Multiple Markov generalized Gaussian processes and their dualities. (English) Zbl 1207.60049
Author’s abstract: A stochastic process describes evolutional random phenomena towards the future. Corresponding with this direction, there will be a process describing, as it were, the backward evolution from future to present. The dual process is a realization of this fact.
We restrict our attention to Gaussian processes for which the multiple Markov property is well investigated. This property expresses the way of the dependence of the involved randomness as time goes by. We can therefore expect that a process with the multiple Markov property would present an exact form of the dual process.
We show that having defined generalized Gaussian process with multiple Markov property, we can construct the dual process satisfying the properties that we claim. The analysis of this can be done in the space of Hida distributions.
We restrict our attention to Gaussian processes for which the multiple Markov property is well investigated. This property expresses the way of the dependence of the involved randomness as time goes by. We can therefore expect that a process with the multiple Markov property would present an exact form of the dual process.
We show that having defined generalized Gaussian process with multiple Markov property, we can construct the dual process satisfying the properties that we claim. The analysis of this can be done in the space of Hida distributions.
Reviewer: Zagorka Lozanov-Crvenković (Novi Sad)
MSC:
| 60H40 | White noise theory |
Keywords:
Gaussian processes; generalized processes; canonical representation; multiple Markov property; dualityReferences:
| [1] | DOI: 10.1214/aoms/1177731234 · Zbl 0060.28907 · doi:10.1214/aoms/1177731234 |
| [2] | Hida T., Mem. Coll. Sci. Univ. Kyoto 34 pp 109– |
| [3] | DOI: 10.1142/9789812812049 · doi:10.1142/9789812812049 |
| [4] | Gel’fand I. M., Generalized Functions 4 (1964) |
| [5] | Itô K., Mem. Univ. Kyoto A 28 |
| [6] | Lions J. L., Non-Homogeneous Boundary Value Problems and Applications, I (1972) |
| [7] | Schwartz L., Théorie des Distributions (1950) |
| [8] | DOI: 10.1142/9789814304061_0026 · doi:10.1142/9789814304061_0026 |
| [9] | Si Si, Introduction to Hida Distributions · doi:10.1142/7103 |
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