related item:: On the maximum of a type of random processes
Description:: This is an article by the author. In this article, we study the asymptotics of tail probability for the suprema of a class of stochastic processes which satisfy the controlled conditional increment condition and have $\alpha$-exponential marginal decays. These conditions are quite weak compared to other assumptions (e.g. being jointly Gaussian, or satisfying Markov property) considered in literature, and are satisfied by a many processes commonly encountered. We start with establishing a Doob type maximal inequality for processes satisfying the controlled conditional condition. Using the Doob type inequality and the $\alpha$-exponential decay condition, we further show that the suprema $\sup_{t \in [0;T ]} |X_t|$ of a stochastic process ${X_t}_t$ has the same tail asymptotics as its margins $X_t, t \in [0; T]$. As an application, we apply our result to derive the existence of strong solutions to supercritical divergence-free stochastic DiPerna--Lions flows.

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Copyright holder:: Liu, X

Licence:: Terms and Conditions of Use for Oxford University Research Archive

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Some contribution to analysis and stochastic analysis

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