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Using stochastic comparison to estimate Green’s functions. (English) Zbl 0744.60093

Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 421-425 (1989).
[For the entire collection see Zbl 0722.00030.]
The author obtains very good estimates for the Green’s function \(g_ D(x,y)\) for a second order elliptic operator \(L\) on \(R^ d\), where \(Lf(x)={1\over2}\sum_{i,j=1}^ d a_{ij}(x){\partial^ 2f\over\partial x_ i\partial x_ j}(x)\), \(a_{ij}\) are bounded, uniformly strictly elliptic and Dini continuous, and \(D\) is a bounded domain. This well-known result is obtained by using probabilistic technique, in fact elementary stochastic calculus. This technique seems to be useful in other situations as well.

MSC:

60J60 Diffusion processes
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35J15 Second-order elliptic equations

Citations:

Zbl 0722.00030
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