## 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

### MSC:

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

Zbl 0688.00009