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In this thesis, we study the existence of stationary solutions for two cases. One is for random difference equations. For this, we prove the existence and uniqueness of the stationary solutions in a finite-dimensional Euclidean space Rd by applying the coupling method. The other one is for semi linear stochastic evolution equations. For this case, we follows Mohammed, Zhang and Zhao [25]'s work. In an infinite-dimensional Hilbert space H, we release the Lipschitz constant restriction by using Arzela-Ascoli compactness argument. And we also weaken the globally bounded condition for F by applying forward and backward Gronwall inequality and coupling method.

Stochastic differential inclusions can be considered as a generalisation of stochastic differential equations. In particular a multivalued mapping describes the set of equations, in which a solution has to be found. This paper presents an existence result for a special parabolic stochastic inclusion. The proof is based on the method of upper and lower solutions. In the deterministic case this method was effectively introduced by S. Carl.

Stochastic Evolution Equations Petr Čoupek Doctoral Thesis Abstract Linear stochastic evolution equations with additive regular Volterra noise are studied in the thesis. Regular Volterra processes need not be Gaussian, Markov or semimartingales, but they admit a certain covariance structure instead. Particular examples cover the fractional Brownian motion of H > 1/2 and, in the non-Gaussian case, the Rosenblatt process. The solution is considered in the mild form, which is given by the variation of constants formula, and takes values either in a separable Hilbert space or the space Lp(D, µ) for large p. In the Hilbert-space setting, existence, space-time regularity and large-time behaviour of the solutions are studied. In the Lp setting, existence and regularity is studied, and in concrete cases of stochastic partial differential equations, the solution is shown to be a space-time continuous random field.

Orientadores: Pedro José Catuogno, Johannes Michael Antonius Maria van Neerven / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-22T15:55:30Z (GMT). No. of bitstreams: 1 AbreuJunior_JamilGomesde_D.pdf: 1681574 bytes, checksum: 280ab5f7ecf646a3ab11f04ca34664e3 (MD5) Previous issue date: 2013 / Resumo: Nesta tese tratamos o problema de caracterizar a existência de medida invariante para equações de evolução estocásticas lineares com ruído aditivo em termos do resolvente associado ao gerador da equação. Este problema foi proposto recentemente na literatura como uma versão estocástica da célebre conjectura de Weiss em teoria de controle para sistemas lineares, que consiste em relacionar admissibilidade de operadores de controle a certas estimativas envolvendo o resolvente do gerador infinitesimal. No contexto estocástico, e no caso em que o gerador da equação é analítico e admite um cálculo funcional do tipo Dunford-Schwartz num espaço de Banach com a propriedade de Pisier, nosso resultado principal consiste de condições analítico-funcionais necessárias e suficientes para existência de medida invariante para o problema de Cauchy estocástico. Em particular, mostramos que existência de medida invariante _e equivalente _a convergência em probabilidade de certa série Gaussiana cujos termos são os resolventes avaliados nos pontos diádicos positivos da reta real, que consideramos como sendo a condição de Weiss estocástica. Há fortes razões para esperar que, _a semelhança do que ocorreu com a conjectura de Weiss clássica, este problema atraia considerável atenção da comunidade acadêmica num futuro próximo / Abstract: In this thesis we consider the problem of characterizing the existence of invariant measure for linear stochastic evolution equations with additive noise in terms of the resolvent operator associated to the generator of the equation. This problem was recently proposed in the literature as a stochastic version of the celebrated Weiss conjecture in linear systems theory, which relates admissibility of control operators to certain estimates involving the resolvent of the infinitesimal generator. In the stochastic setting and when the generator is analytic and admits a bounded functional calculus in a Banach space with Pisier property, our main result consists of necessary and sufficient functional analytic conditions for the existence of an invariant measure for the stochastic Cauchy problem. In particular, we show that existence of invariant measure is equivalent to convergence in probability of a certain Gaussian series whose terms are the resolvents evaluated at the positive dyadic points of the real line, which we consider as being the stochastic Weiss condition. There are strong reasons to expect that, similarly to what happened to the classical Weiss conjecture, this work will attract considerable attention of the academic community in the near future / Doutorado / Matematica / Doutor em Matemática

Topology design of distributed local area networks (DLANs) can be classified as an NP-hard problem. Intelligent algorithms, such as evolutionary and swarm intelligence techniques, are candidate approaches to address this problem and to produce desirable solutions. DLAN topology design consists of several conflicting objectives such as minimization of cost, minimization of network delay, minimization of the number of hops between two nodes, and maximization of reliability. It is possible to combine these objectives in a single-objective function, provided that the trade-offs among these objectives are adhered to. This thesis proposes a strategy and a new aggregation operator based on fuzzy logic to combine the four objectives in a single-objective function. The thesis also investigates the use of a number of evolutionary algorithms such as stochastic evolution, simulated evolution, and simulated annealing. A number of hybrid variants of the above algorithms are also proposed. Furthermore, the applicability of swarm intelligence techniques such as ant colony optimization and particle swarm optimization to topology design has been investigated. All proposed techniques have been evaluated empirically with respect to their algorithm parameters. Results suggest that simulated annealing produced the best results among all proposed algorithms. In addition, the hybrid variants of simulated annealing, simulated evolution, and stochastic evolution generated better results than their respective basic algorithms. Moreover, a comparison of ant colony optimization and particle swarm optimization shows that the latter generated better results than the former. / Thesis (PhD)--University of Pretoria, 2009. / Computer Science / unrestricted

Stochastic differential inclusions can be considered as a generalisation of stochastic differential equations. In particular a multivalued mapping describes the set of equations, in which a solution has to be found. This paper presents an existence result for a special parabolic stochastic inclusion. The proof is based on the method of upper and lower solutions. In the deterministic case this method was effectively introduced by S. Carl.

Filtering for Stochastic Evolution Equations Vít Kubelka Doctoral thesis Abstract Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss-Volterra process observed at finitely many points of the domain and to delayed SPDEs driven by white noise. Subsequently, the continuous dependence of the filter and observation error on parameters which may be present both in the signal and the obser- vation process is proved. These results are applied to signals governed by stochastic heat equations driven by distributed or pointwise fractional noise. The observation process may be a noisy observation of the signal at given points in the domain, the position of which may depend on the parameter. 1

Controlled linear stochastic evolution equations driven by Lévy processes are studied in the Hilbert space setting. The control operator may be unbounded which makes the results obtained in the abstract setting applicable to parabolic SPDEs with boundary or point control. The first part contains some preliminary technical results, notably a version of Itô formula which is applicable to weak/mild solutions of controlled equations. In the second part, the ergodic control problem is solved: The feedback form of the optimal control and the formula for the optimal cost are found. The control problem is solved in the mean-value sense and, under selective conditions, in the pathwise sense. As examples, various parabolic type controlled SPDEs are studied. 1

Stochastic partial differential equations have proven useful in many applied areas of mathematics, such as physics or mathematical finance. A major part of such equations consists of linear equations with additive noise. In certain cases, however, the drift part of the differential equation additionally contains a possibly problematic non-linear term, which makes it unsolvable by the standard methods and even a solution in the mild sense may be out of reach. In such situations, we may still find a solution in the weak sense by employing a suitable transformation of the probability space. This thesis deals with semilinear stochastic evolution equations in a separable Hilbert space, where the driving process is an element of a large class of processes - so called Volterra processes, which can be understood as a generalisation of the Wiener process and may be of use to model a wide range of phenomena. The weak solutions, however, have been studied so far only for equations with the cylindrical fractional Brownian motion as the driving process. In this thesis, we introduce a generalisation of the Girsanov theorem for cylindrical Gaussian Volterra processes and give, in full generality, sufficient conditions for the existence of a weak solution and the uniqueness of the equation in law. Further, we introduce...