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.

Continuity properties of Schrödinger semigroups with magnetic fieldsmentioned.) – Michael Tinker Aug 26 '13 at 13:26