Onsager-Machlup Functional for SDE with Time-Varying Fractional Noise
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909899443666944 |
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| author | Zhu, Yanbin Jiang, Xiaomeng Li, Yong |
| author_facet | Zhu, Yanbin Jiang, Xiaomeng Li, Yong |
| contents | In this paper, we derive the Onsager-Machlup functional for stochastic differential equations driven by time-varying fractional noise of the form X(t) = x0 + integral from 0 to t b_s(X(s)) ds + integral from 0 to t sigma_s dB^H(s), where B^H denotes fractional Brownian motion with Hurst parameter H. Our main results are established for H in (1/4, 1) by extending small ball probability estimates and the Girsanov theorem for fractional Brownian motion to the setting with time-dependent coefficients. Regarding the choice of norms, for 1/4 < H < 1/2 the analysis is valid under the supremum norm and Holder norms of order 0 < beta < H - 1/4. For 1/2 < H < 1 the analysis applies to Holder norms of order beta satisfying H - 1/2 < beta < H - 1/4. In the case H = 1/2, the admissible norms depend on the spatial regularity of the drift coefficient b: specifically, if b is n-times continuously differentiable, then Holder norms of order 0 < beta < 1/2 - 1/(2n) are permissible. To validate our theoretical findings, we perform numerical simulations for a classical double-well potential system, illustrating how time-varying fractional noise influences transition dynamics between metastable states. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09300 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Onsager-Machlup Functional for SDE with Time-Varying Fractional Noise Zhu, Yanbin Jiang, Xiaomeng Li, Yong Probability 60H10, 60G22, 60F10 In this paper, we derive the Onsager-Machlup functional for stochastic differential equations driven by time-varying fractional noise of the form X(t) = x0 + integral from 0 to t b_s(X(s)) ds + integral from 0 to t sigma_s dB^H(s), where B^H denotes fractional Brownian motion with Hurst parameter H. Our main results are established for H in (1/4, 1) by extending small ball probability estimates and the Girsanov theorem for fractional Brownian motion to the setting with time-dependent coefficients. Regarding the choice of norms, for 1/4 < H < 1/2 the analysis is valid under the supremum norm and Holder norms of order 0 < beta < H - 1/4. For 1/2 < H < 1 the analysis applies to Holder norms of order beta satisfying H - 1/2 < beta < H - 1/4. In the case H = 1/2, the admissible norms depend on the spatial regularity of the drift coefficient b: specifically, if b is n-times continuously differentiable, then Holder norms of order 0 < beta < 1/2 - 1/(2n) are permissible. To validate our theoretical findings, we perform numerical simulations for a classical double-well potential system, illustrating how time-varying fractional noise influences transition dynamics between metastable states. |
| title | Onsager-Machlup Functional for SDE with Time-Varying Fractional Noise |
| topic | Probability 60H10, 60G22, 60F10 |
| url | https://arxiv.org/abs/2511.09300 |