Let $\Omega\subset \mathbb R^d$ be a bounded open (and connected) set. Consider $E\subset \Omega$ and $x\in \Omega\setminus E$. Denote by $W^x$ the standard Brownian motion starting at $x$, i.e. $W^x_0=x$. Define respectively
$$\tau_+:=\{t\ge 0: W^x_t\in \partial \Omega\} \quad \mbox{and}\quad \tau_-:=\{t\ge 0: W^x_t\in \partial E\}.$$
Here $W^x$ is used to model some particle which escapes successfully if $\tau_+>\tau-$ and is trapped if $\tau_+<\tau-$. Using PDE method we may compute the probability $\mathbb P[\tau_+>\tau_-]$. My question is as follows: If $E\subset \Omega$ is some random set, is there any reference on the computation of the above probability?
PS: An example is to take $E=B(Z,R):=\{y\in\mathbb R^d: |y-Z|<R\}$, where $Z$ uniformly distributed in $\Omega$ and conditioning on $Z$, $R$ is uniformly distributed in $[0,d(Z)]$ where $d(Z):=\{|y-Z|: y\in\partial \Omega\}$. An obvious generalisation is to consider a finite (random) number of traps $E_1, E_2,\ldots$
