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Propagation of chaos and the Burgers equation. (English) Zbl 0554.35104
For \(N=2,3,...\), let \(L_ N=\Delta_ N+(1/N)\sum_{1\leq i<j\leq N}\delta (x_ i-x_ j)(\partial /\partial x_ i+\partial /\partial x_ j)\), where \(\Delta_ N\) is the N-dimensional Laplacian, and consider the equation \(\partial F_ N/\partial t=L_ NF_ N\), with the initial condition \(F_ N(0;x_ 1x_ 2,...,x_ N)=f_ 0(x_ 1)...f_ 0(x_ N),\) \(f_ 0\) being normalized to \(\int^{- \infty}_{+\infty}f_ 0(x)dx=1.\) The equation \(\partial F_ N/\partial t=L_ NF_ N\) describes the motion of N Brownian particles on the line with certain singular interactions. It had been conjectured by McKean that the limits \[ f_ s(t;x_ 1,x_ 2,...,x_ s)=\lim_{N\to \infty}\int^{\infty}_{-\infty}...\int^{\infty}_{-\infty}F_ N(t;x_ 1,x_ 2,...,x_ N)dx_{s+1}...dx_ N \] all exist, that \(f_ 1\) satisfies the Burgers equation \(\partial f/\partial t=\partial^ 2f/\partial x^ 2+2f\quad \partial f/\partial x\) with initial condition \(f_ 1(0;x)=f_ 0(x)\) and that \(f_ s(t;x_ 1,...,x_ s)=f_ 1(t;x_ 1)...f_ s(t,x_ s),\) \(\forall\) s (”propagation of chaos”). The authors give a sketch of the proof of these assertions. They also construct, formally, an operator \(Q_ N\) on the symmetric functions on \(R^ N\) which intertwines \(\Delta_ N\) with \(\Delta_ N+(1/N)\sum_{1\leq i<j\leq N}\delta (x_ i-x_ j)(\partial /\partial x_ i+\partial /\partial x_ j)\) and introduce a variant of the Hopf-Cole transformation.
Reviewer: O.Liess

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35B32 Bifurcations in context of PDEs
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