Let (left(B_{t} ight)_{t geqslant 0}) be a (mathrm{BM}^{d}) and set (u(t, x):=P_{t} u(x)). Adapt the proof of

Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}\) and set \(u(t, x):=P_{t} u(x)\). Adapt the proof of Proposition 7.3.g) and show that for \(u \in \mathcal{B}_{b}\left(\mathbb{R}^{d}\right)\) the function \(u(t, \cdot) \in \mathcal{C}^{\infty}\) and \(u \in \mathcal{C}^{1,2}\) (i.e. once continuously differentiable in \(t\), twice continuously differentiable in \(x\) ).

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 (B+)120 and Coo Coo (Rd), Cb = 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 = ullo for all u Bb. c) Pt is positivity preserving: for u e Bb we have u>0 d) P+ is sub-Markovian: for u = B, we have 0 < u <1 Pu > 0; 0 < Pu < 1; e) Pt has the Feller property: if u e Coo then Pru Coo; f) Pt is strongly continuous on Co.: limt-o ||Ptu - ullo = 0 for all u Coo; g) Pt has the strong Feller property: if u e Bb, then Pru Ch.