From 1f3bf1f786f98095a21cd6b615ddec99c2a014da Mon Sep 17 00:00:00 2001
From: Urbain Vaes <urbain@vaes.uk>
Date: Thu, 21 Sep 2023 19:47:38 +0200
Subject: Minor changes

---
 main.tex | 140 +++++++++++++++++++++++++++++++++++++++------------------------
 1 file changed, 86 insertions(+), 54 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index c6117c6..7596b40 100755
--- a/main.tex
+++ b/main.tex
@@ -219,6 +219,92 @@
   \end{minipage}
 \end{frame}
 
+\begin{frame}
+  {Nonequilibrium overdamped Langevin dynamics}
+  In general, can we prove existence of and convergence to an invariant measure for
+  \[
+      \d q_t = - \grad V(q_t) \, \d t + \alert{\eta F} +  \sqrt{2} \, \d W_t \, ?
+  \]
+
+  \begin{itemize}
+      \item 
+          If the state space is compact (e.g. $\torus^d$),
+          then we can apply Doeblin's theorem.
+
+      \item
+          If not, we can apply its generization, Harris' theorem.
+  \end{itemize}
+
+  \medskip
+  Fix $t > 0$ and denote by $p\colon \mathcal E \times \mathcal B(\mathcal E)$ the Markov transition kernel
+  \[
+      p(x, A) := \proba \left[ q_t \in A \, \middle| \, q_0 = x \right].
+  \]
+  For an observable $\phi \colon \mathcal E \to \real$ and a probability measure $\mu$,
+  we let
+  \[
+      (\mathcal P \phi)(x) := \int_{\mathcal E} \phi(y) \, p(x, \d y),
+      \qquad
+      (\mathcal P^{\dagger} \mu)(A)  := \int_{A} p(x, A) \, \mu(\d x).
+  \]
+  Note that $\mathcal P$ and $\mathcal P^{\dagger}$ are formally $L^2$ adjoints:
+  \[
+      \int_{\mathcal E} (\mathcal P \phi) \, \d \mu = \int_{\mathcal E} \phi \, \d (\mathcal P^\dagger \mu).
+  \]
+\end{frame}
+
+\begin{frame}
+  {Existence of an invariant measure (1/2)}
+  Remember that the set of probability measures with TV distance $d(\placeholder, \placeholder)$ is complete.
+  \begin{theorem}
+      [Doeblin's theorem]
+      If there exists $\alpha \in (0, 1)$ and a probability measure $\eta$ such that
+      \[
+          \mathcal P^{\dagger} \mu \geq \alpha \eta,
+      \]
+      then there exists $\mu_{\infty}$ such that $\mathcal P^{\dagger} \mu_{\infty} = \mu_{\infty}$.
+      Furthermore $d(\mathcal P^{\dagger^n}  \mu, \mu_{\infty}) \leq \alpha^n d(\mu, \mu_{\infty})$.
+  \end{theorem}
+
+  \emph{Sketch of proof.} Define the Markov transition kernel
+  \[
+      \widetilde {p}(x, \placeholder) := \frac{1}{1-\alpha} p(x, \placeholder) - \frac{\alpha}{1 - \alpha} \eta(\placeholder), 
+  \]
+  Let $\mathcal F$ denote the set of measurable functions $\phi \colon \mathcal E \to [-1, 1]$.
+  We have
+  \begin{align*}
+      d(\mathcal P^\dagger \mu, \mathcal P^\dagger \nu)
+      &= \sup_{\phi \in \mathcal F} \int_{\mathcal E} \phi(q) (\mathcal P^{\dagger} \mu - \mathcal P^{\dagger} \nu) (\d q)
+      = \sup_{\phi \in \mathcal F} \int_{\mathcal E} \mathcal P \phi(q) \bigl(\mu - \nu\bigr) (\d q) \\
+      &= (1 - \alpha) \sup_{\phi \in \mathcal F} \int_{\mathcal E} \widetilde {\mathcal P} \phi(q) (\mu - \nu) (\d q)
+      \leq (1 - \alpha) \, d(\mu, \nu).
+  \end{align*}
+  Conclude using Banach's fixed point theorem.
+\end{frame}
+
+\begin{frame}
+  {Existence of an invariant measure (2/2)}
+  \begin{itemize}
+      \item
+          Suppose that $\phi$ is uniformly bounded. Then
+          \begin{align*}
+              \left\lvert \mathcal P^n \phi(x) - \overline \phi \right\rvert
+              &= \int_{\mathcal E} \mathcal P^n (\phi - \overline \phi) \, \d(\delta_x - \mu_{\infty})
+              = \int_{\mathcal E} (\phi - \overline \phi) \, (\mathcal P^{\dagger n} \delta_x -  \mathcal P^{\dagger n} \mu_{\infty}) (\d q) \\
+              &\leq \norm{\phi - \overline \phi}_{L^{\infty}} (1-\alpha)^n d(\delta_x, \mu_{\infty}) 
+              \leq 2 \norm{\phi - \overline \phi}_{L^{\infty}} (1 - \alpha)^n.
+          \end{align*}
+
+
+      \item
+          In molecular dynamics, this theorem can be employed for showing existence of and convergence to the invariant measure,
+          provided that the \blue{state space is compact}.
+
+      \item
+          For \alert{noncompact state spaces}, an extension called \emph{Harris' theorem}
+  \end{itemize}
+\end{frame}
+
 \begin{frame}
   {Existence of an invariant measure for noneq.\ dynamics}
   Consider the paradigmatic dynamics
@@ -263,60 +349,6 @@
   Then apply the main theorem from~\footfullcite{MR2857021}.
 \end{frame}
 
-\begin{frame}
-  {Existence of an invariant measure (1/2)}
-  For a Markov transition kernel~$\mathcal P\colon \mathcal E \times \mathcal B(\mathcal E) \to [0, 1]$, let
-  \[
-      (\mathcal P \phi)(x) := \int_{\mathcal E} \phi(y) \, \mathcal P(x, \d y),
-      \qquad
-      (\mathcal P^{\dagger} \mu)(A)  := \int_{A} \mathcal P(x, A) \, \mu(\d x).
-  \]
-  \begin{theorem}
-      [Doeblin's theorem]
-      If there exists $\alpha \in (0, 1)$ and a probability measure $\eta$ such that
-      \[
-          \mathcal P^{\dagger} \mu \geq \alpha \eta,
-      \]
-      then there exists $\mu_{\infty}$ such that $\mathcal P^{\dagger} \mu_{\infty} = \mu_{\infty}$.
-      Furthermore $d(\mathcal P^{\dagger^n}  \mu, \mu_{\infty}) \leq \alpha^n d(\mu, \mu_{\infty})$.
-  \end{theorem}
-
-  \emph{Sketch of proof.}
-  Use Banach's fixed point theorem. Define the Markov transition
-  \[
-      \widetilde {\mathcal P}(x, \placeholder) := \frac{1}{1-\alpha} \mathcal P(x, \placeholder) - \frac{\alpha}{1 - \alpha} \eta(\placeholder).
-  \]
-  Let $V$ denote the set of measurable functions $\phi \colon \mathcal E \to [-1, 1]$.
-  We have
-  \begin{align*}
-      d(\mathcal P^\dagger \mu, \mathcal P^\dagger \nu)
-      &= \sup_{\phi \in V} \int_{\mathcal E} \phi(q) (\mathcal P^{\dagger} \mu - \mathcal P^{\dagger} \nu) (\d q)
-      = \sup_{\phi \in V} \int_{\mathcal E} \mathcal P \phi(q) \bigl(\mu - \nu\bigr) (\d q) \\
-      &= (1 - \alpha) \sup_{\phi \in V} \int_{\mathcal E} \widetilde {\mathcal P} \phi(q) (\mu - \nu) (\d q)
-      \leq (1 - \alpha) \, d(\mu, \nu).
-  \end{align*}
-\end{frame}
-
-\begin{frame}
-  {Existence of an invariant measure (2/2)}
-  \begin{itemize}
-      \item
-          Suppose that $\phi$ is uniformly bounded and let $\overline \phi = \int_{\mathcal E} \phi \, \d \mu_{\infty}$. Then
-          \[
-              \Bigl\lVert \mathcal P \left(\phi - \overline \phi\right) \Bigr\rVert_{L^\infty}
-              = (1 - \alpha) \Bigl\lVert \widetilde {\mathcal P} (\phi - \overline \phi) \Bigr\rVert_{L^{\infty}}
-              \leq (1 - \alpha) \Bigl\lVert  \phi - \overline \phi \Bigr\rVert_{L^{\infty}},
-          \]
-
-
-      \item
-          In molecular dynamics, this theorem can be employed for showing existence of and convergence to the invariant measure,
-          provided that the \blue{state space is compact}.
-
-      \item
-          For \alert{noncompact state spaces}, an extension called \emph{Harris' theorem}
-  \end{itemize}
-\end{frame}
 
 \begin{frame}
     {Linear response of nonequilibrium dynamics (1)}
-- 
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