Moments, intermittency and growth indices for the nonlinear fractional stochastic heat equation

Abstract

We study the nonlinear fractional stochastic heat equation in the spatial domain \({\mathbb {R}}\) driven by space-time white noise. The initial condition is taken to be a measure on \({\mathbb {R}}\), such as the Dirac delta function, but this measure may also have non-compact support. Existence and uniqueness, as well as upper and lower bounds on all pth moments \((p\ge 2)\), are obtained. These bounds are uniform in the spatial variable, which answers an open problem mentioned in Conus and Khoshnevisan (Probab Theory Relat Fields 152(3–4):681–701, 2012). We improve the weak intermittency statement by Foondun and Khoshnevisan (Electron J Probab 14(21):548–568, 2009), and we show that the growth indices (of linear type) introduced in Conus and Khoshnevisan (Probab Theory Relat Fields 152(3–4):681–701, 2012) are infinite. We introduce the notion of “growth indices of exponential type” in order to characterize the manner in which high peaks propagate away from the origin, and we show that the presence of a fractional differential operator leads to significantly different behavior compared with the standard stochastic heat equation.