Coarse graining of stochastic differential equations: averaging and projection method
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911010781134848 |
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| author | Duong, Manh Hong Hartmann, Carsten Ottobre, Michela |
| author_facet | Duong, Manh Hong Hartmann, Carsten Ottobre, Michela |
| contents | We study coarse-graining methods for stochastic differential equations. In particular we consider averaging and a type of projection operator method, sometimes referred to as effective dynamic via conditional expectations. The projection method (PM) we consider is related to the ``mimicking marginals'' coarse graining approach proposed by Gyöngy. The first contribution of this paper is to provide further theoretical background for the PM and a rigorous link to the Gyöngy method. Moreover, we compare PM and averaging. While averaging applies to systems with time scale separation, the PM can in principle be applied irrespective of this. However it is often assumed that the two methods coincide in presence of scale separation. The second contribution of this paper is to make this statement precise, provide sufficient conditions under which these two methods coincide and then show -- via examples and counterexamples -- that this needs not be the case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_14939 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coarse graining of stochastic differential equations: averaging and projection method Duong, Manh Hong Hartmann, Carsten Ottobre, Michela Probability Mathematical Physics Dynamical Systems 60H10, 34F05, 60J60, 34K33, 35B40, 82C31 We study coarse-graining methods for stochastic differential equations. In particular we consider averaging and a type of projection operator method, sometimes referred to as effective dynamic via conditional expectations. The projection method (PM) we consider is related to the ``mimicking marginals'' coarse graining approach proposed by Gyöngy. The first contribution of this paper is to provide further theoretical background for the PM and a rigorous link to the Gyöngy method. Moreover, we compare PM and averaging. While averaging applies to systems with time scale separation, the PM can in principle be applied irrespective of this. However it is often assumed that the two methods coincide in presence of scale separation. The second contribution of this paper is to make this statement precise, provide sufficient conditions under which these two methods coincide and then show -- via examples and counterexamples -- that this needs not be the case. |
| title | Coarse graining of stochastic differential equations: averaging and projection method |
| topic | Probability Mathematical Physics Dynamical Systems 60H10, 34F05, 60J60, 34K33, 35B40, 82C31 |
| url | https://arxiv.org/abs/2506.14939 |