Moment bounds for a class of fractional stochastic heat equations

Abstract

We consider fractional stochastic heat equations of the form $\frac{\partial u_{t}(x)}{\partial t}=-(-\Delta)^{\alpha/2}u_{t}(x)+\lambda\sigma(u_{t}(x))\dot{F}(t,x)$. Here, $\dot{F}$ denotes the noise term. Under suitable assumptions, we show that the second moment of the solution grows exponentially with time. Since we do not assume that the initial condition is bounded below, this solves an open problem stated in [Probab. Theory Related Fields 152 (2012) 681–701]. Along the way, we prove a number of other interesting results about continuity properties and noise excitation indices. These extend and complement results in [Stochastic Process. Appl. 124 (2014) 3429–3440], [Khoshnevisan and Kim (2013)] and [Khoshnevisan and Kim (2014)].