2013/12/18 by Rodrigo Bañuelos, Bañuelos, Rodrigo, Adam Osekowski +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #math.AP #math.FA #math.PR
paper · pdf · doi:10.48550/arxiv.1312.5038
arxiv created 2013/12/18 · arxiv updated 2013/12/19
Let X be a supermartingale starting from 0 which has only nonnegative jumps. For each 0<p<1 we determine the best constants cp, Cp and \mathfrakcp such that supt≥ 0||Xt||p≤ Cp||-inft≥ 0Xt||p, ||supt≥ 0Xt||p≤ cp||-inft≥ 0Xt||p and ||supt≥ 0|Xt| ||p≤ \mathfrakcp||-inft≥ 0Xt||p. The estimates are shown to be sharp if X is assumed to be a stopped one-dimensional Brownian motion. The inequalities are deduced from the existence of special functions, enjoying certain majorization and convexity-type properties. Some applications concerning harmonic functions on Euclidean domains are indicated.