= Feynman-Kac formula with a bounded potential
{c}
{title2=$u_t=\tfrac12u_{xx}+Vu$}
For bounded $V$ and bounded initial data $f$, a sufficiently regular <classical solution> bounded on every finite time slab satisfies
$$
u(t,x)=\mathbb E_x\left[f(B_t)\exp\!\left(\int_0^tV(B_s)ds\right)\right].
$$
To prove this, the <Itô formula> and <Itô product rule> make $u(T-t,B_t)e^{\int_0^tV(B_s)ds}$ a <local martingale>. Its deterministic bound on $[0,T]$ makes it a true <martingale>; taking <expectations> at the two endpoints gives the formula. This also gives uniqueness in the finite-horizon bounded class. The exponent has the same sign as the potential in the equation.
Back to article page
