Convergence of solutions of one–dimensional stochastic heat equations1

Abstract

We consider one-dimensional stochastic heat equations of the following form: where X(t) is an C([-m, m])-valued Stochastic Process and {Bt t≥0} denotes the so–called cylindrical Brownian motion on the real, separable Hilbert space H = L 2[-m m]. For the case that σ is a multiplication operator we prove the weak convergence of solutions of a stochastic heat equation on the interval [-m m] to the solution of the corresponding equation on the whole real line as m→∞ For the case that σ is a particular operator (depending only on m) we show convergence of the solutions to a stationary Gaussian limit process that can serve as model of stationary freeway traffic flow