  The main goal of this work is to prove that every non-negative {\it strong
solution} $u(x,t)$ to the problem $$
  u_t+(-\Delta)^{\alpha/2}u=0 \ \quad\mbox{for }
  (x,t)\in\mathbb{R}^{n}\times(0,T), \quad 0<\alpha<2, $$
  can be written as $$u(x,t)=\int_{\mathbb{R}^{n}}{P_{t}(x-y)u(y,0)\, dy},$$
where $$P_{t}(x)=\frac{1}{t^{n/\alpha}}P\left(\frac{x}{t^{1/\alpha}}\right), $$
and $$ P(x):=\int_{\mathbb{R}^{n}}{e^{ix\cdot\xi-|\xi|^{\alpha}}d\xi}. $$ This
result shows uniqueness in the setting of non-negative solutions and extends
some classical results for the heat equation by D. V. Widder in \cite{W0} to
the nonlocal diffusion framework.
