Let (left(T_{t} ight)_{t geqslant 0}) be a Feller semigroup, i.e. a strongly continuous, positivity preserving sub-Markovian semigroup

Let \(\left(T_{t}\right)_{t \geqslant 0}\) be a Feller semigroup, i.e. a strongly continuous, positivity preserving sub-Markovian semigroup on \(\left(\mathcal{C}_{\infty}\left(\mathbb{R}^{d}\right),\|\cdot\|_{\infty}\right)\), cf. Remark 7.4.

a) Show that \((t, x) \mapsto T_{t} u(x)\) is jointly continuous for \(t \geqslant 0\) and \(x \in \mathbb{R}^{d}\).

b) Conclude that for every compact set \(K \subset \mathbb{R}^{d}\) the function \((t, x) \mapsto T_{t} \mathbb{1}_{K}(x)\) is measurable.

c) Show that \((t, x, u) \mapsto T_{t} u(x)\) is jointly continuous for all \(t \geqslant 0, x \in \mathbb{R}^{d}\) and \(u \in \mathcal{C}_{\infty}\left(\mathbb{R}^{d}\right)\).

Data From 7.4 Remark 

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7.4 Remark. A semigroup of operators (Pt)tao on Bb (Rd) which satisfies a)-d) is called a Markov semigroup; b)-d) is called a sub-Markov semigroup; b)-f) is called a Feller semigroup; b)-d), g) is called a strong Feller semigroup. Proof of Proposition 73. a) We have P+1(x) = E* 1R(B) = P(B+ Rd) = 1 for all X Rd. b) We have |Pru(x)| = | E* u(Bt)| < | \u(B+)| dp* < ||u||.. for all x = Rd.