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.


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


Zbl 0722.00030
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