Optimal Domains for L0-valued Operators Via Stochastic Measures

Abstract

Abstract. We study extension of operators, where E is an F-function space and L 0 ([0, 1]) the space of measurable functions with the topology of convergence in measure, to domains larger than E, and we study the properties of such domains. The main tool is the integration of scalar functions with respect to stochastic measures and the corresponding spaces of integrable functions.

Stratonovich-type integral with respect to a general stochastic measure

“…For a deterministic measurable function g : X → R and a SM µ, an integral of the form X g dµ is defined and studied in [10,Chapter 7], see also [2], [12]. In particular, every bounded measurable g is integrable with respect to any µ.…”

“…Integrals of deterministic functions with respect to general stochastic measures are well studied, see [10], [2], [12]. Stochastic partial differential equations with this integral were considered in [14], [16].…”

L 1(μ) for vector-valued measure μ

“…For vector lattices (Riesz spaces) and topologies on them, we mostly follow Aliprantis and Burkinshaw [2]. A reader who finds the above description of the integral too sketchy may consult the quoted paper [5] of Curbera and Delgado or Turpin in 1975 [31].…”

“…Following the terminology of Curbera and Delgado in [5], we say that X satisfies the bounded multiplier test if the following condition holds:…”

Mild solution of the heat equation with a general stochastic measure

“…If W H (t) is a fractional Brownian motion with Hurst index H > 1/2 and f : [7] states conditions under which a process with independent increments generates a stochastic measure. For deterministic measurable functions g : X → R, an integral of the form X g dµ is studied in [12] (see also [7,Chapter 7], [2]). The construction of this integral is standard, uses an approximation by simple functions and is based on results of [16,17,19,20].…”