TY - BOOK

T1 - Rough paths and Lévy processes As Perturbations of Differential Equations

AU - Fahim, Kistosil

N1 - embargoed until null

PY - 1800

Y1 - 1800

N2 - In the first topic of this thesis, we investigate the existence of the solution of partial differential equations driven by rough paths. We note that the theory of rough path introduced by Terry Lyons in his seminal work as an extension of the classical theory of controlled differential equations. Currently, this theory has developed intensively, see the work of Gubinelli, Friz, Hairer and Friz. In this thesis, we give two different PDEs that are Volterra equations and Landau–Lifshitz–Gilbert equations (abbreviation: LLGEs). Firstly, we show existence and uniqueness of solution for the semilinear Volterra equations driven by a rough path perturbation. We give a maximal regularity result of the Ornstein-Uhlenbeck process with memory term driven by a rough path using the Nagy Dilation theorem. Secondly, this thesis contains the work on Wong-Zakai approximation for Landau-Lifshitz-Gilbert equations (LLGEs) driven by Geometric rough paths. We adapt Lyon’s rough paths theory to study LLGEs driven by geometric rough paths in one dimension, with non-zero exchange energy only. The key ingredients for the construction of the solution and its corresponding convergence results are maximal regularity property and the geometric rough path theory. In the second topic of this thesis, we study the stochastic differential equation driven by Lévy processes. The Lévy process is a fundamental class of stochastic processes. We note that Poisson processes, compound Poisson process, Brownian motion, and stable processes are essential examples of Lévy processes. The Lévy process introduced by Paul Lévy in the 1930s and currently many researches have discussed properties of their distributions and behaviours of their sample functions. The important class of stochastic processes are obtained as generalizations of the class of Lévy processes. In this thesis, we present analytic properties of nonlocal transition semigroups, the so-called Markovian semigroup, associated with a class of stochastic differential equations in R^d driven by pure jump-type Lévy processes. We show under which conditions the semigroup will be analytic on the Besov space $B_{p,q}^ m(R^d)$ with $1\le p, q<\infty$ and $m\in R$. Moreover, this thesis presents some applications by proving the substantial Feller property and give weak error estimates for approximating schemes of the stochastic differential equations over the Besov space $B_{\infty,\infty}^ m(R^d)$.

AB - In the first topic of this thesis, we investigate the existence of the solution of partial differential equations driven by rough paths. We note that the theory of rough path introduced by Terry Lyons in his seminal work as an extension of the classical theory of controlled differential equations. Currently, this theory has developed intensively, see the work of Gubinelli, Friz, Hairer and Friz. In this thesis, we give two different PDEs that are Volterra equations and Landau–Lifshitz–Gilbert equations (abbreviation: LLGEs). Firstly, we show existence and uniqueness of solution for the semilinear Volterra equations driven by a rough path perturbation. We give a maximal regularity result of the Ornstein-Uhlenbeck process with memory term driven by a rough path using the Nagy Dilation theorem. Secondly, this thesis contains the work on Wong-Zakai approximation for Landau-Lifshitz-Gilbert equations (LLGEs) driven by Geometric rough paths. We adapt Lyon’s rough paths theory to study LLGEs driven by geometric rough paths in one dimension, with non-zero exchange energy only. The key ingredients for the construction of the solution and its corresponding convergence results are maximal regularity property and the geometric rough path theory. In the second topic of this thesis, we study the stochastic differential equation driven by Lévy processes. The Lévy process is a fundamental class of stochastic processes. We note that Poisson processes, compound Poisson process, Brownian motion, and stable processes are essential examples of Lévy processes. The Lévy process introduced by Paul Lévy in the 1930s and currently many researches have discussed properties of their distributions and behaviours of their sample functions. The important class of stochastic processes are obtained as generalizations of the class of Lévy processes. In this thesis, we present analytic properties of nonlocal transition semigroups, the so-called Markovian semigroup, associated with a class of stochastic differential equations in R^d driven by pure jump-type Lévy processes. We show under which conditions the semigroup will be analytic on the Besov space $B_{p,q}^ m(R^d)$ with $1\le p, q<\infty$ and $m\in R$. Moreover, this thesis presents some applications by proving the substantial Feller property and give weak error estimates for approximating schemes of the stochastic differential equations over the Besov space $B_{\infty,\infty}^ m(R^d)$.

KW - Rough paths theory

KW - Volterra equation

KW - Mild solution

KW - Resolvent

KW - Partial differential equation

KW - Landau–Lifshitz–Gilbert equations

KW - Wong-Zakai approximation

KW - Ferromagnetism

KW - Stochastic integral of jump type

KW - Poisson random measures

KW - Markovian semigroup

KW - Pseudo-differential operators

KW - Non-local operators

KW - Theorie der rauhen Pfade

KW - Volterra-Gleichungen

KW - Milde Lösung

KW - Resolvente

KW - Partielle Differentialgleichungen

KW - Landau-Lifshitz-Gilbert-Gleichungen

KW - Wong-Zakai-Approximation

KW - Ferromagnetismus

KW - Stochastisches Integral vom Sprung-Typ

KW - Poisson'sche Zufallsmaße

KW - Markov'sche Halbgruppe

KW - Pseudodifferentialoperatoren

KW - nicht lokale Operatoren

M3 - Doctoral Thesis

ER -