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The Ito-Stratonovich theory of stochastic integration and stochastic differential equations has several shortcomings, especially when it comes to existence and consistency with the theory of Lebesque-Stieltjes integration and ordinary differential equations. An attempt is made firstly, to isolate the path property, possessed by almost all Brownian paths, that makes the stochastic theory of integration work. Secondly, to construct a new concept of solutions for differential equations, which would have the required consistency and continuity properties, within a class of deterministic noise functions, large enough to include almost all Brownian paths. The algebraic structure of iterated path integrals for smooth paths leads to a formal definition of a solution for a differential equation in terms of generalized path integrals for more general noises. This suggests a way of constructing solutions to differential equations in a large class of paths as limits of operators. The concept of the driving noise is extended to include the generalized path integrals of the noise. Less stringent conditions on the Holder continuity of the path can be compensated by giving more of its iterated integrals. Sufficient conditions for the solution to exist are proved in some special cases, and it is proved that almost all paths of Brownian motion as well as some other stochastic processes can be included in the theory.

This thesis is concerned with stochastic heat equation with memory and nonlinear energy supply. The main motivation to study such systems comes from Thermodynamics, see [85]. The main objective of this work is to study the existence and uniqueness of solutions to such equations and to investigate some fundamental properties of solutions like continuous dependence on initial conditions. In our approach we follow the seminal papers by Da Prato and Clement [10], where the stochastic heat equation with memory is tranformed into an integral equation in a function space and the so-called mild solutions are studied. In the aforementioned papers only linear equations with additive noise were investigated. The main contribution of this work is the extension of this approach to nonlinear equations. Our main tools are the theory of stochastic convolutions as developed in [33] and the theory of resolvent kernels for deterministic linear heat equations with memory, see[10]. Since the solution at time t depends on the whole history of the process up to time t, the resolvent kernel does not define a semigroup of operators in the state space of the process and therefore a ???standard??? theory of stochastic evolution equations as presented in the monograph [33] does not apply. A more delicate analysis of the resolvent kernles and the associated stochastic convolutions is needed. We will describe now content of this thesis in more detail. Introductory Chapters 1 and 2 collect some basic and essentially well known facts about the Wiener process, stochastic integrals, stochastic convolutions and integral kernels. However, some results in Chapter 2 dealing with stochastic convolution with respect to non-homogenous Wiener process are extensions of the existing theory. The main results of this thesis are presented in Chapters 3 and 4. In Chapter 3 we prove the existence and uniqueness of solutions to heat equations with additive noise and either Lipschitz or dissipative nonlinearities. In both cases we prove the continuous dependence of solutions on initial conditions. In Chapter 4 we prove the existence and uniqueness of solutions and continuous dependence on initial conditions for equations with multiplicative noise. The diffusion coefficients defined by unbounded operators are allowed.

This thesis is concerned with stochastic heat equation with memory and nonlinear energy supply. The main motivation to study such systems comes from Thermodynamics, see [85]. The main objective of this work is to study the existence and uniqueness of solutions to such equations and to investigate some fundamental properties of solutions like continuous dependence on initial conditions. In our approach we follow the seminal papers by Da Prato and Clement [10], where the stochastic heat equation with memory is tranformed into an integral equation in a function space and the so-called mild solutions are studied. In the aforementioned papers only linear equations with additive noise were investigated. The main contribution of this work is the extension of this approach to nonlinear equations. Our main tools are the theory of stochastic convolutions as developed in [33] and the theory of resolvent kernels for deterministic linear heat equations with memory, see[10]. Since the solution at time t depends on the whole history of the process up to time t, the resolvent kernel does not define a semigroup of operators in the state space of the process and therefore a ???standard??? theory of stochastic evolution equations as presented in the monograph [33] does not apply. A more delicate analysis of the resolvent kernles and the associated stochastic convolutions is needed. We will describe now content of this thesis in more detail. Introductory Chapters 1 and 2 collect some basic and essentially well known facts about the Wiener process, stochastic integrals, stochastic convolutions and integral kernels. However, some results in Chapter 2 dealing with stochastic convolution with respect to non-homogenous Wiener process are extensions of the existing theory. The main results of this thesis are presented in Chapters 3 and 4. In Chapter 3 we prove the existence and uniqueness of solutions to heat equations with additive noise and either Lipschitz or dissipative nonlinearities. In both cases we prove the continuous dependence of solutions on initial conditions. In Chapter 4 we prove the existence and uniqueness of solutions and continuous dependence on initial conditions for equations with multiplicative noise. The diffusion coefficients defined by unbounded operators are allowed.

Let H be a separable Hilbert space. Suppose (Ω, F, Ft, P) is a complete stochastic basis with a right continuous filtration and {Wt,t ∈ R} is an H-valued cylindrical Brownian motion with respect to {Ω, F, Ft, P). U(t, s) denotes an almost strong evolution operator generated by a family of unbounded closed linear operators on H. Consider the semilinear stochastic integral equation [formula omitted] where • f is of monotone type, i.e., ft(.) = f(t, w,.) : H → H is semimonotone, demicon-tinuous, uniformly bounded, and for each x ∈ H, ft(x) is a stochastic process which satisfies certain measurability conditions. • gs(.) is a uniformly-Lipschitz predictable functional with values in the space of Hilbert-Schmidt operators on H. • Vt is a cadlag adapted process with values in H. • X₀ is a random variable. We obtain existence, uniqueness, boundedness of the solution of this equation. We show the solution of this equation changes continuously when one or all of X₀, f, g, and V are varied. We apply this result to find stationary solutions of certain equations, and to study the associated large deviation principles. Let {Zt,t ∈ R} be an H-valued semimartingale. We prove an Ito-type inequality and a Burkholder-type inequality for stochastic convolution [formula omitted]. These are the main tools for our study of the above stochastic integral equation. / Science, Faculty of / Mathematics, Department of / Graduate

A novel statistical technique introduced by Besieris is used to study solutions of the nonlinear stochastic complex parabolic equation in the presence of two profiles. Specifically, the randomly modulated linear potential and the randomly perturbed quadratic focusing medium. In the former, a class of solutions is shown to admit an exact statistical description in terms of the moments of the wave function. In the latter, all even-order moments are computed exactly, whereas the odd-order moments are solved asymptotically. Lastly, it is shown that this statistical technique is isomorphic to mappings of nonconstant coefficient partial differential equations to constant coefficient equations. A generalization of this mapping and its inherent restrictions are discussed. / Ph. D. / incomplete_metadata

In this thesis, we study the existence of pathwise random periodic solutions to both the semilinear stochastic differential equations with linear multiplicative noise and the semilinear stochastic partial differential equations with linear multiplicative noise in a Hilbert space. We identify them as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general cases, and then perform the argument of the relative compactness of Wiener-Sobolev spaces in C([0, T],L2Ω,Rd)) or C([0, T],L2(Ω x O)) and Schauder's fixed point theorem to show the existence of a solution of the coupled stochastic forward-backward infinite horizon integral equations.