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We consider a degenerate elliptic Kolmogorov--type operator arising from second order stochastic differential equations in $\mathbb R^{n}$ perturbed by noise. We study a realization of such an operator in $L^2$ spaces with respect to an explicit invariant measure, and we prove that it is $m$-dissipative.

Maximal dissipativity of a class of elliptic degenerate operators in weighted $L^2$ spaces