Feynman--Kac formula for the heat equation with a one-center point interaction in $d=3$
We study Schrödinger operators with a one-center point interaction, formally defined by \begin{align*} -Δ_α=-Δ+α\,δ_0(\cdot), \end{align*} for $α\in\mathbb{R}$, and the associated heat equation \begin{align} \partial_t u=\tfrac{1}{2}Δ_α u,\quad u(0,x)=u_0(x)\in C_c^{\infty}(\mathbb{R}^3\setminus\{0\}).\label{eq:HEapp} \end{align} Here $Δ$ denotes the Laplacian (self-adjoint on $L^2(\mathbb{R}^3)$) and $δ_x$ the Dirac measure at $x$. The operator $-Δ_α$ can be realized either as a self-adjoint extension of $-Δ|_{C_0^{\infty}(\mathbb{R}^3\setminus\{0\})}$ in $L^2(\mathbb{R}^3)$, or as the norm-resolvent limit of $-Δ+λ_\varepsilon V(\cdot/\varepsilon)$ for suitable $λ_\varepsilon$ and $V:\mathbb{R}^3\to\mathbb{R}$. In this paper we construct, for each $t>0$ and $x\in\mathbb{R}^3\setminus\{0\}$, a probability law on path space and a normalizing function $G_t^α(x)$ giving the following probabilistic representation of the solution to the associated equation: \begin{align*} u(t,x)=G_t^α(x)\,\mathbb{E}\bigl[u_0\bigl(W^{t,x}(t)\bigr)\bigr], \end{align*} where $\{W^{t,x}(s):0\le s\le t\}$ is a continuous process depending on $(t,x,α)$. The result provides a Feynman--Kac type formula for the heat equation with a one-point interaction in three dimensions.