Random motion of strings and related stochastic evolution equations. (English) Zbl 0531.60095

Funaki, Tadahisa

Nagoya Math. J. 89, 129-193 (1983).

This is a very nice, clearly written, comprehensive paper about the Einstein-Smoluchowski type equation (*) \(dX_ t(z)=dB_ t(z)- frac{1}{2}\nabla H(X_ t,z)dt\), \(z\in [0,1]\), for the Hamiltonian H of a d-dimensional elastic string \(X\in {\mathcal C}([0,1],{\mathbb{R}}^ d)\) with potential U and tension \(\kappa\) : \(H(X)=\frac{\kappa}{2}\int^{1}_{0}| \frac{\partial X}{\partial z}|^ 2(z)dz+\int^{1}_{0}U(X(z))dz. B_ t=B_ t(\sigma)\) is an \(L^ 2([0,1])\)-valued Brownian motion, and formally \(\nabla H(X,z)=- \kappa \frac{\partial^ 2}{\partial z^ 2}X(z)+\nabla U(X(z)).\) Under some conditions on U, (*) has a unique continuous solution, which can be approximated by a spatially discrete version of (*).
The limit of the tension \(\kappa\to \infty\) yields - in absence of boundary conditions - the motion of a horizontally straight rod, moving like a Brownian motion in a potential U; if fixed at 0 by \(X_ t(0)\equiv a_ 0\), it stays \(\equiv a_ 0\); if fixed at 0 and 1 by \(X_ t(0)=a_ 0\), \(X_ t(1)=a_ 1\), it gives \(X_ t(z)\equiv(1- z)a_ 0+za_ 1\). Following Kolmogoroff’s characterization, the Gibbs measures, formally written as \(\exp(-H(X))dX=\exp(-\int U(X(z))dz)dP_ z\) where \(P_ z\) is the Wiener measure in \(z\in [0,1]\) with diffusion term \(\kappa^{-1}\), are the reversible measures of \(X_ t(z)\). The converse, however, is not shown. (Notice that the first term of H is just the entropy of a Brownian motion in z with diffusion coefficient \(\kappa^{-1}.)\) It follows that in two dimensions and without a potential (\(U\equiv const.)\), the process is recurrent and meets all points in \({\mathbb{R}}^ 2\). An appendix proves that (*) is obtained as a limit (\(\beta\to \infty)\) of the following Ornstein-Uhlenbeck process: \(dX_ t(z)=V_ t(z)dt\), \(dV_ t(z)=\beta(dB_ t(z)-frac{1}{2}\nabla H(X_ t,z)dt-V_ t(z)dt).\)

Reviewer: Th.Eisele

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60G60	Random fields
60J65	Brownian motion

Keywords:

evolution equations; sequence of polygonal approximations; infinite- dimensional Einstein-Smoluchowski equation; elastic string; Gibbs measures; Wiener measure; Ornstein-Uhlenbeck process

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