The polynomial stability problem of stochastic delay differential equations has been studied in recent years. In contrast, there are relatively few works on stochastic partial differential equations with pantograph delay. The present paper is devoted to investigating large-time asymptotic properties of solutions for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. We first show that the mild solutions of stochastic pantograph delay evolution equations with nonlinear multiplicative noise tend to zero with general decay rate (including both polynomial and logarithmic rates) in the $p$th moment and almost sure senses. The analysis is based on the Banach fixed point theorem and various estimates involving the gamma function. Moreover, by using a generalized version of the factorization formula and exploiting an approximation technique and a convergence analysis, we construct the nontrivial equilibrium solution, defined for $t\in ℝ$, for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. In particular, the uniqueness, Hölder regularity in time and general stability, in the $p$th moment and almost sure senses, of the nontrivial equilibrium solution are established.

近年来，人们对随机时滞微分方程的多项式稳定性问题进行了研究。相比之下，关于受电弓延迟的随机偏微分方程的研究相对较少。本文致力于研究具有非线性乘性噪声的随机受电弓时滞演化方程解的大时间渐近性质。我们首先证明，具有非线性乘性噪声的随机受电弓时滞演化方程的温和解在一般衰减率（包括多项式和对数率）下趋于零。$p$那一刻，几乎可以肯定的感觉。该分析基于巴纳赫不动点定理和涉及伽玛函数的各种估计。此外，通过使用分解公式的广义版本并利用近似技术和收敛分析，我们构造了非平凡平衡解，定义为$t\epsilon ℝ$，用于具有非线性乘性噪声的随机受电弓延迟演化方程。特别是，在时间上的唯一性、霍尔德规律性和总体稳定性，$p$建立了非平凡平衡解的第 时刻和几乎确定的意义。