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Slabá řešení stochastických diferenciálních rovnic
Hofmanová, Martina ; Seidler, Jan (advisor) ; Maslowski, Bohdan (referee)
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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Degenerate Parabolic Stochastic Partial Differential Equations
Hofmanová, Martina ; Seidler, Jan (advisor) ; Perthame, Benoit (referee) ; Flandoli, Franco (referee)
In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyper- bolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochas- tic forcing and study their approximations in the sense of Bhatnagar-Gross- Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkohod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition. 1
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Degenerate Parabolic Stochastic Partial Differential Equations
Hofmanová, Martina
In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyper- bolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochas- tic forcing and study their approximations in the sense of Bhatnagar-Gross- Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkohod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition. 1
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Degenerate Parabolic Stochastic Partial Differential Equations
Hofmanová, Martina
In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyper- bolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochas- tic forcing and study their approximations in the sense of Bhatnagar-Gross- Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkohod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition. 1
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Degenerate Parabolic Stochastic Partial Differential Equations
Hofmanová, Martina ; Seidler, Jan (advisor) ; Perthame, Benoit (referee) ; Flandoli, Franco (referee)
In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyper- bolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochas- tic forcing and study their approximations in the sense of Bhatnagar-Gross- Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkohod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition. 1
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Winter roosts of Rooks (Corvus frugilegus) and associated Jackdaws (Corvus monedula): social behaviour and interseasonal dynamics
Hofmanová, Martina ; Musilová, Zuzana (advisor) ; Šťastný, Karel (referee)
The presented work summarizes the literary knowledge of the collective winter sleepovers for Rook Corvus frugilegus and Jackdaw Corvus monedula, particularly focusing on the behavior during the day to revaluate the potential benefits of the causes of mass sleepover. Both species are migratory birds which have migrated throughout most of Europe. Rook is from Corvus genus and it is very kind, very social, preferring nesting gathering food in the colonies. It is synanthropy, which shows strong binding to areas inhabited by humans. He uses mass sleepovers and regularly fly between sleapover and food source. The work also includes analysis of own results mid season dynamics of Rooks and Jackdaws in traditional sleepover base in Kralupy nad Vltavou and their neighborhood. The results show a decrease in Rooks and Jackdaws. Among the main consequences of the decline of these specieslikely include changes in the landscape caused by natural phenomena in the form of floods, loss of agricultural land farmed soil, urban modernization, increasing development and by changing of climatic 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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