Limiting measure and stationarity of solutions to stochastic evolution equations with Volterra noise

Abstract

Large-time behavior of solutions to stochastic evolution equations driven by two-sided regular cylindrical Volterra processes is studied. The solution is understood in the mild sense and takes values in a separable Hilbert space.

Sufficient conditions for the existence of a limiting measure and strict stationarity of the solution process are found and an example for which these conditions are also necessary is provided. The results are further applied to the heat equation perturbed by the two-sided Rosenblatt process.

Keywords

• Volterra process
• stochastic evolution equation
• limiting measure
• stationary increments
• Rosenblatt process