Let $(X_t)_{t\geq0}$ be a simple symmetric random walk in $\mathbb{Z}^d$, in continuous time (holding times are $Exp(1)$ random variables). Moreover, suppose $X_t$ is killed after walking for a time $T_{\lambda}\sim Exp(\lambda)$.
For such a walk $(X_t)_{t\geq0}$ starting from the origin, I am interested in the probability that $X_t$ hits site $y\in\mathbb{Z}^d$ before being killed, i.e.
$$ \mathbb{P}_0(\tau_y<T_{\lambda}), $$
where $\tau_y=\inf\{t\geq0\,:\,X_t=y\}$ is the hitting time of site $y\in\mathbb{Z}^d$.
Such a quantity has already been considered in the literature, for example in "Random Walk: A Modern Introduction" by Lawler and Limic [RW], where it is called "first visit generating function", and denoted by
$$ F(x,y;\lambda)=\mathbb{P}_x(\tau_y<T_{\lambda}). $$
The only difference is that in [RW] time is discrete, $\tau_y=\min\{n\geq1\,:\,X_n=y\}$ and $T_\lambda$ is a Geometric random variable such that $\mathbb{P}(T>j)=\lambda^j$. The name "generating function" is due to the fact that in discrete time, as shown in [RW],
$$ F(x,y;\lambda)=\sum_{n=1}^\infty\lambda^nf_n(x,y), $$
where $f_n(x,y)=\mathbb{P}_x(\tau_y=n)$.
Going back to continuous time, after some standard computations one gets that
$$ \mathbb{P}_0(\tau_y<T_{\lambda})=\frac{\sum_{n=0}^\infty\mathbb{P}_0(X_n=y)(\lambda+1)^{-n}}{\sum_{n=0}^\infty\mathbb{P}_0(X_n=0)(\lambda+1)^{-n}} $$ where now the walk $(X_n)$ on the right had side is actually in discrete time. The function appearing at the numerator and denominator on the rhs has also been already considered in the literature, for example in [RW] is called "Green's generating function" and denoted by
$$ G(x,y;\xi)=\sum_{n=0}^\infty\mathbb{P}_x(X_n=y)\xi^{n}. $$
As shown in [RW], for positive $\xi\leq1$, $G(x,y;\xi)$ is the expected number of visits of the random walk (in discrete time) to $y$, starting from $x$, before being killed (killing time $T_\xi$ is such that $\mathbb{P}(T_{\xi}>j)=\xi^j$).
With this notation, one gets that
$$ \mathbb{P}_0(\tau_y<T_{\lambda})=\frac{G(0,y;(\lambda+1)^{-1}))}{G(0,0;(\lambda+1)^{-1})}. $$
I am looking for ways to work out the dependence of the rhs above from $y$ and $\lambda$, and hence for estimates on the Green's generating function $G(x,y;\xi)$.
Are there some know results that might be helpful anybody could point me to?