Let $A=\sum_{i,j=1}^Na_{ij}(x)D_{ij}+\sum_{i=1}^Nb_i(x)D_i$ be an elliptic differential operator with unbounded coefficients on $\R^N$ and assume that the associated Feller semigroup $(T(t))_{t\geq 0}$ has an invariant measure $\mu$. Then $(T(t))_{t\geq 0}$ extends to a strongly continuous semigroup $(T_p(t))_{t\geq 0}$ on $L^p(\mu)=L^p(\R^N,\mu)$ for every $1\leq p\leq \infty$. We prove that, under mild conditions on the coefficients of $A$, the space of test functions $C_c^\infty(\R^N)$ is a core for the generator $(A_p,D_p)$ of $(T_p(t))_{t\geq 0}$ in $L^p(\mu)$ for $1\leq p<\infty$.

### Cores for Feller semigroups with an invariant measure

#### Abstract

Let $A=\sum_{i,j=1}^Na_{ij}(x)D_{ij}+\sum_{i=1}^Nb_i(x)D_i$ be an elliptic differential operator with unbounded coefficients on $\R^N$ and assume that the associated Feller semigroup $(T(t))_{t\geq 0}$ has an invariant measure $\mu$. Then $(T(t))_{t\geq 0}$ extends to a strongly continuous semigroup $(T_p(t))_{t\geq 0}$ on $L^p(\mu)=L^p(\R^N,\mu)$ for every $1\leq p\leq \infty$. We prove that, under mild conditions on the coefficients of $A$, the space of test functions $C_c^\infty(\R^N)$ is a core for the generator $(A_p,D_p)$ of $(T_p(t))_{t\geq 0}$ in $L^p(\mu)$ for $1\leq p<\infty$.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11587/102082
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