On the derivation of conservation laws for stochastic dynamics. (English) Zbl 0699.60097

Analysis, et cetera, Res. Pap. in Honor of J. Moser’s 60th Birthd., 677-694 (1990).
[For the entire collection see Zbl 0688.00009.]
The paper gives an introduction into the problem of deriving macroscopic evolution equations from a microscopic dynamic. It is shown, that if we replace deterministic microscopic evolution equations by a stochastic evolution mechanism it is indeed possible to derive the macroscopic evolution equation from the microscopic dynamics. The paper discusses this derivation for the following model:
Let \(\{x_1(t),\ldots,x_ N(t)\}\) evolve according to \[ dx_ i(t) = dz_{i,i-1} - dz_{i,i+1}, \qquad dz_{i,i+1} = [\psi (x_ i(t)-\psi (x_{i+1}(t))]dt+\alpha \beta_{i,i+1}(t), \] where \((\beta_{i,i+1}(t))\) are \(N\) independent Brownian motions and \(\psi = 2^{-1}(d/dx)\Phi\). The initial state is defined by a profile \(a(\cdot)\) as follows. The \((x_ i)_{I=1,\ldots,N}\) are independent, uniformly distributed in \([a_ i-1/2, a_ i+1/2]\) with \(a=a_ 0(i/N)\). In the limit \(N\to \infty\), rescaling time by \(N^2\) and space by \(N^{-1}\), we obtain that the density \(a(t,\theta)\) evolves according to: \[ (\partial /\partial t) a(t,\theta) = 2^{-1}(\partial /\partial \theta)^ 2 [h'(a(t,\theta))],\qquad a(0,\theta) = a_ 0(\theta), \] with \(h'\) being determined as the solution of an equation involving the potential \(\Phi\). The author explains in particular the role of entropy and large deviation estimates to establish the result quoted above.
Reviewer: A.Greven


60K35 Interacting random processes; statistical mechanics type models; percolation theory


Zbl 0688.00009