Show that under the assumptions of Proposition 23.5 we can interchange integration and differentiation: \(\frac{\partial^{2}}{\partial x_{j} \partial x_{k}} \int p(t, x, y) u(y) d y=\int \frac{\partial^{2}}{\partial x_{j} \partial x_{k}} p(t, x, y) u(y) d y\) and that the resulting function is in \(\mathcal{C}_{\infty}\left(\mathbb{R}^{d}\right)\).

Data From 23.5 Proposition 

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23.5 Proposition (backward equation. Kolmogorov 1931). Let (X+) to denote a diffusion with generator (A, D(A)) such that Alex (Rd) is given by (23.1) with bounded drift and diffusion coefficients b = C(Rd, Rd) and a = C(Rd, Rdxd). Assume that (X+)+20 has a transition density p(t, x, y), t> 0, x, y e Rd, satisfying P, ' ((0,00) Rdx Rd), p(t,.,.), at ap(t,,) ap(t,) dp(t,.,.) Coo(Rd x Rd) for all t>0. Then Kolmogorov's first or backward equation holds: op(t, x, y) at L(x, Dx)p(t, x, y) for all t> 0, x, y e Rd. (23.5) Proof. Denote by L(x, Dx) the second-order differential operator given by (23.1). By assumption, A = L(x, D) on eco (Rd). Since p(t, x, y) is the transition density, we know that the transition semigroup is given by Ttu(x) = [p(t,x,y)u(y) dy.