Profile cut-off phenomenon for the ergodic Feller root process
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915142430621696 |
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| author | Barrera, Gerardo Esquivel, Liliana |
| author_facet | Barrera, Gerardo Esquivel, Liliana |
| contents | The present manuscript is devoted to the study of the convergence to equilibrium as the noise intensity $\varepsilon>0$ tends to zero for ergodic random systems out of equilibrium of the type \begin{align*} \mathrm{d} X^{\varepsilon}_t(x) = (\mathfrak{b}-\mathfrak{a} X^{\varepsilon}_t(x))\mathrm{d} t+\varepsilon \sqrt{X^{\varepsilon}_t(x)}\mathrm{d} B_t, \quad X^{\varepsilon}_0(x) = x, \quad t\geqslant 0, \end{align*} where $x\geqslant 0$, $\mathfrak{a}>0$ and $\mathfrak{b}>0$ are constants, and $(B_t)_{t \geqslant 0}$ is a one dimensional standard Brownian motion. More precisely, we show the strongest notion of asymptotic profile cut-off phenomenon in the total variation distance and in the renormalized Wasserstein distance when $\varepsilon$ tends to zero with explicit cut-off time, explicit time window, and explicit profile function. In addition, asymptotics of the so-called mixing times are given explicitly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Profile cut-off phenomenon for the ergodic Feller root process Barrera, Gerardo Esquivel, Liliana Probability 60H10, 37A25, 60B10, 34D10 The present manuscript is devoted to the study of the convergence to equilibrium as the noise intensity $\varepsilon>0$ tends to zero for ergodic random systems out of equilibrium of the type \begin{align*} \mathrm{d} X^{\varepsilon}_t(x) = (\mathfrak{b}-\mathfrak{a} X^{\varepsilon}_t(x))\mathrm{d} t+\varepsilon \sqrt{X^{\varepsilon}_t(x)}\mathrm{d} B_t, \quad X^{\varepsilon}_0(x) = x, \quad t\geqslant 0, \end{align*} where $x\geqslant 0$, $\mathfrak{a}>0$ and $\mathfrak{b}>0$ are constants, and $(B_t)_{t \geqslant 0}$ is a one dimensional standard Brownian motion. More precisely, we show the strongest notion of asymptotic profile cut-off phenomenon in the total variation distance and in the renormalized Wasserstein distance when $\varepsilon$ tends to zero with explicit cut-off time, explicit time window, and explicit profile function. In addition, asymptotics of the so-called mixing times are given explicitly. |
| title | Profile cut-off phenomenon for the ergodic Feller root process |
| topic | Probability 60H10, 37A25, 60B10, 34D10 |
| url | https://arxiv.org/abs/2402.15457 |