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Markov SemigroupWe consider the family of convolution operators on $L^{\infty}(\mathbb R^d)$ indexed by $t \geq 0$ and given for every $f \in L^{\infty}(\mathbb R^d)$ by$P_t f(x)=E_x(f(X_t))=\int_{R^d} f(x+y)\mathbb P(X_t\in dy)$ The Semigroup $(P_t, t\geq 0)$ has the Feller property, that is for every $f \in l_0$:

• $P_t f \in l_0$ for every $t\geq 0$,
• $lim_{t\to 0} P_t f= f$ (uniformly.) Read the rest of this entry »