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Russo Francesco - One of the best experts on this subject based on the ideXlab platform.
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BSDEs with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples.
HAL CCSD, 2021Co-Authors: Barrasso Adrien, Russo FrancescoAbstract:Let $(\mathbb{P}^{s,x})_{(s,x)\in[0,T]\times E}$ be a family of probability measures, where $E$ is a Polish Space,defined on the canonical probability Space ${\mathbb D}([0,T],E)$ of $E$-valued cadlag functions. We suppose that a martingale problem with respect to a time-inhomogeneous generator $a$ is well-posed. We consider also an associated semilinear {\it Pseudo-PDE} with generator $a$ for which we introduce a notion of so called {\it decoupled mild} solution and study the equivalence with the notion of martingale solution introduced in a companion paper. We also investigate well-posedness for decoupled mild solutions and their relations with a special class of BSDEs without driving martingale. The notion of decoupled mild solution is a good candidate to replace the notion of viscosity solution which is not always suitable when the map $a$ is not a PDE operator
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BSDEs with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples
2020Co-Authors: Barrasso Adrien, Russo FrancescoAbstract:Let $(\mathbb{P}^{s,x})\_{(s,x)\in[0,T]\times E}$ be a family of probability measures, where $E$ is a Polish Space,defined on the canonical probability Space ${\mathbb D}([0,T],E)$ of $E$-valued cadlag functions. We suppose that a martingale problem with respect to a time-inhomogeneous generator $a$ is well-posed. We consider also an associated semilinear {\it Pseudo-PDE} with generator $a$ for which we introduce a notion of so called {\it decoupled mild} solution and study the equivalence with the notion of martingale solution introduced in a companion paper. We also investigate well-posedness for decoupled mild solutions and their relations with a special class of BSDEs without driving martingale. The notion of decoupled mild solution is a good candidate to replace the notion of viscosity solution which is not always suitable when the map $a$ is not a PDE operator.Comment: arXiv admin note: text overlap with arXiv:1701.0289
Barrasso Adrien - One of the best experts on this subject based on the ideXlab platform.
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BSDEs with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples.
HAL CCSD, 2021Co-Authors: Barrasso Adrien, Russo FrancescoAbstract:Let $(\mathbb{P}^{s,x})_{(s,x)\in[0,T]\times E}$ be a family of probability measures, where $E$ is a Polish Space,defined on the canonical probability Space ${\mathbb D}([0,T],E)$ of $E$-valued cadlag functions. We suppose that a martingale problem with respect to a time-inhomogeneous generator $a$ is well-posed. We consider also an associated semilinear {\it Pseudo-PDE} with generator $a$ for which we introduce a notion of so called {\it decoupled mild} solution and study the equivalence with the notion of martingale solution introduced in a companion paper. We also investigate well-posedness for decoupled mild solutions and their relations with a special class of BSDEs without driving martingale. The notion of decoupled mild solution is a good candidate to replace the notion of viscosity solution which is not always suitable when the map $a$ is not a PDE operator
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BSDEs with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples
2020Co-Authors: Barrasso Adrien, Russo FrancescoAbstract:Let $(\mathbb{P}^{s,x})\_{(s,x)\in[0,T]\times E}$ be a family of probability measures, where $E$ is a Polish Space,defined on the canonical probability Space ${\mathbb D}([0,T],E)$ of $E$-valued cadlag functions. We suppose that a martingale problem with respect to a time-inhomogeneous generator $a$ is well-posed. We consider also an associated semilinear {\it Pseudo-PDE} with generator $a$ for which we introduce a notion of so called {\it decoupled mild} solution and study the equivalence with the notion of martingale solution introduced in a companion paper. We also investigate well-posedness for decoupled mild solutions and their relations with a special class of BSDEs without driving martingale. The notion of decoupled mild solution is a good candidate to replace the notion of viscosity solution which is not always suitable when the map $a$ is not a PDE operator.Comment: arXiv admin note: text overlap with arXiv:1701.0289
Pandelis Dodos - One of the best experts on this subject based on the ideXlab platform.
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Codings of separable compact subsets of the first Baire class
2016Co-Authors: Pandelis DodosAbstract:Abstract. Let X be a Polish Space and K a separable compact subset of the first Baire class on X. For every sequence f = (fn)n dense in K, the descriptive set-theoretic properties of the set Lf = {L ∈ [N] : (fn)n∈L is pointwise convergent} are analyzed. It is shown that if K is not first countable, then Lf is Π11-complete. This can also happen even if K is a pre-metric compactum of degree at most two, in the sense of S. Todorčević. However, if K is of degree exactly two, then Lf is always Borel. A deep result of G. Debs implies that Lf contains a Borel cofinal set and this gives a tree-representation of K. We show that classical ordinal assignments of Baire-1 functions are actually Π11-ranks on K. We also provide an example of a Σ11 Ramsey-null subset A of [N] for which there does not exist a Borel set B ⊇ A such that the difference B \ A is Ramsey-null. 1
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dichotomies of the set of test measures of a haar null set
arXiv: Functional Analysis, 2010Co-Authors: Pandelis DodosAbstract:We prove that if $X$ is a Polish Space and $F$ is a face of $P(X)$ with the Baire property, then $F$ is either a meager or a co-meager subset of $P(X)$. As a consequence we show that for every abelian Polish group $X$ and every analytic Haar-null set $A\subseteq X$, the set of test measures $T(A)$ of $A$ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null set $F\subseteq X$ with $T(F)$ is meager. Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every $\sigma$-compact subgroup $G$ of $X$ there exists a $G$-invariant $F_\sigma$ subset of $X$ which is neither prevalent nor Haar-null.
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dichotomies of the set of test measures of a haar null set
Israel Journal of Mathematics, 2004Co-Authors: Pandelis DodosAbstract:We prove that ifX is a Polish Space andF a face ofP(X) with the Baire property, thenF is either a meager or a co-meager subset ofP(X). As a consequence we show that for every abelian Polish groupX and every analytic Haar-null set Λ⊆X, the set of test measuresT(Λ) of Λ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null setF⊆X withT(F) meager, Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every σ-compact subgroupG ofX there exists aG-invariantF σ subset ofX which is neither prevalent nor Haar-null.
Huaizhong Zhao - One of the best experts on this subject based on the ideXlab platform.
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Random periodic processes, periodic measures and ergodicity
2020Co-Authors: Chunrong Feng, Huaizhong ZhaoAbstract:© 2020 The Author(s) Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish Space. In the Markovian case, the idea of Poincaré sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only has equally placed simple eigenvalues including 0 on the imaginary axis, then the periodic measure is PS-ergodic and has positive minimum period. Conversely if the periodic measure with the positive minimum period is PS-mixing, then the infinitesimal generator only has equally placed simple eigenvalues (infinitely many) including 0 on the imaginary axis. Moreover, under the spectral gap condition, PS-mixing of the periodic measure is proved. The “equivalence” of random periodic processes and periodic measures is established. This is a new class of ergodic random processes. Random periodic paths of stochastic perturbation of the periodic motion of an ODE is obtained
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Random Periodic Processes, Periodic Measures and Strong Law of Large Numbers
2014Co-Authors: Chunrong Feng, Huaizhong ZhaoAbstract:Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish Space. In the Markovian case, the idea of Poincare sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only has equally placed simple eigenvalues including $0$ on the imaginary axis, then the periodic measure is PS-ergodic and has positive minimum period. Conversely if the periodic measure with the positive minimum period is PS-mixing, then the infinitesimal generator only has equally placed simple eigenvalues (infinitely many) including $0$ on the imaginary axis. Moreover, under the spectral gap condition, PS-mixing of the periodic measure is proved. The ``equivalence" of random periodic processes and periodic measures is established. This is a new class of ergodic random processes. Random periodic paths of stochastic perturbation of the periodic motion of an ODE is obtained.
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Relationship between Random Periodic Paths, Periodic Measures, and Invariant Measures
2014Co-Authors: Chunrong Feng, Yong Liu, Huaizhong ZhaoAbstract:Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish Space. In the Markovian case, the idea of Poincare sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only has equally placed simple eigenvalues including $0$ on the imaginary axis, then the periodic measure is PS-ergodic and has positive minimum period. Conversely if the periodic measure with the positive minimum period is PS-mixing, then the infinitesimal generator only has equally placed simple eigenvalues (infinitely many) including $0$ on the imaginary axis. Moreover, under the spectral gap condition, PS-mixing of the periodic measure is proved. The ``equivalence" of random periodic processes and periodic measures is established. This is a new class of ergodic random processes. Random periodic paths of stochastic perturbation of the periodic motion of an ODE is obtained.
Aaron Hill - One of the best experts on this subject based on the ideXlab platform.
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Topological isomorphism for rank-1 systems
Journal d'Analyse Mathématique, 2016Co-Authors: Su Gao, Aaron HillAbstract:We define the Polish Space R of non-degenerate rank-1 systems. Each non-degenerate rank-1 system can be viewed as a measure-preserving transformation of an atomless, σ -finite measure Space and as a homeomorphism of a Cantor Space. We completely characterize when two non-degenerate rank-1 systems are topologically isomorphic. We also analyze the complexity of the topological isomorphism relation on R, showing that it is $${F_\sigma }$$ as a subset of R × R and bi-reducible to E _0. We also explicitly describe when a non-degenerate rank-1 system is topologically isomorphic to its inverse.
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topological isomorphism for rank 1 systems
arXiv: Dynamical Systems, 2012Co-Authors: Su Gao, Aaron HillAbstract:We define the Polish Space $\mathcal{R}$ of non-degenerate rank-1 systems. Each non-degenerate rank-1 system can be viewed as a measure-preserving transformation of an atomless, $\sigma$-finite measure Space and as a homeomorphism of a Cantor Space. We completely characterize when two non-degenerate rank-1 systems are topologically isomorphic. We also analyze the complexity of the topological isomorphism relation on $\mathcal{R}$, showing that it is $F_{\sigma}$ as a subset of $\mathcal{R} \times \mathcal{R}$ and bi-reducible to $E_0$. We also explicitly describe when a non-degenerate rank-1 system is topologically isomorphic to its inverse.
