Assume that in Lemma 8.1 the initial datum satisfies (f in mathcal{C}_{infty}left(mathbb{R}^{d} ight)) but not necessarily (f

Assume that in Lemma 8.1 the initial datum satisfies \(f \in \mathcal{C}_{\infty}\left(\mathbb{R}^{d}\right)\) but not necessarily \(f \in \mathfrak{D}(A)\). Consider (8.3) with \(P_{\epsilon} f\) instead of \(f\). Discuss the limit \(\epsilon \rightarrow 0\).

Data From Lemma 8.1

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8.1 Lemma. Let (Bt)to be a BMd, f = D(A) and u(t, x) = Exf(Br). Then u(t, x) is the unique bounded solution in 1,2 ((0, co) x Rd) ne([0, co) x Rd) and u(t,.) C (Rd) of the heat equation with initial value f, ru(t,x) - Axu(t, x) = 0, u(0, x) = f(x). (8.3a) (8.3b) Proof. Since f = D(A) c Co. (Rd), a variation of the calculation in the proof of Proposi- Ex. 8.1 tlon 7.3.g) shows that the u(t, x) = P+f(x) is in e,2 ((0, co) x Rd) and u(t,.) e (Rd). It remains to show uniqueness. Let u(t, x) be a bounded solution of (8.3) such that ue e,2((0,00) x Rd) ne([0, co) x Rd) and u(t,) = 2, (Rd). Thus, u(t,.) = D(A) and the Laplace transform is given by VA(x) = - u(t, x) e-t dt = lim lim fu(t, x) e-t dt. 0