Suppose $B_t$ is a standard Brownian motion on $\mathbb{R}$ and let $L_t$ be its local time at zero. Let $p_t(x,y)$ be the transition density of $B_t$, i.e. $p_t(x,y) = \frac{1}{\sqrt{2\pi}}\exp\left(- \frac{(x-y)^2}{2t} \right)$ and let $u^\alpha(x,y)$ be the potential density of $B_t$, i.e. $u^\alpha(x,y) = \int_0^\infty e^{-\alpha t} p_t(x,y)dt$. It is well known that $u^0(x,y) \equiv \infty$, in other words, there is no Greens function for 1-d heat equation.
Now, let $B^k_t$ be the 1-d Brownian motion killed when $L_t>e$ where $e$ is an independent exponential random variable. It is known that the transition semi-group of $B_t^k$ is given by $$ P_t^kf(x) = \mathbb{E}_x\left[e^{-L_t}f(B_t) \right],\quad f \in L^2(\mathbb{R}),\, t \ge 0. $$
My question is, 1) does the killed process $B^k_t$ has a Greens function? That is, if $u_k^\alpha(x,y)$ is the $\alpha$-potential density of $B^k_t$, which exists by Fukashima, does $u^0_k(x,y)$ exist? 2) if it exists, can it be expressed in terms of some known functions?
More generally, if $({\cal E},{\cal F})$ is a recurrent Dirichlet form on some nice space $X$, (thinking about BM in $\mathbb{T}^3$ where $\mathbb{T}$ is the unit circle), and $\nu$ is a finite smooth measure with finite energy integral, (such as $\delta_{x = y}(dx,dy,dz)$ on $\mathbb{T}^3$), can we compute the potential density of the Dirichlet form $$ {\cal E}^\nu(u,v) := {\cal E}(u,v) + (u,v)_\nu, \quad u,v \in {\cal F}. $$ where $(u,v)_\mu := \int_{X} u(x)v(x) \nu(dx)$.
Yes. In fact $$ u_k^0(x,y) = u^0_0(x,y)+c^{-1},\qquad \qquad (1) $$ where $u^0_0$ is the Green's function for the Brownian motion killed upon first hitting $0$, and the constant $c$ is given by $$ c=\int_{\Bbb R} \Bbb E_x[L_1]\,dx. $$ This came from a little of the theory of Brownian excursions from $0$.
In general, if $X$ admits potential densities and the CAF $A$ associated with $\nu$ satisfies $\Bbb P_x[A_\infty>0]=1$, then the killed process $X^k$ is transient and will admit potential densities. Something like (1) will hold. The first term on the right of (1) should be replaced by the the potential density of $X$ killed on hitting the fine support, $F$, of $A$. The second term won't be constant, but will involve the theory of the excursions of $X$ from $F$.