On pg. 133 of *Heat Kernels and Spectral Theory*, Davies is studying the heat kernel $K(x,y,t)$ of the operator $H = -\Delta + |x|^{\alpha}$ for $\alpha > 0$. He wishes to prove a lower bound, and writes, "If $H\_{B}$ is the operator obtained from $H$ by imposing Dirichlet boundary conditions on the surface of the ball with center $x$ and radius $1$, then
$$K(x,x,t) \geq K\_{B}(x,x,t)$$
for all $t > 0$." (This I see follows from the parabolic maximum principle and nonnegativity of $K$.) He then continues, "Moreover,
$$|y|^{\alpha} \leq (|x| + 1)^{\alpha}$$
for all $y \in B$ so
$$K\_{B}(x,x,t) \geq K\_{0}(x,x,t)\exp\left\{-(|x| + 1)^{\alpha}t\right\}$$
for all $t > 0$ where $K\_{0}$ is the heat kernel of $-\Delta$ on $B$ with Dirichlet boundary conditions."


This is where he loses me. I'm aware that for, say, $V\_{1}, V\_{2} \in L^{\infty}(\mathbb{R}^{n})$ with $V\_{1} \leq V\_{2}$, the Feynman-Kac formula implies that the operator $H\_{1} = -\Delta + V\_{1}$ has heat kernel greater than or equal to that of $H\_{2} = -\Delta + V\_{2}$.


But I'm not sure if something analogous is the justification in Davies' argument, because I can't find a version of the Feynman-Kac formula that would apply in this setting.


Does anyone happen to know the proper explanation? Thank you.



Is there a Feynman-Kac formula applicable to Dirichlet problems for Schrödinger operators?
Yes, if $H = -\Delta + V$ on a smooth domain $D$ with Dirichlet boundary conditions, then the solution to $\partial\_t u = - H u$ with initial condition $u\_0$ is given by
$$
u(t,x) = E\_x \Bigl[ \exp\Bigl(-\int\_0^t V(X\_s)\,ds\Bigr) u\_0(X\_s) 1\_{t < \tau}\Bigr]\;,
$$
where $X$ is a Brownian motion started at $x$ and $\tau$ is the first time when $X$ hits the boundary of $D$. I don't have a reference off the top of my head, but you can see this quite easily by arguing that $H$ is the limit (in resolvent sense) of $H\_n$ where $H\_n$ is defined on all of $\mathbf{R}^d$ and has potential $V\_n$ which is equal to $V$ inside $D$ and $n$ outside $D$. (The intuition is that the Dirichlet boundary condition really corresponds to having a potential that is infinite outside of $D$.) For Neumann boundary conditions, you get a similar formula by using a reflected Brownian motion instead of a killed Brownian motion.


