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$C$-existence families, $C$-semigroups and their associated abstract Cauchy problems in complete random normed modules

Xia Zhang, Leilei Wei, Ming Liu

TL;DR

This work extends classical semigroup theory to complete random normed modules by introducing $(mild)\,C$-existence families and $C$-semigroups, and by proving that these structures yield (mild) solutions to abstract Cauchy problems in the random setting. It establishes precise conditions under which mild solutions exist and, with additional assumptions, are unique, linking the existence framework to the behavior of locally a.s. bounded and exponentially bounded $C$-semigroups and their generators. The paper also details the relationships among $C$-existence families, $C$-semigroups, and their associated Cauchy problems, and concludes with a stochastic differential equation application that demonstrates how to construct explicit solutions via $V(t)C^{-1}$. These results broaden the applicability of operator semigroup methods to random environments and provide a robust toolkit for random evolution equations.

Abstract

In this paper, we first introduce the notion of a (mild) $C$-existence family in complete random normed modules, then we prove that a (mild) $C$-existence family can guarantee the existence of the (mild) solutions of the associated abstract Cauchy problem in the random setting. Second, we investigate several important properties peculiar to locally almost surely bounded $C$-semigroups in complete random normed modules, which are not involved in the classical theory of $C$-semigroups. Finally, based on the above work, some relations among $C$-existence families, $C$-semigroups and their associated abstract Cauchy problems in complete random normed modules are established, which extend and improve some known results. Besides, an application to a type of stochastic differential equations is also given.

$C$-existence families, $C$-semigroups and their associated abstract Cauchy problems in complete random normed modules

TL;DR

This work extends classical semigroup theory to complete random normed modules by introducing -existence families and -semigroups, and by proving that these structures yield (mild) solutions to abstract Cauchy problems in the random setting. It establishes precise conditions under which mild solutions exist and, with additional assumptions, are unique, linking the existence framework to the behavior of locally a.s. bounded and exponentially bounded -semigroups and their generators. The paper also details the relationships among -existence families, -semigroups, and their associated Cauchy problems, and concludes with a stochastic differential equation application that demonstrates how to construct explicit solutions via . These results broaden the applicability of operator semigroup methods to random environments and provide a robust toolkit for random evolution equations.

Abstract

In this paper, we first introduce the notion of a (mild) -existence family in complete random normed modules, then we prove that a (mild) -existence family can guarantee the existence of the (mild) solutions of the associated abstract Cauchy problem in the random setting. Second, we investigate several important properties peculiar to locally almost surely bounded -semigroups in complete random normed modules, which are not involved in the classical theory of -semigroups. Finally, based on the above work, some relations among -existence families, -semigroups and their associated abstract Cauchy problems in complete random normed modules are established, which extend and improve some known results. Besides, an application to a type of stochastic differential equations is also given.

Paper Structure

This paper contains 6 sections, 14 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C$-existence families in complete $RN$ modules
  4. $C$-semigroups in complete $RN$ modules
  5. Relations among $C$--existence families, $C$--semigroups and their associated abstract Cauchy problems in complete $RN$ modules
  6. An application to a type of stochastic differential equations

Key Result

Proposition 2.1

Let $(X_{1},\|\cdot\|_{1})$ and $(X_{2},\|\cdot\|_{2})$ be two $RN$ modules over $K$ with base $(\Omega,{\mathcal{F}},P)$ and $T$ a linear operator from $X_{1}$ to $X_{2}$. Then the following statements hold. (1) $T$ belongs to $B\left(X_1, X_2\right)$ if and only if $T$ is a continuous module homom

Theorems & Definitions (29)

  • Proposition 2.1: Guorelations
  • Proposition 2.2: Guozhang
  • Proposition 2.3: Guozhang
  • Proposition 2.4: Z-L
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • ...and 19 more