From 406f82e9d3f3f2a364b7b0e65b7877c440a107fc Mon Sep 17 00:00:00 2001
From: Urbain Vaes <urbain@vaes.uk>
Date: Mon, 25 Sep 2023 12:31:05 +0200
Subject: Minor change

---
 main.tex | 212 +++++++++++++++++++++++++++++++++++++--------------------------
 1 file changed, 126 insertions(+), 86 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index c89b161..ceb383b 100755
--- a/main.tex
+++ b/main.tex
@@ -149,8 +149,8 @@
   \begin{itemize}
     \item Equilibrium vs nonequilibrium dynamics;
     \item Definition and computation of the mobility;
-    \item Computation of other transport coefficients.
-    \item Error analysis
+    \item Computation of other transport coefficients;
+    \item Error analysis.
   \end{itemize}
 \end{frame}
 
@@ -435,14 +435,14 @@
             The \textbf{minorization condition} is satisfied.
             Indeed for $t > 0$
             \begin{align*}
-                p(x, A)
-                &= \expect \left[ q_t \in A \, \middle| \, q_0 = x \right]
-                = \expect \left[ \mathds 1_{A} \left(x +  W_t \right) M_t \right]
-                && M_t = \text{Girsanov weight} \\
-                &= \proba \left[ x + W_t \in A \right] \expect \left[ M_t \, | \, \{x + W_t \in A\} \right] \\
-                &\geq C \proba \left[ x + W_t \in A \right] \geq C \lambda(A) && \lambda := \text{Lebesgue measure}.
+                \mathcal P^{\dagger}\mu (A)
+                &= \expect \left[ q_t \in A \, \middle| \, q_0 \sim \mu \right]
+                =  \int_{\mathcal E} \int_{A} p_t(x, y) \, \mu(\d x) \, 
+                && p_t = \text{transition pdf} \\
+                &\geq \left( \inf_{(x,y) \in \mathcal E^2} p_t(x, y) \right) \lambda(A) && \lambda := \text{Lebesgue measure}.
             \end{align*}
-            and additionally ${\rm Law} (q_t)$ is smooth by parabolic regularity.
+            The infimum is achieved by parabolic regularity, 
+            and achieved by {\blue Harnack's inequality}.
         \item
             \textbf{Decay of the semigroup}:
             For $t \in [0, \infty)$ and $\varphi \in L^{\infty}_*$, it holds that
@@ -604,51 +604,90 @@
 
 \begin{frame}
     {Elements of proof}
-    Let us introduce
+    Let $\Pi_0$ denote the following projection operator
     \[
-        H^1_{p}(\psi_0) =
-        \Bigl\{ \varphi \in L^2(\psi_0) : \grad_p \varphi \in L^2(\psi) \Bigr\},
-        \qquad  \| \varphi \|_{H^1_{p}(\psi_0)}^2 = \| \varphi \|_{L^2(\psi_0)}^2 + \| \nabla_p \varphi \|_{L^2(\psi_0)}^2.
+        \Pi_0 f := f - \int_{\mathcal E} f \, \psi_0
     \]
     \vspace{-.3cm}
     \begin{itemize}
         \itemsep.2cm
-        \item
-            The operator {\blue $\widetilde {\mathcal L}^*\colon H^1_p(\psi_0) \to L^2_0(\psi)$} is well-defined and bounded.
-            Indeed
+        \item 
+            The operator $\mathcal L_0^{-1}$ is a well defined bounded operator on $L_0^2(\psi_0)$ \\
+
+            ({\red Hypocoercivity} + {\red hypoelliptic regularization})
+
+        \item Since $\dps \gamma \| \nabla_p \varphi \|^2_{L^2(\psi_0)} = -\langle \mathcal L_0 \varphi,\varphi \rangle_{L^2(\psi_0)}$, it follows that
+            \vspace{-0.2cm}
             \[
-                \lVert \widetilde {\mathcal L}^* \varphi \rVert_{L^2_0(\psi_0)}^2
-                = \ip{\nabla_p^* F \varphi}{\nabla_p^* F \varphi}_{L^2_0(\psi_0)}
-                \leq \lVert \varphi \rVert_{H^1_p(\psi_0)}^2
+                \| \widetilde {\mathcal L} \varphi \|^2_{L^2(\psi_0)} \leq \| \nabla_p \varphi \|^2_{L^2(\psi_0)} \leq \frac{1}{\gamma} \| \mathcal L_0 \varphi \|_{L^2(\psi_0)} \| \varphi \|_{L^2(\psi_0)}
             \]
-            and
+            Thus {\blue $\Pi_0 \widetilde {\mathcal L} \mathcal L_0^{-1}$ is bounded on $L^2_0(\psi_0)$}.
             \[
-                \int_{\mathcal E} \widetilde {\mathcal L}^* \phi \, \psi_0
-                = \int_{\mathcal E} \nabla_p^* (F \phi) \, \psi_0 = 0.
+                \| \widetilde {\mathcal L} \mathcal L_0^{-1} \varphi \|^2_{L^2(\psi_0)}\leq \frac{\beta}{\gamma} \| \varphi \|_{L^2(\psi_0)} \| \mathcal L_0^{-1} \varphi \|_{L^2(\psi_0)}.
             \]
-        \item
-            The operator {\blue $(\mathcal L_0^*)^{-1} \colon L^2_0(\psi_0) \to H^1_p(\psi_0)$} is well-defined and bounded,
-            by {\red hypocoercivity} and {\red hypoelliptic regularization}.
-            % In particular, for $\phi = (\mathcal L_0^*)^{-1} \varphi$
-            % \begin{align*}
-            %     \| \phi \|_{L^2(\psi_0)}^2
-            %     +  \| \nabla_p \phi \|_{L^2(\psi_0)}^2
-            %     &= \|(\mathcal L_0^*)^{-1} \varphi \|_{L^2(\psi_0)}^2
-            %     + \frac{1}{\gamma} \ip{-\mathcal L_0^* \phi}{\phi}_{L^2(\psi_0)} \\
-            %     &\leq \frac{1}{\gamma} \norm{(\mathcal L_0^*)^{-1}}_{\mathcal B\bigl(L^2(\psi_0)\bigr)}^2
-            %     \norm{\varphi}_{L^2(\psi_0)}
-            % \end{align*}
+
+
+        \item It follows that $(\widetilde {\mathcal L} \mathcal L_0^{-1})^* \Pi_0 = (\widetilde {\mathcal L} \mathcal L_0^{-1})^*$ is also bounded on $L^2_0(\psi_0)$
+
+            \medskip
 
         \item Invariance of $f_\eta$ by $\mathcal L_\eta^* = \mathcal L_0^* + \eta \wcL^*$
             \vspace{-0.2cm}
             \[
-                \mathcal L_\eta^* f_\eta = \mathcal L_0^* \left(1 + \eta (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \right) f_\eta = \mathcal L_0^* \mathbf{1} = 0.
+                \mathcal L_\eta^* f_\eta = \mathcal L_0^* \left(1 + \eta (\wcL \mathcal L_0^{-1})^* \right) f_\eta = \mathcal L_0^* \mathbf{1} = 0
             \]
 
         \item {\red Prove that $f_\eta \geq 0$}.
     \end{itemize}
 \end{frame}
 
+% \begin{frame}
+%     {Elements of proof}
+%     Let us introduce
+%     \[
+%         H^1_{p}(\psi_0) =
+%         \Bigl\{ \varphi \in L^2(\psi_0) : \grad_p \varphi \in L^2(\psi) \Bigr\},
+%         \qquad  \| \varphi \|_{H^1_{p}(\psi_0)}^2 = \| \varphi \|_{L^2(\psi_0)}^2 + \| \nabla_p \varphi \|_{L^2(\psi_0)}^2.
+%     \]
+%     \vspace{-.3cm}
+%     \begin{itemize}
+%         \itemsep.2cm
+%         \item
+%             The operator {\blue $\widetilde {\mathcal L}^*\colon H^1_p(\psi_0) \to L^2_0(\psi)$} is well-defined and bounded.
+%             Indeed
+%             \[
+%                 \lVert \widetilde {\mathcal L}^* \varphi \rVert_{L^2_0(\psi_0)}^2
+%                 = \ip{\nabla_p^* F \varphi}{\nabla_p^* F \varphi}_{L^2_0(\psi_0)}
+%                 \leq \lVert \varphi \rVert_{H^1_p(\psi_0)}^2
+%             \]
+%             and
+%             \[
+%                 \int_{\mathcal E} \widetilde {\mathcal L}^* \phi \, \psi_0
+%                 = \int_{\mathcal E} \nabla_p^* (F \phi) \, \psi_0 = 0.
+%             \]
+%         \item
+%             The operator {\blue $(\mathcal L_0^*)^{-1} \colon L^2_0(\psi_0) \to H^1_p(\psi_0)$} is well-defined and bounded,
+%             by {\red hypocoercivity} and {\red hypoelliptic regularization}.
+%             % In particular, for $\phi = (\mathcal L_0^*)^{-1} \varphi$
+%             % \begin{align*}
+%             %     \| \phi \|_{L^2(\psi_0)}^2
+%             %     +  \| \nabla_p \phi \|_{L^2(\psi_0)}^2
+%             %     &= \|(\mathcal L_0^*)^{-1} \varphi \|_{L^2(\psi_0)}^2
+%             %     + \frac{1}{\gamma} \ip{-\mathcal L_0^* \phi}{\phi}_{L^2(\psi_0)} \\
+%             %     &\leq \frac{1}{\gamma} \norm{(\mathcal L_0^*)^{-1}}_{\mathcal B\bigl(L^2(\psi_0)\bigr)}^2
+%             %     \norm{\varphi}_{L^2(\psi_0)}
+%             % \end{align*}
+% 
+%         \item Invariance of $f_\eta$ by $\mathcal L_\eta^* = \mathcal L_0^* + \eta \wcL^*$
+%             \vspace{-0.2cm}
+%             \[
+%                 \mathcal L_\eta^* f_\eta = \mathcal L_0^* \left(1 + \eta (\mathcal L_0^*)^{-1} \widetilde {\mathcal L}^* \right) f_\eta = \mathcal L_0^* \mathbf{1} = 0.
+%             \]
+% 
+%         \item {\red Prove that $f_\eta \geq 0$}.
+%     \end{itemize}
+% \end{frame}
+
 \begin{frame}
     {Perturbation expansion for {\yellow $\eta$ sufficiently small} (3/3)}
 
@@ -1064,7 +1103,7 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
 
 
 \begin{frame}
-    {Shear viscosity in fluids (2)}
+    {Shear viscosity in fluids (3)}
     Assume pairwise interactions
     \[
         V(q) = \sum_{1 \leq i < j \leq N} \mathcal V(\abs{q_i - q_j}).
@@ -1092,7 +1131,7 @@ The Hamiltonian of the system is the sum of the potential and kinetic energies:
 
 
 \begin{frame}
-    {Shear viscosity in fluids (3)}
+    {Shear viscosity in fluids (4)}
 
 \bu {\blue Linear response}:
 \[
@@ -1362,58 +1401,10 @@ We verify the error estimate for $\varphi \in \mathrm{Ran}(P_\dt-\I)$.
 \begin{block}{}
 Suggests $f_{p+1} = -\left( \mathcal A_1^* \right)^{-1} g_{p+1}$
 \end{block}
-
 \end{frame}
 
 \begin{frame}
-    {Numerical estimators and associated challenges}
-    \begin{itemize}
-        \item
-            Estimator of linear response (observable~$R$ with equilibrium average~0)
-            \[
-                \widehat{A}_{\eta,t} = \frac{1}{\eta t}\int_0^t R(q_s^\eta,p_s^\eta) \, ds \xrightarrow[t\to+\infty]{\mathrm{a.s.}}
-                \alpha_\eta := \frac1\eta \int_{\mathcal E} R \, f_\eta \, d\mu = \alpha + \mathcal O(\eta)
-            \]
-            {\bf Issues with linear response methods:}
-            \begin{itemize}
-                \item Statistical error with {\red asymptotic variance $\mathcal O(\eta^{-2})$}
-                \item Bias $\mathcal O(\eta)$ due to $\eta \neq 0$
-                \item Bias from finite integration time
-            \end{itemize}
-
-    \end{itemize}
-\end{frame}
-
-\begin{frame}\frametitle{Analysis of variance / finite integration time bias}
-
-  \bu {\bf Statistical error} dictated by {\blue Central Limit Theorem}:
-  \[
-  \sqrt{t} \left(\widehat{A}_{\eta,t} - \alpha_\eta \right) \xrightarrow[t \to +\infty]{\mathrm{law}} \mathcal{N}\left(0,\frac{\sigma_{R,\eta}^2}{\eta^2}\right),
-  \qquad
-  \sigma_{R,\eta}^2 = \sigma_{R,0}^2 + \mathcal O(\eta)
-  \]
-  so $\dps \widehat{A}_{\eta,t} = \alpha_\eta + \mathcal O_{\rm P}\left(\frac{1}{\eta \sqrt{t}}\right)$ $\to$ requires {\red long simulation times} $t \sim \eta^{-2}$
-  
-  \bigskip
-  
-  \bu {\bf Finite time integration bias}: $\dps \left| \mathbb{E}\left(\widehat{A}_{\eta,t}\right) - \alpha_\eta \right| \leq \frac{K}{\eta t}$ \\
-  Bias due to $t < +\infty$ is $\dps \mathcal O\left(\frac{1}{\eta t}\right)$ $\to$ typically {\red smaller than statistical error}
-
-%\bigskip
- %\bu Bias~$\mathcal O(\eta)$ and statistical error equilibrated for~$t \sim \eta^{-3}$
-
-\bigskip
-
-\bu Key equality for the proofs: introduce $\dps -\left(\mathcal{L}+\eta\widetilde{\mathcal{L}}\right) \mathscr{R}_\eta = R - \int_\mathcal{E} R f_\eta \, d\mu$
-\[
-\widehat{A}_{\eta,t} - \frac1\eta \!\int_{\mathcal{E}} \!R f_\eta \, d\mu = \frac{\mathscr{R}_\eta(q_0^\eta,p_0^\eta) - \mathscr{R}_\eta(q_t^\eta,p_t^\eta)}{\eta t} + \frac{\sqrt{2\gamma}}{\eta t\sqrt{\beta}} \int_0^t \!\!\nabla_p \mathscr{R}_\eta(q_s^\eta,p_s^\eta)^T dW_s
-\]
-
-\end{frame}
-
-
-
-\begin{frame}\frametitle{Examples of splitting schemes for Langevin dynamics (1)}
+    {Examples of splitting schemes for Langevin dynamics (1)}
 
 \bu Example: Langevin dynamics, discretized using a {\blue splitting} strategy
 \[
@@ -1475,6 +1466,55 @@ p^{n+1} & = \alpha_{\dt/2} \widetilde{p}^{n+1} + \sqrt{\frac{1-\alpha_{\dt}}{\be
 \end{frame}
 
 
+\begin{frame}
+    {Numerical estimators and associated challenges}
+    \begin{itemize}
+        \item
+            Estimator of linear response (observable~$R$ with equilibrium average~0)
+            \[
+                \widehat{A}_{\eta,t} = \frac{1}{\eta t}\int_0^t R(q_s^\eta,p_s^\eta) \, ds \xrightarrow[t\to+\infty]{\mathrm{a.s.}}
+                \alpha_\eta := \frac1\eta \int_{\mathcal E} R \, f_\eta \, d\mu = \alpha + \mathcal O(\eta)
+            \]
+            {\bf Issues with linear response methods:}
+            \begin{itemize}
+                \item Statistical error with {\red asymptotic variance $\mathcal O(\eta^{-2})$}
+                \item Bias $\mathcal O(\eta)$ due to $\eta \neq 0$
+                \item Bias from finite integration time
+            \end{itemize}
+
+    \end{itemize}
+\end{frame}
+
+\begin{frame}\frametitle{Analysis of variance / finite integration time bias}
+
+  \bu {\bf Statistical error} dictated by {\blue Central Limit Theorem}:
+  \[
+  \sqrt{t} \left(\widehat{A}_{\eta,t} - \alpha_\eta \right) \xrightarrow[t \to +\infty]{\mathrm{law}} \mathcal{N}\left(0,\frac{\sigma_{R,\eta}^2}{\eta^2}\right),
+  \qquad
+  \sigma_{R,\eta}^2 = \sigma_{R,0}^2 + \mathcal O(\eta)
+  \]
+  so $\dps \widehat{A}_{\eta,t} = \alpha_\eta + \mathcal O_{\rm P}\left(\frac{1}{\eta \sqrt{t}}\right)$ $\to$ requires {\red long simulation times} $t \sim \eta^{-2}$
+  
+  \bigskip
+  
+  \bu {\bf Finite time integration bias}: $\dps \left| \mathbb{E}\left(\widehat{A}_{\eta,t}\right) - \alpha_\eta \right| \leq \frac{K}{\eta t}$ \\
+  Bias due to $t < +\infty$ is $\dps \mathcal O\left(\frac{1}{\eta t}\right)$ $\to$ typically {\red smaller than statistical error}
+
+%\bigskip
+ %\bu Bias~$\mathcal O(\eta)$ and statistical error equilibrated for~$t \sim \eta^{-3}$
+
+\bigskip
+
+\bu Key equality for the proofs: introduce $\dps -\left(\mathcal{L}+\eta\widetilde{\mathcal{L}}\right) \mathscr{R}_\eta = R - \int_\mathcal{E} R f_\eta \, d\mu$
+\[
+\widehat{A}_{\eta,t} - \frac1\eta \!\int_{\mathcal{E}} \!R f_\eta \, d\mu = \frac{\mathscr{R}_\eta(q_0^\eta,p_0^\eta) - \mathscr{R}_\eta(q_t^\eta,p_t^\eta)}{\eta t} + \frac{\sqrt{2\gamma}}{\eta t\sqrt{\beta}} \int_0^t \!\!\nabla_p \mathscr{R}_\eta(q_s^\eta,p_s^\eta)^T dW_s
+\]
+
+\end{frame}
+
+
+
+
 \begin{frame}\frametitle{Error estimates on linear response}
 
 \begin{block}{Error estimates for nonequilibrium dynamics}
-- 
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