Let $K_\alpha(t,x)$ be the (generalised or fractional) Heat kernel which corresponds to the fractional Heat equation (I'm not sure that's the right name) in $\mathbb R^n$ $$ u_t=(-\Delta)^\alpha u, \quad \alpha\in(0,1]. $$

A fact, which appears to be very well-known, states that: $$ K_\alpha(t,x)>0, \quad \text{for all $x\in\mathbb R^n,\, t>0$ and $\alpha\in(0,1]$ }. $$ It is so well-known that no paper where I have encountered this fact gives any reference.

I am looking for a reference where I could see the proof of this fact.


If I guess right you are interested in the monotonicity of the operator. That is, if $u_0 \ge 0$ then $u \ge 0$. This can proved directly, by other means.

From this fact, due to the expression of $u$, and since this is true for every $u_0$, I guess you can prove directly that $K_\alpha \ge 0$.


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