From Metropolis to diffusions: Gibbs states and optimal scaling.
Let \(\Pi = (\pi _ {W}(a,\,dx),\; W\subseteq \mathbb Z^ {d} \;\text{finite}, a\in \mathbb R^ {\mathbb Z^ {d}})\) be a specification, that is, a consistent family of finite-volume conditional Gibbs measures for a finite-range Hamiltonian \(H\). Suppose that \(\xi \) is the corresponding infinite-volume Gibbs measure, let \(\xi \) be translation invariant. Given a subset \(V_ {n}\subseteq \mathbb Z^ {d}\) of cardinality \(n\) and a boundary condition \(z\), let \((X_ {t}(V_ {n},z),\;t\geq 0)\) be the random walk Metropolis chain for \(\pi _ {V_ {n}}(z,\cdot )\). It was shown by the second author and \textit{A. F. M. Smith} [Stochastic Processes Appl. 49, 207-216 (1994; Zbl 0803.60067)] that \(X_ {t}(V_ {n},z)\) converges weakly to \(\pi _ {V_ {n}}(z, \cdot )\) as \(t\to \infty \). The behaviour of the algorithm as \(V_ {n}\uparrow \mathbb Z^ {d}\) is studied. In particular, choose the proposal variance \(\sigma ^ {2}_ {n} = ln^ {-1}\). Under suitable assumptions on \(H\) and \(\xi \) it is proven that \(X_ {[nt]}(V_ {n},z)\) converges weakly as \(n\to \infty \) to an infinite-dimensional diffusion \(Z_ {t}\) on a Hilbert space \(E=L^ 2(\mathbb Z^ {d},\rho )\). The measure \(\rho \) is given by \(\rho (\{k\}) = (\sum _ {j\in \mathbb Z^ {d}} \exp (-| j| ) )^ {-1}\exp (-| k| )\), \(Z\) solves the equation \[ dZ_ {t} = -\tfrac {l}2 v(Z_ {t})\nabla H(Z_ {t})\,dt + \sqrt {lv(Z_ {t})}\,dB_ {t}, \] driven by a Brownian motion \(B\) in \(E\) and with the initial condition \(Z_ 0 = \xi \) in law. The coefficient \(v\) is defined in terms of the second derivative of the Hamiltonian \(H\).
scientific article; zbMATH DE number 2102348