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Infinite products
Zavadilová, Barbora ; Seidler, Jan (advisor) ; Kaluža, Jan (referee)
We study theory of infinite products, summarize basic definitions, terms and theorems, which we apply to some elementary examples in this work. Moreover we concern with convergence of products with real factors. We also prove required and sufficient condition of convergence or absolute convergence of infinite product. The main objective of the work is derivation of sine product formula, where we also prove the theorem of exchanging limit and infinite product. These results we use for deduction cosine product formula. 1
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Stochastic evolution equations with multiplicative fractional noise
Šnupárková, Jana ; Maslowski, Bohdan (advisor) ; Hlubinka, Daniel (referee) ; Seidler, Jan (referee)
Title: Stochastic evolution equations with multiplicative fractional noise Author: Jana Šnupárková Departement: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc. Supervisor's e-mail address: maslow@karlin.mff.cuni.cz Abstract: The fractional Gaussian noise is a formal derivative of a fractional Brownian motion with Hurst parameter H ∈ (0, 1). An explicit formula for a solution to stochastic differential equations with a multiplicative fractional Gaussian noise in a separable Hilbert space is given. The large time behaviour of the solution is studied. In addition, equations of this type with a nonlinear perturbation of a drift part are investigated in the case H > 1/2. Keywords: Fractional Brownian Motion, Stochastic Differential Equations in Hilbert Space, Explicit Formula for Solution
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Invariant measures for dissipative stochastic differential equations
Lavička, Karel ; Seidler, Jan (advisor) ; Maslowski, Bohdan (referee)
The main topic of this Thesis is a new simplified proof of the Sunyach theorem that provides suffici- ent conditions for existence and uniqueness of an invariant measure for a Markov kernel on a complete separable metric space equipped with its Borel σ-algebra. Weak convergence of measures following from Sunyach's theorem is strengthened to convergence in the total variation norm provided that the Markov kernel is strong Feller. Furthermore, sufficient conditions for geometric ergodicity are stated. Another topic treated is the strong Feller property: its characterization by absolute measurability and uniform integrability and derivation of some other sufficient conditions.
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Slabá řešení stochastických diferenciálních rovnic
Hofmanová, Martina ; Maslowski, Bohdan (referee) ; Seidler, Jan (advisor)
In the present work we study a stochastic di fferential equation with coefficients continuous in x having in this variable linear growth. As a main result we show that there exists a weak solution to this equation by a new, more elementary method. Standard methods are based either on the concept of the weak solution or equivalently on solving a martingale problem. However, both approaches employ the integral representation theorem for martingales, whose proof becomes rather complicated in dimension greater than one. By a simple modi cation of the usual procedure, one can identify the weak solution elementary, with no need to apply the above mentioned theorem. In the preliminaries we summarize some auxiliary results: namely, some properties of the space of continuous functions as the space of trajectories are established and an important theorem which allows us to approximate continuous function by functions Lipschitz continuous is proved.
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