From 06ebedae24304577f71880412ba33d0ecf0d10a4 Mon Sep 17 00:00:00 2001
From: Urbain Vaes <urbain@vaes.uk>
Date: Thu, 21 Sep 2023 11:43:21 +0200
Subject: Save changes

---
 main.tex | 84 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 76 insertions(+), 8 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 04a91b6..ba7d319 100755
--- a/main.tex
+++ b/main.tex
@@ -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}
 
@@ -138,6 +139,27 @@
   \end{theorem}
 \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
@@ -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}
 
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+% vim: ts=4 sw=4
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