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 $$ u_t+(-Δ)^{α/2}u=0 \ \quad\mbox{for } (x,t)\in\mathbb{R}^{n}\times(0,T), \quad 0<α<2, $$ can be written as $u(x,t)=\int_{\mathbb{R}^{n}}{P_{t}(x-y)u(y,0)\, dy},$ where $P_{t}(x)=\frac{1}{t^{n/α}}P\left(\frac{x}{t^{1/α}}\right),$ and $P(x):=\int_{\mathbb{R}^{n}}{e^{ix\cdotξ-|ξ|^α}dξ}.$ 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.

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