We consider one-dimensional stochastic heat equations of the following form:whereX(t) is anC([-m, m])-valued Stochastic Process and {Btt≥0} denotes the so–called cylindrical Brownian motion on the real, separable Hilbert spaceH=L2[-mm]. For the case that σ is a multiplication operator we prove the weak convergence of solutions of a stochastic heat equation on the interval [-mm] to the solution of the corresponding equation on the whole real line asm→∞ For the case that σ is a particular operator (depending only onm) we show convergence of the solutions to a stationary Gaussian limit process that can serve as model of stationary freeway traffic flow