TY - JOUR
T1 - On some properties of a class of fractional stochastic heat equations
AU - Liu, Wei
AU - Tian, Kuanhou
AU - Foondun, Mohammud
PY - 2016/5/19
Y1 - 2016/5/19
N2 - We consider nonlinear parabolic stochastic equations of the form ∂tu=Lu+λσ(u)ξ˙∂tu=Lu+λσ(u)ξ˙ on the ball B(0,R)B(0,R) , where ξ˙ξ˙ denotes some Gaussian noise and σσ is Lipschitz continuous. Here LL corresponds to a symmetric αα -stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on σσ , we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014).
AB - We consider nonlinear parabolic stochastic equations of the form ∂tu=Lu+λσ(u)ξ˙∂tu=Lu+λσ(u)ξ˙ on the ball B(0,R)B(0,R) , where ξ˙ξ˙ denotes some Gaussian noise and σσ is Lipschitz continuous. Here LL corresponds to a symmetric αα -stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on σσ , we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014).
KW - fractional laplacian
KW - heat kernel
KW - stochastic heat equation
KW - stochastic partial differential equations
UR - http://www.scopus.com/inward/record.url?scp=84969776674&partnerID=8YFLogxK
UR - http://link.springer.com/journal/10959
U2 - 10.1007/s10959-016-0684-6
DO - 10.1007/s10959-016-0684-6
M3 - Article
AN - SCOPUS:84969776674
SN - 0894-9840
JO - Journal of Theoretical Probability
JF - Journal of Theoretical Probability
ER -