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Diffstat (limited to 'main.tex')
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@@ -220,6 +220,92 @@ \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 \begin{align*} @@ -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)} |