Christian Fonseca-Mora

Bio: Christian Fonseca-Mora is an academic researcher from University of Costa Rica. The author has an hindex of 2, co-authored 2 publications receiving 19 citations.

Existence of Continuous and Càdlàg Versions for Cylindrical Processes in the Dual of a Nuclear Space

Abstract: Let $$\Phi $$ be a nuclear space and let $$\Phi '_{\beta }$$ denote its strong dual. In this paper, we introduce sufficient conditions for a cylindrical process in $$\Phi '$$ to have a version that is a $$\Phi '_{\beta }$$ -valued continuous or cadlag process. We also establish sufficient conditions for the existence of such a version taking values and having finite moments in a Hilbert space continuously embedded in $$\Phi '_{\beta }$$ . Finally, we apply our results to the study of properties of cylindrical martingales in $$\Phi '$$ .

Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space

Abstract: Let $$\Phi $$ be a nuclear space and let $$\Phi '_{\beta }$$ denote its strong dual. In this work, we prove the existence of cadlag versions, the Levy–Ito decomposition and the Levy–Khintchine formula for $$\Phi '_{\beta }$$-valued Levy processes. Moreover, we give a characterization for Levy measures on $$\Phi '_{\beta }$$ and provide conditions for the existence of regular versions to cylindrical Levy processes in $$\Phi '$$. Furthermore, under the assumption that $$\Phi $$ is a barrelled nuclear space we establish a one-to-one correspondence between infinitely divisible measures on $$\Phi '_{\beta }$$ and Levy processes in $$\Phi '_{\beta }$$. Finally, we prove the Levy–Khintchine formula for infinitely divisible measures on $$\Phi '_{\beta }$$.

Stochastic integration and stochastic PDEs driven by jumps on the dual of a nuclear space

Abstract: We develop a novel theory of weak and strong stochastic integration for cylindrical martingale-valued measures taking values in the dual of a nuclear space. This is applied to develop a theory of SPDEs with rather general coefficients. In particular, we can then study SPDEs driven by general Levy processes in this context.

L\'{e}vy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space

Abstract: Let $\Phi$ be a nuclear space and let $\Phi'_{\beta}$ denote its strong dual. In this work we establish the one-to-one correspondence between infinitely divisible measures on $\Phi'_{\beta}$ and Levy processes taking values in $\Phi'_{\beta}$. Moreover, we prove the Levy-Ito decomposition, the Levy-Khintchine formula and the existence of cadlag versions for $\Phi'_{\beta}$-valued Levy processes. A characterization for Levy measures on $\Phi'_{\beta}$ is also established. Finally, we prove the Levy-Khintchine formula for infinitely divisible measures on $\Phi'_{\beta}$.

Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space

Abstract: Let $$\Phi $$ be a nuclear space and let $$\Phi '_{\beta }$$ denote its strong dual. In this work, we prove the existence of cadlag versions, the Levy–Ito decomposition and the Levy–Khintchine formula for $$\Phi '_{\beta }$$-valued Levy processes. Moreover, we give a characterization for Levy measures on $$\Phi '_{\beta }$$ and provide conditions for the existence of regular versions to cylindrical Levy processes in $$\Phi '$$. Furthermore, under the assumption that $$\Phi $$ is a barrelled nuclear space we establish a one-to-one correspondence between infinitely divisible measures on $$\Phi '_{\beta }$$ and Levy processes in $$\Phi '_{\beta }$$. Finally, we prove the Levy–Khintchine formula for infinitely divisible measures on $$\Phi '_{\beta }$$.

Modelling Lévy space-time white noises

Abstract: Based on the theory of independently scattered random measures, we introduce a natural generalisation of Gaussian space-time white noise to a Levy-type setting, which we call Levy-valued random measures. We determine the subclass of cylindrical Levy processes which correspond to Levy-valued random measures, and describe the elements of this subclass uniquely by their characteristic function. We embed the Levy-valued random measure, or the corresponding cylindrical Levy process, in the space of general and tempered distributions. For the latter case, we show that this embedding is possible if and only if a certain integrability condition is satisfied. Similar to existing definitions, we introduce Levy-valued additive sheets, and show that integrating a Levy-valued random measure in space defines a Levy-valued additive sheet. This relation is manifested by the result, that a Levy-valued random measure can be viewed as the weak derivative of a Levy-valued additive sheet in the space of distributions.