Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ellipticity and Ergodicity   Let $S=\{S_t\}_{t\geq0}$ be the submarkovian semigroup on $L_2(\Ri^d)$
generated by a self-adjoint, second-order, divergence-form, elliptic operator
$H$ with Lipschitz continuous coefficients $c_{ij}$. Further let $\Omega$ be an
open subset of $\Ri^d$. Under the assumption that $C_c^\infty(\Ri^d)$ is a core
for $H$ we prove that $S$ leaves $L_2(\Omega)$ invariant if, and only if, it is
invariant under the flows generated by the vector fields
$Y_i=\sum^d_{j=1}c_{ij}\partial_j$.

1
The categories of this paper are: cs
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
7
The categories of this paper are: hep-th
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
15
The categories of this paper are: nucl-ex