Diffusivity, rescaling, and the heat kernel compared with the Poisson kernels

Assume Countable Choice for the kernel identities cited below.

With diffusivity parameter $\kappa >0$, the equation $u_t-\kappa \mathrm{\Delta }u=0$ is converted into the $\kappa =1$ heat equation of The heat operator, the heat equation, and the Cauchy problem by $v(x,t):=u(x,t\mathrm{/}\kappa )$, and its kernel is ${\mathrm{\Gamma }}_{\kappa }(x,t)=(4\pi \kappa t{)}^{-n\mathrm{/}2}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(4\kappa t)}$; the kernel $\mathrm{\Gamma }$ of The heat kernel on ${\mathbb{R}}^n$ and its causal extension is the case $\kappa =1$. The rescaling is the chain rule: ${\mathrm{\partial }}_{t}v(x,t)={\kappa }^{-1}{\mathrm{\partial }}_{t}u(x,t\mathrm{/}\kappa )$ while ${\mathrm{\Delta }}_{x}v(x,t)=(\mathrm{\Delta }u)(x,t\mathrm{/}\kappa )$, so $v_t=\mathrm{\Delta }v$ is equivalent to $u_t=\kappa \mathrm{\Delta }u$; the identity ${\mathrm{\Gamma }}_{\kappa }(x,t)=\mathrm{\Gamma }(x,\kappa t)$ gives the unit-mass normalisation of Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel.

For $n\ge 3$, the ball Poisson kernel $P_{R,a}(x,y)=(R^2-\mathrm{\mid }x-a{\mathrm{\mid }}^2)\mathrm{/}(R{\omega }_{n-1}\mathrm{\mid }x-y{\mathrm{\mid }}^n)$ of Poisson kernel of a Euclidean ball is a boundary-value kernel: its two slots are an interior point and a boundary point, it carries no time parameter, and its dependence on $x$ is not translation-invariant convolution, so it is not the heat kernel in other notation. For the half-space kernel of Poisson kernel and bounded Dirichlet problem on a half-space, a boundary mode $e^{2\pi i\xi \cdot x^{\mathrm{\prime }}}$ has bounded harmonic extension $e^{-2\pi \mathrm{\mid }\xi \mathrm{\mid }t}{e}^{2\pi i\xi \cdot x^{\mathrm{\prime }}}$: direct differentiation shows it is harmonic and the bounded Dirichlet uniqueness identifies it with the Poisson integral. Its derivative at $t=0$ is $-2\pi \mathrm{\mid }\xi \mathrm{\mid }$ times that mode, whereas ${\mathrm{\Delta }}_{x^{\mathrm{\prime }}}$ multiplies the mode by $-4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2$. This is the Fourier-symbol meaning of the Poisson generator $-\sqrt{-\mathrm{\Delta }}$, and distinguishes its normal-time family from the heat semigroup. In three spatial dimensions the homogeneous wave representation uses spherical means at radius $ct$, rather than a positive Gaussian convolution; this comparison is dimension-specific. A source's diffusivity normalisation must be matched before its kernel formula is quoted.