Let $X_t \in \mathbb{R}^d$ satisfy
$$dX_t = \sqrt{2D}dW_t$$
where $W_t$ is standard Brownian motion. Assume we impose a reflecting boundary condition at the boundary of the ball of unit radius (centered at the origin). Assuming $|X_0| \in (\epsilon, 1)$ for $\epsilon \in (0, 1)$, calculate the mean first passage time for $X_t$ to reach the ball of radius $\epsilon$ (centered at the origin).
If $T_1(x)$ is the mean first passage time when $|X_0| = x$, I think I saw that that we must have $\mathscr{L}T_1 = -1$ with boundary condition $T_1(x) = 0$ for $x$ on the boundary, with $\mathscr{L}$ being the infinitesimal generator of the process. For Brownian motion scaled as the problem suggested I calculated that $\mathscr{L} = D\sum_{i = 1}^{d} \frac{\partial^2}{\partial x_i^2}$, but I'm not even sure this is right given the reflecting nature of the process. I honestly don't know how to account for the reflection at the boundary.
The problem suggests we will need to use radial coordinates to convert a PDE to an ODE. I don't understand PDEs at have very weak ODE knowledge so I don't have a clue what the problem is suggesting or how to solve the problem it suggests. I don't know how to convert to radial coordinates for a $d$-dimensional object.
For the reflecting boundary condition, simply add a Neumann boundary condition $\partial T_1 / \partial \mathbf{n} = 0$ on $\{ |x| = 1 \}$. Remembering also the Dirichlet boundary datum $T_1(x) = 0$ at the absorbing circle of radius $\epsilon$ (i.e., at $\{|x| = \epsilon\}$). Together with the PDE $\Delta T_1 = -1/D$, these lead you to a Poisson's equation with the mixed Dirichlet-Neumann boundary condition. I believe you can apply the Feynman-Kac formula to obtain a probabilistic solution of this problem at hand.