A Widder's type Theorem for the heat equation with nonlocal diffusion

The main goal of this work is to prove that every non-negative {\it strong solution} $u(x,t)$ to the problem $$u_t+(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}u=0\text{quadfor}(x,t)\in {\mathbb{R}}^n\times (0,T),0<\alpha <2,$$ can be written as $$u(x,t)={\int }_{{\mathbb{R}}^n}{P}_t(x-y)u(y,0)dy,$$ where $$P_t(x)=\frac{1}{t^{n\mathrm{/}\alpha }}P\left(\frac{x}{t^{1\mathrm{/}\alpha }}\right),$$ and $$P(x):={\int }_{{\mathbb{R}}^n}{e}^{ix\cdot \xi -\mathrm{\mid }\xi {\mathrm{\mid }}^{\alpha }}d\xi \mathrm{.}$$ 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.