An inequality for potentials and the 'hot-spots' conjecture

Abstract

Consider a nonnegative continuous potential V on the half disk D + = {z = x + iy: y > 0, |z| < 1} for which r 2 V(re iθ ) is nondecreasing as a function of r for every fixed 0 < θ < π. We prove an inequality for the distribution of the random variable ∫ τD+ 0 V(B s ) ds where B s is the Brownian motion reflected on the top portion of the boundary and killed on the lower portion and T D+ is its lifetime. This inequality, by the conformal invariance of Brownian motion, implies a result of Pascu [13] on hot-spots for certain symmetric convex domains.