Space-time duality for fractional powers of operators and graph Laplacians

We develop an operator–theoretic framework for spatially nonlocal diffusion. Our
main result establishes a space–time duality: for any α > 0, the semigroup orbit u(t) = e−tAαφ solves both the space–fractional evolution ∂tu = −Aαu and the time–fractional problem D1/αt u =Au, where D1/αt denotes the right-sided Liouville derivative. This duality explains when spatial nonlocality may be reinterpreted as temporal memory, and for α = 1/p, p ∈ N, yields a classical
p-th order-in-time formulation. We apply the theory to diffusion on graphs driven by fractional Laplacians Lα and by transformed d-path Laplacians (Estrada operators) ˜Lc. We show that ∂tu = −Lαu is equivalent to D1/αt u = Lu, whereas ˜Lc are purely spatially nonlocal and cannot be realized as fractional powers of local Laplacians. In finite dimensions every diffusive Laplacian admits a fractional root, though generically such roots are dense and hence nonlocal. These results provide structural criteria distinguishing mixed from intrinsically nonlocal diffusion on networks.

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