Currently, I am reading a paper about Brownian Motion. There is the following statement that I am desperate to understand:
Let $\mu(y)\equiv\mathbb E_y[\tau_{B_r(x)}]$, [...] then $\Delta\mu=-2$ on $\mathbb T_2\setminus B_r(x).$
Now, I do not have much knowledge about stochastic differential equations thus I basically only came up with this (trivial) thought:
- $\Delta\mu=0$ on $B_r(x)$ since $\mu\equiv 0$ on $B_r(x)$.
I'd be really happy about some advice, even a good book I could read to get familiar with this type of questions. If you need more details about the setup, please let me know!