An Integral Inequality and Its Application to a Problem of Stabilization by Noise

We prove the integral inequality [formula] for nonnegative functions b and g with g(∞) = 0 as well as various extensions. We apply it to g(x) = exp(−αx2), α > 0, thereby answering the question of whether an explosive scalar equation dy/dt = b(y(t)) can be made nonexplosive by adding homogeneous white noise in the negative.