Show that Lemma 8.10 remains true for any \(d\)-dimensional Feller process \(X_{t}\) with continuous paths and generator \(L=\sum_{j, k=1}^{d} a_{j k}(x) \partial_{j} \partial_{k}+\sum_{j=1}^{d} b_{j}(x) \partial_{j}\) such that \(a_{11}(x) \geqslant a_{0}>0\) and \(\left|b_{1}(x)\right| \leqslant b_{0}

Set \(u(x)=e^{-x_{1}^{2} / / r^{2}}\) and multiply it with a smooth cut-off function \(\chi \in \mathcal{C}_{c}^{2}\left(\mathbb{R}^{d}\right),\left.\chi\right|_{\mathbb{B}(0, r)} \equiv 1\), to make sure it is in \(\mathfrak{D}(L)\). Then calculate \(L u(x)=L(u \chi)(x)\) for \(x \in \mathbb{B}(0, r)\).

Data From Lemma 8.10

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8.10 Lemma. Let (B) to be a BMd and D C Rd be an open and bounded set. Then E* TDc 0 such that Dc ue (Rd) c D(A) and, for all |x| e-1/4 =: K > 0. 2 2r2 4r2 Therefore, we can use Dynkin's formula (7.30) with = T^n to get E(T ^n) < Au(B) dr = E u(Br^n) = u(x) = 2|||| = 2. By Fatou's lemma, E*T