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From Metropolis to diffusions: Gibbs states and optimal scaling. (English) Zbl 1047.60065
Let $\Pi = (\pi \sb {W}(a,\,dx),\; W\subseteq \Bbb Z\sp {d} \ \text{finite}, a\in \Bbb R\sp {\Bbb Z\sp {d}})$ 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 $V\sb {n}\subseteq \Bbb Z\sp {d}$ of cardinality $n$ and a boundary condition $z$, let $(X\sb {t}(V\sb {n},z),\;t\ge 0)$ be the random walk Metropolis chain for $\pi \sb {V\sb {n}}(z,\cdot )$. It was shown by the second author and {\it A. F. M. Smith} [Stochastic Processes Appl. 49, 207-216 (1994; Zbl 0803.60067)] that $X\sb {t}(V\sb {n},z)$ converges weakly to $\pi \sb {V\sb {n}}(z, \cdot )$ as $t\to \infty $. The behaviour of the algorithm as $V\sb {n}\uparrow \Bbb Z\sp {d}$ is studied. In particular, choose the proposal variance $\sigma \sp {2}\sb {n} = ln\sp {-1}$. Under suitable assumptions on $H$ and $\xi $ it is proven that $X\sb {[nt]}(V\sb {n},z)$ converges weakly as $n\to \infty $ to an infinite-dimensional diffusion $Z\sb {t}$ on a Hilbert space $E=L\sp 2(\Bbb Z\sp {d},\rho )$. The measure $\rho $ is given by $\rho (\{k\}) = (\sum \sb {j\in \Bbb Z\sp {d}} \exp (-\vert j\vert ) )\sp {-1}\exp (-\vert k\vert )$, $Z$ solves the equation $$dZ\sb {t} = -\tfrac {l}2 v(Z\sb {t})\nabla H(Z\sb {t})\,dt + \sqrt {lv(Z\sb {t})}\,dB\sb {t},$$ driven by a Brownian motion $B$ in $E$ and with the initial condition $Z\sb 0 = \xi $ in law. The coefficient $v$ is defined in terms of the second derivative of the Hamiltonian $H$.

60J05Discrete-time Markov processes on general state spaces
65C05Monte Carlo methods
60J22Computational methods in Markov chains
Full Text: DOI
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