Existence of stationary measures for partially damped SDEs with generic, Euler-type nonlinearities
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929432932909056 |
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| author | Bedrossian, Jacob Blumenthal, Alex Callis, Keagan Liss, Kyle |
| author_facet | Bedrossian, Jacob Blumenthal, Alex Callis, Keagan Liss, Kyle |
| contents | We study nonlinear energy transfer and the existence of stationary measures in a class of degenerately forced SDEs on $\mathbb R^d$ with a quadratic, conservative nonlinearity $B(x,x)$ constrained to possess various properties common to finite-dimensional fluid models and a linear damping term $-Ax$ that acts only on a proper subset of phase space in the sense that $\mathrm{dim}(\mathrm{ker}A) \gg 1$. Existence of a stationary measure is straightforward if $\mathrm{ker}A = \{0\}$, but when the kernel of $A$ is nontrivial a stationary measure can exist only if the nonlinearity transfers enough energy from the undamped modes to the damped modes. We develop a set of sufficient dynamical conditions on $B$ that guarantees the existence of a stationary measure and prove that they hold ``generically'' within our constraint class of nonlinearities provided that $\mathrm{dim}(\mathrm{ker}A) < 2d/3$ and the stochastic forcing acts directly on at least two degrees of freedom. We also show that the restriction $\mathrm{dim}(\mathrm{ker}A) < 2d/3$ can be removed if one allows the nonlinearity to change by a small amount at discrete times. In particular, for a Markov chain obtained by evolving our SDE on approximately unit random time intervals and slightly perturbing the nonlinearity within our constraint class at each timestep, we prove that there exists a stationary measure whenever just a single mode is damped. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16592 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of stationary measures for partially damped SDEs with generic, Euler-type nonlinearities Bedrossian, Jacob Blumenthal, Alex Callis, Keagan Liss, Kyle Probability Analysis of PDEs We study nonlinear energy transfer and the existence of stationary measures in a class of degenerately forced SDEs on $\mathbb R^d$ with a quadratic, conservative nonlinearity $B(x,x)$ constrained to possess various properties common to finite-dimensional fluid models and a linear damping term $-Ax$ that acts only on a proper subset of phase space in the sense that $\mathrm{dim}(\mathrm{ker}A) \gg 1$. Existence of a stationary measure is straightforward if $\mathrm{ker}A = \{0\}$, but when the kernel of $A$ is nontrivial a stationary measure can exist only if the nonlinearity transfers enough energy from the undamped modes to the damped modes. We develop a set of sufficient dynamical conditions on $B$ that guarantees the existence of a stationary measure and prove that they hold ``generically'' within our constraint class of nonlinearities provided that $\mathrm{dim}(\mathrm{ker}A) < 2d/3$ and the stochastic forcing acts directly on at least two degrees of freedom. We also show that the restriction $\mathrm{dim}(\mathrm{ker}A) < 2d/3$ can be removed if one allows the nonlinearity to change by a small amount at discrete times. In particular, for a Markov chain obtained by evolving our SDE on approximately unit random time intervals and slightly perturbing the nonlinearity within our constraint class at each timestep, we prove that there exists a stationary measure whenever just a single mode is damped. |
| title | Existence of stationary measures for partially damped SDEs with generic, Euler-type nonlinearities |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2407.16592 |