Negative regularity mixing for random volume preserving diffeomorphisms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912085359722496 |
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| author | Bedrossian, Jacob Flynn, Patrick Punshon-Smith, Sam |
| author_facet | Bedrossian, Jacob Flynn, Patrick Punshon-Smith, Sam |
| contents | We consider the negative regularity mixing properties of random volume preserving diffeomorphisms on a compact manifold without boundary. We give general criteria so that the associated random transfer operator mixes $H^{-δ}$ observables exponentially fast in $H^{-δ}$ (with a deterministic rate), a property that is false in the deterministic setting. The criteria apply to a wide variety of random diffeomorphisms, such as discrete-time iid random diffeomorphisms, the solution maps of suitable classes of stochastic differential equations, and to the case of advection-diffusion by solutions of the stochastic incompressible Navier-Stokes equations on $\mathbb T^2$. In the latter case, we show that the zero diffusivity passive scalar with a stochastic source possesses a unique stationary measure describing "ideal" scalar turbulence. The proof is based on techniques inspired by the use of pseudodifferential operators and anisotropic Sobolev spaces in the deterministic setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_19251 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Negative regularity mixing for random volume preserving diffeomorphisms Bedrossian, Jacob Flynn, Patrick Punshon-Smith, Sam Analysis of PDEs Dynamical Systems Probability 35 (Primary), 60, 37, 76 (Secondary) We consider the negative regularity mixing properties of random volume preserving diffeomorphisms on a compact manifold without boundary. We give general criteria so that the associated random transfer operator mixes $H^{-δ}$ observables exponentially fast in $H^{-δ}$ (with a deterministic rate), a property that is false in the deterministic setting. The criteria apply to a wide variety of random diffeomorphisms, such as discrete-time iid random diffeomorphisms, the solution maps of suitable classes of stochastic differential equations, and to the case of advection-diffusion by solutions of the stochastic incompressible Navier-Stokes equations on $\mathbb T^2$. In the latter case, we show that the zero diffusivity passive scalar with a stochastic source possesses a unique stationary measure describing "ideal" scalar turbulence. The proof is based on techniques inspired by the use of pseudodifferential operators and anisotropic Sobolev spaces in the deterministic setting. |
| title | Negative regularity mixing for random volume preserving diffeomorphisms |
| topic | Analysis of PDEs Dynamical Systems Probability 35 (Primary), 60, 37, 76 (Secondary) |
| url | https://arxiv.org/abs/2410.19251 |