when consider the fractional laplacian $(-\triangle)^\alpha$,is there an essential difference between $0<\alpha<1$ and $\alpha>1$ ? As far as I'm concern,the higer order laplacian ($\alpha>1$ ) ,unlike the lower case, has little connection with stochastic process.(lack of positivy.) Since most paper i have met is the case of $0<\alpha<1$.And i wonder how things change when considering the higer order laplacian?



Higher order fractional laplacian
The fractional Laplacian can be represented as a hypersingular integral operator, an integral operator with a singularity higher than the space dimension. Its convergence is provided by a regularization whose form depends on the order. From its form the difference in positivity properties is immediately clear. See S. G. Samko, Hypersingular Integrals and their applications, Taylor and Francis, London, 2002.


On the other hand, there exists a theory of parabolic equations with fractional Laplacians (and more general pseudodifferential operators) of any order. See S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Birkhauser, Basel, 2004. 


