---
abstract: 'In this paper, we introduce a class of stochastic partial differential equations (SPDEs) with fractional time-derivatives, and study the $L_2$-theory of the equations. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping.'
address:
- 'Department of Mathematics, University of Washington, Seattle, WA 98195, USA'
- 'Department of Mathematics, Korea University, 1 Anam-dong, Sungbuk-gu, Seoul, 136-701, Republic of Korea'
- 'Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, San56-1 Shinrim-dong Kwanak-gu, Seoul 151-747, Republic of Korea'
author:
- '[**Zhen-Qing Chen**]{}'
- '[**Kyeong-Hun Kim**]{}'
- '[**Panki Kim**]{}'
title: Fractional time stochastic partial differential equations
---

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**Introduction**
================

Fractional calculus has attracted lots of attentions in several fields including mathematics, physics, chemistry, engineering, hydrology and even finance and social sciences. The classical heat equation $\partial_t u=\Delta u$ describes heat propagation in homogeneous medium. The time-fractional diffusion equation $\partial^\beta_t u=\Delta  u$ with $0<\beta<1$ has been widely used to model the anomalous diffusions exhibiting subdiffusive behavior, due to particle sticking and trapping phenomena. Here the fractional time derivative $\partial^{\beta}_t$ is the Caputo derivative of order $\beta\in (0,1)$, defined 