Let (left(B_{t} ight)_{t geqslant 0}) be a (mathrm{BM}^{d}) with transition semigroup (left(P_{t} ight)_{t geqslant 0}). Show, using

Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}\) with transition semigroup \(\left(P_{t}\right)_{t \geqslant 0}\). Show, using arguments similar to those in the proof of Proposition 7.3.f), that \((t, x, u) \mapsto P_{t} u(x)\) is continuous. As usual, we equip \([0, \infty) \times \mathbb{R}^{d} \times \mathcal{C}_{\infty}\left(\mathbb{R}^{d}\right)\) with the norm \(|t|+|x|+\|u\|_{\infty}\).

Data From Proposition 7.3

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7.3 Proposition. Let (Pt)to be the transition semigroup of a d-dimensional Brownian motion (Bt)tzo and Coo Coo (Rd), C = C(Rd) and Bb Bb (Rd). For all t > 0 = a) Pt is conservative: P+1 = 1. b) Pt is a contraction on Bb: ||Ptulo 0; 0 < Pu < 1; c) P+ is positivity preserving: for u Bb we have u 0 d) P+ is sub-Markovlan: for u Bb we have 0 < u <1 e) Pt has the Feller property: if u = Co. then Ptu Coo; f) Pt is strongly continuous on Co.: limt-o Ptu-ulco = 0 for all u Coo; g) Pt has the strong Feller property: if u = Bb, then Pru Ch.