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| Author(s): |
Jamil Gomes de Abreu Júnior
Total Authors: 1
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| Document type: | Doctoral Thesis |
| Press: | Campinas, SP. |
| Institution: | Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica |
| Defense date: | 2013-02-05 |
| Examining board members: |
Pedro Jose Catuogno;
Alberto Masayoshi Faria Ohashi;
Daniel Marinho Pellegrino;
Edson Alberto Coayla Teran;
Jorge Mujica
|
| Advisor: | Johannes Michael Antonius Maria van Neerven; Pedro Jose Catuogno |
| 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 (AU) | |
| FAPESP's process: | 07/08220-9 - Stochastic partial differential equations and Lie Groups |
| Grantee: | Jamil Gomes de Abreu Júnior |
| Support Opportunities: | Scholarships in Brazil - Doctorate (Direct) |