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| -rwxr-xr-x | main.bib | 19 | ||||
| -rwxr-xr-x | main.tex | 84 |
2 files changed, 94 insertions, 9 deletions
@@ -433,4 +433,21 @@ MRREVIEWER = {Yves Derriennic}, URL = {https://doi-org.extranet.enpc.fr/10.1007/BF00531822}, } - +@incollection {MR2857021, + AUTHOR = {Hairer, Martin and Mattingly, Jonathan C.}, + TITLE = {Yet another look at {H}arris' ergodic theorem for {M}arkov + chains}, + BOOKTITLE = {Seminar on {S}tochastic {A}nalysis, {R}andom {F}ields and + {A}pplications {VI}}, + SERIES = {Progr. Probab.}, + VOLUME = {63}, + PAGES = {109--117}, + PUBLISHER = {Birkh\"{a}user/Springer Basel AG, Basel}, + YEAR = {2011}, + ISBN = {978-3-0348-0020-4}, + MRCLASS = {60J05 (37A30 37A50 47D07)}, + MRNUMBER = {2857021}, +MRREVIEWER = {Wojciech\ Bartoszek}, + DOI = {10.1007/978-3-0348-0021-1\_7}, + URL = {https://doi.org/10.1007/978-3-0348-0021-1_7}, +} @@ -1,5 +1,6 @@ -\documentclass[10pt]{beamer} +\documentclass[9pt]{beamer} \renewcommand{\emph}[1]{\textcolor{blue}{#1}} +\newcommand{\blue}[1]{\textcolor{blue}{#1}} \newif\iflong \longfalse @@ -7,7 +8,7 @@ \input{macros} \newcommand{\highlight}[2]{% - \colorbox{#1!20}{$\displaystyle#2$}} +\colorbox{#1!20}{$\displaystyle#2$}} \newcommand{\hiat}[4]{% \only<#1>{\highlight{#3}{#4}}% @@ -27,13 +28,13 @@ } \author{% - Urbain Vaes \texorpdfstring{\\\texttt{urbain.vaes@inria.fr}}{} + Urbain Vaes \texorpdfstring{\\\texttt{urbain.vaes@inria.fr}}{} } \institute{% - MATHERIALS -- Inria Paris - \textcolor{blue}{\&} CERMICS -- - École des Ponts ParisTech + MATHERIALS -- Inria Paris + \textcolor{blue}{\&} CERMICS -- + École des Ponts ParisTech } \date{\today} @@ -62,7 +63,7 @@ \item \fullcite{MR3509213} \item \fullcite{pavliotis2011applied} \item Lecture notes by Gabriel Stoltz on computational statistical physics: - \url{http://cermics.enpc.fr/~stoltz/Cours/intro_phys_stat.pdf} + \url{http://cermics.enpc.fr/~stoltz/Cours/intro_phys_stat.pdf} \end{itemize} \end{frame} @@ -139,6 +140,27 @@ \end{frame} \begin{frame} + {Invariant distribution in dimension 1} + For the equilibrium overdamped Langevin dynamics + \[ + \d q_t = - V'(q_t) \, \d t + \sqrt{2} \, \d W_t, + \] + the invariant probability distribution is given by~$Z^{-1} \e^{-V(q)} \, \d q$. + For the perturbed dynamics + \[ + \d q_t = - V'(q_t) \, \d t + \blue{\eta} + \sqrt{2} \, \d W_t, + \] + the invariant probability distribution~$\rho_{\eta}$ solves the Fokker--Planck equation + \[ + \frac{\d}{\d q}\left( \left(\frac{\d V}{\d q} - \eta\right) \rho_{\eta} + \frac{\d \rho_{\eta}}{\d q} \right) = 0, + \] + which can be solved as + \[ + \rho_{\eta}(q) \propto \int_{\torus} \e^{V(q+y) - V(q) - \eta y} \, \d y. + \] +\end{frame} + +\begin{frame} {Existence of an invariant measure for noneq.\ dynamics} Consider the paradigmatic dynamics \begin{align*} @@ -147,8 +169,54 @@ \end{align*} where $(q_t, p_t) = \torus^d \times \real^d$ and $F \in \real^d$ with $\abs{F} = 1$ is a given direction. + \begin{figure}[ht] + \centering + \includegraphics[width=0.39\linewidth]{figures/intro_position.pdf} + \includegraphics[width=0.39\linewidth]{figures/intro_velocity.pdf} + \caption{% + Marginals of the steady state solution of the Langevin dynamics with forcing + } + \end{figure} +\end{frame} + +\begin{frame} + {Existence of an invariant distribution} + + \begin{theorem} + Fix~$\eta_* > 0$ and $n \geq 2$, + and let $\mathcal K_n(q, p) := 1 + \abs{p}^n$. + For any $\eta \in [- \eta_*, \eta_*]$, + there exists a unique invariant probability measure, + with a smooth density~$\psi_{\eta}(q, p)$ with respect to the Lebesgue measure. + Furthermore there exists $C = C(n, \eta_*) > 0$ and $\lambda = \lambda(n, \eta_*) > 0$ such that + \[ + \forall \phi \in L^{\infty}_{\mathcal K_n}(\mathcal E), \qquad + \left\lVert \e^{t \mathcal L_n} \phi - \int_{\mathcal E} \phi \, \psi_{\eta} \right\rVert_{L^{\infty}_{\mathcal K_n}} + \leq C \e^{-\lambda t} \left\lVert \phi - \int_{\mathcal E} \phi \, \psi_{\eta} \right\rVert_{L^{\infty}_{\mathcal K_n}} + \] + \end{theorem} + + \textbf{Idea of the proof.} + Show that + \begin{align*} + \mathcal L \mathcal K_n &\leq - c_1 \mathcal K_n(q, p) + c_2, + \end{align*} + for $c_1 > 0$ and $c_2 > 0$. + Then apply the main theorem from~\footfullcite{MR2857021}. +\end{frame} + +\begin{frame} + {Existence of an invariant measure} + \[ + d(P \mu, P \nu) + \leq + \] +\end{frame} + +\begin{frame} + \end{frame} \end{document} -% vim: ts=2 sw=2 +% vim: ts=4 sw=4 |