Asymptotic properties of the heat equation driven by stochastic measure

For the heat equations driven by a stochastic measure on $[-L,L]$ with the Dirichlet boundary condition, we prove that solutions tend to the solution of the heat equations defined on $\mathbb{R}$ as $L\to \mathrm{\infty }$. The estimate of the convergence rate is obtained. For a stochastic measure, we assume the $\sigma$-additivity in probability only.