when consider the fractional laplacian $(-\triangle)^\alpha$,i wonder how biger (-\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 $(-\triangle)^\alpha$.And 0<\alpha<1$.And i wonder how things change when we consider considering the higer order laplacian?

when consider the fractional laplacian $(-\triangle)^\alpha$,i wonder how biger difference between $0<\alpha<1$ and $\alpha>1$ . Since most paper i have met is the case of $(-\triangle)^\alpha$.And i wonder how things change when we consider the higer order laplacian?