= Positive-potential growth from Brownian recurrence
Suppose bounded $V\geq0$ is bounded below by a positive constant on some nonempty open interval, and $f\geq C>0$. The <Feynman-Kac formula with a bounded potential> gives $u(t,x)\geq C\mathbb E_x e^{\int_0^tV(B_s)ds}$. <Infinite occupation time of one-dimensional Brownian motion> makes that integral tend to infinity <almost surely>. The <monotone convergence theorem> then gives $u(t,x)\to\infty$ for every fixed $x$. Thus no globally bounded <classical solution> exists under these hypotheses, although boundedness on every finite time slab is compatible with the conclusion.
Back to article page
