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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.04926 |
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| _version_ | 1866911772960620544 |
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| author | Oberguggenberger, Michael |
| author_facet | Oberguggenberger, Michael |
| contents | The paper addresses the question whether a random functional, a map from a set $E$ into the space of real-valued measurable functions on a probability space, has a measurable version with values in ${\mathbb R}^E$. Similarly, one may ask whether linear random functionals have versions in the algebraic dual. Most importantly, it can be asked which locally convex topological vector spaces $E$ have the ``regularity property'' that any linear random functional on $E$ has a version with values in the dual $E'$, an important issue in the theory of generalized stochastic processes. It has been shown by Itô and Nawata that this is the case when $E$ is nuclear. However, the question of uniqueness has only been partially answered. We build up a framework where these and related questions can be clarified in terms of spaces and mappings. We study classes of spaces $E$ (beyond nuclear spaces) with the said regularity property, prove a seemingly new uniqueness result and exhibit various examples and counterexamples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04926 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized stochastic processes revisited Oberguggenberger, Michael Functional Analysis Probability 60G20 (Primary), 46A99 (Secondary) The paper addresses the question whether a random functional, a map from a set $E$ into the space of real-valued measurable functions on a probability space, has a measurable version with values in ${\mathbb R}^E$. Similarly, one may ask whether linear random functionals have versions in the algebraic dual. Most importantly, it can be asked which locally convex topological vector spaces $E$ have the ``regularity property'' that any linear random functional on $E$ has a version with values in the dual $E'$, an important issue in the theory of generalized stochastic processes. It has been shown by Itô and Nawata that this is the case when $E$ is nuclear. However, the question of uniqueness has only been partially answered. We build up a framework where these and related questions can be clarified in terms of spaces and mappings. We study classes of spaces $E$ (beyond nuclear spaces) with the said regularity property, prove a seemingly new uniqueness result and exhibit various examples and counterexamples. |
| title | Generalized stochastic processes revisited |
| topic | Functional Analysis Probability 60G20 (Primary), 46A99 (Secondary) |
| url | https://arxiv.org/abs/2402.04926 |