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Probability conserving elliptic operators. (Russian) Zbl 0661.35022
The main result of this work is as follows: Let $$a_{kj}(x)\in C^{\infty}({\mathbb{R}}^ e)$$ $$(k,j=1,...,l)$$ be such real functions on $${\mathbb{R}}^ l$$, that $$a_{kj}(x)=a_{jk}(x)$$ $$\forall k,j=1,...,l$$, $$\forall x\in {\mathbb{R}}^ l$$, $$0<\sum_{k,j}a_{kj}(x)\xi_ k\xi_ j$$ $$(\forall x\in {\mathbb{R}}^ l)$$ $$(\forall \xi \in {\mathbb{R}}^ l$$, $$\xi\neq 0)$$ $$(\exists R>0)$$ $$\sum_{k,j}a_{kj}(x)x_ kx_ j/| x|^ 2\leq const | x|^ 2\ln | x|$$ $$(\forall x:| x| >R)$$. Then for all $$\lambda >0$$ the operator ($$\lambda$$ I- $$\sum_{j,k}(\partial /\partial x_ k)a_{kj}(\partial /\partial x_ j))\upharpoonright C_ 0^{\infty}({\mathbb{R}}^ l)$$ has a dense range in the space $$L^ 1({\mathbb{R}}^ l)$$ and (1) $$e^{-tH_ 1}=1$$ where $$H_ 1$$ is the closure of $$-\sum^{l}_{k,j=1}(\partial /\partial x_ k)a_{kj}\partial /\partial x_ j$$ in $$L^ 1({\mathbb{R}}^ l).$$
The property (1) means that the operator $$H_ 1$$ conserves the probability.
Reviewer: V.S.Rabinovich

##### MSC:
 35J10 Schrödinger operator, Schrödinger equation 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) 60J60 Diffusion processes
##### Keywords:
symmetric elliptic operator; diffusion process