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Diffusion semigroups corresponding to uniformly elliptic divergence form operators. I: Aronson’s estimate for elliptic operators in divergence form. (English) Zbl 0651.47031
Séminaire de probabilités XXII, Strasbourg/France, Lect. Notes Math. 1321, 316-347 (1988).
[For the entire collection see Zbl 0635.00013.]
If \(L=\bar V\cdot (a\bar V)\) is a second order partial differential operator, where a: \(R^ N\to R^ N\otimes R^ N\) is a measurable, symmetric matrix-valued function which satisfies the elipticity condition \(\lambda\) \(I\leq a(\cdot)\leq \frac{1}{\lambda}I\) for some \(\lambda\in (0,1]\). Then there is the unique Feller continuous Markov semigroup on \(C_ b(R^ N)\) such that \[ [P_ t\Phi](x)-\Phi (x)=\int^{t}_{0}[P_ sL\Phi](x)ds\quad (t,x)\in [0,\infty)\times R^ N \] and a function \(p\in \cup^{\infty}_{n=1}C_ b^{\infty}([1/n,n]\times R^ N\times R^ N\); \((0,+\infty))\) satisfying \[ [P_ t\Phi](x)=\int_{R^ N}\Phi (y)p(t,x,y)dy. \] A very good estimate by Aronson is that there is an \(M=M(\lambda,N)\in [1,+\infty)\) such that \[ \frac{1}{Mt^{N/2}}\exp (-M| y-x|^ 2/t)\leq p(t,x,y)\leq \frac{M}{t^{N/2}}\exp (-| y-x|^ 2/Mt). \] This is a summary paper about the application of Aronson’s Estimate. In chapter I, by use of the estimate various upper bounds and lower bounds are obtained. In Chapter II, the estimate is applied to Harmonic Analysis and perturbation of divergence form operators.
Reviewer: Wu Liangsen

47D07 Markov semigroups and applications to diffusion processes
35J15 Second-order elliptic equations
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