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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.00243 |
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| _version_ | 1866909598031544320 |
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| author | Hartmann, Carsten Neureither, Lara Sharma, Upanshu |
| author_facet | Hartmann, Carsten Neureither, Lara Sharma, Upanshu |
| contents | This paper deals with the realisation of affine constraints on nonreversible stochastic differential equations (SDE) by strong confining forces. We prove that the confined dynamics converges pathwise and on bounded time intervals to the solution of a projected SDE in the limit of infinitely strong confinement, where the projection is explicitly given and depends on the choice of the confinement force. We present results for linear Ornstein-Uhlenbeck (OU) processes, but they straightforwardly generalise to nonlinear SDEs. Moreover, for linear OU processes that admit a unique invariant measure, we discuss conditions under which the limit also preserves the long-term properties of the SDE. More precisely, we discuss choices for the design of the confinement force which in the limit yield a projected dynamics with invariant measure that agrees with the conditional invariant measure of the unconstrained processes for the given constraint. The theoretical findings are illustrated with suitable numerical examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00243 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affine constraints in non-reversible diffusions with degenerate noise Hartmann, Carsten Neureither, Lara Sharma, Upanshu Probability 60H10, 60J60 This paper deals with the realisation of affine constraints on nonreversible stochastic differential equations (SDE) by strong confining forces. We prove that the confined dynamics converges pathwise and on bounded time intervals to the solution of a projected SDE in the limit of infinitely strong confinement, where the projection is explicitly given and depends on the choice of the confinement force. We present results for linear Ornstein-Uhlenbeck (OU) processes, but they straightforwardly generalise to nonlinear SDEs. Moreover, for linear OU processes that admit a unique invariant measure, we discuss conditions under which the limit also preserves the long-term properties of the SDE. More precisely, we discuss choices for the design of the confinement force which in the limit yield a projected dynamics with invariant measure that agrees with the conditional invariant measure of the unconstrained processes for the given constraint. The theoretical findings are illustrated with suitable numerical examples. |
| title | Affine constraints in non-reversible diffusions with degenerate noise |
| topic | Probability 60H10, 60J60 |
| url | https://arxiv.org/abs/2505.00243 |