MOMENT BOUNDS FOR A CLASS OF FRACTIONAL STOCHASTIC HEAT EQUATIONS

• 成果类型: Article
• 署名作者: Foondun, Mohammud; Liu, Wei; Omaba, Mcsylvester
• 署名单位: University of Strathclyde; Shanghai Normal University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1108
• 发表日期: 2017
• 页码: 2131-2153
• 关键词: partial-differential-equations; nonlinear noise excitation

摘要

We consider fractional stochastic heat equations of the form partial derivative u(t)(x)/partial derivative t = -(-Delta)(alpha/2)u(t)(x) + lambda sigma(u(t)(x))(F) over dot (t, x). Here, (F) over dot 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)].