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