Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes

We prove a new Burkholder–Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if (S(t, s)) _⩽_s_≤_t_⩽_T is a C-evolution family of contractions on a 2-smooth Banach space X and (Wt)t∈[0,T] is a...

Maximal inequalities for stochastic convolutions in 2-smooth Banach spaces and applications to stochastic evolution equations

This paper presents a survey of maximal inequalities for stochastic convolutions in 2-smooth Banach spaces and their applications to stochastic evolution equations. This article is part of the theme issue 'Semigroup applications everywhere'.