Heat kernels and semigroup smoothing

On Euclidean d-space, consider uₜ=κΔu with constant κ>0 and initial data u₀ in L¹. The Gaussian kernel displayed below has total integral one, is positive everywhere for t>0, and solves the heat equation away from its initial singularity. Convolving it with u₀ defines a smooth solution at every positive time. The initial condition is attained in L¹ as t decreases to zero; an arbitrary integrable datum need not have a pointwise value or a classical derivative at every initial point.

If u₀ is nonnegative and not zero as an L¹ function, positivity of the kernel makes u(x,t)>0 at every spatial point for every positive time. This is infinite propagation speed, even though the distant values can be extremely small. It is a mathematical feature of the diffusion model, not a statement that every underlying microscopic mechanism transmits signals infinitely fast. Integrating the convolution preserves the total mass when u₀ is integrable. For signed data, absolute integrability justifies the corresponding exchange of integrals.

Convolution of two centered Gaussians adds their variances, so evolving for time s and then time t equals evolving for time s+t. Writing S(t)u₀=Gκ(t)*u₀ gives S(t)S(s)=S(t+s), with S(0) the identity. Positivity and unit kernel mass imply contraction in each Lᵖ norm for 1≤p≤∞. Strong continuity at zero holds on Lᵖ for p<∞. It does not hold in the norm of all L∞ functions; a jump can remain a fixed essential-supremum distance from any smooth approximation near the jump.

The kernel also gives quantitative improvement between norms. Its supremum times the initial L¹ norm bounds the positive-time supremum of the solution. Derivatives can be placed on the Gaussian to obtain further estimates, with factors reflecting inverse powers of the diffusion length. These bounds become singular as t approaches zero, which is consistent with smoothing rough data instantly without pretending the rough initial datum was already smooth. They provide a priori control on the constructed solution, not backward stability for reconstructing an earlier state.

The same Gaussian cannot simply be used unchanged on every domain. Periodic heat flow uses a periodized kernel or Fourier series. Homogeneous Dirichlet flow loses heat through an absorbing boundary, whereas homogeneous Neumann flow preserves mass in a bounded insulated region. Their eigenfunctions, long-time limits, and possible zero modes differ. An expression that solves the differential equation in the interior can still fail the prescribed boundary condition, so the spatial operator and its domain belong in the definition of the evolution.

Smoothing is strongly directional in time. A Fourier component with frequency magnitude ξ is multiplied by exp(−κ|ξ|²t), strongly suppressing fine scales. Reversing the multiplier amplifies measurement errors exponentially at high frequency, making the backward heat problem unstable in ordinary data norms. This does not contradict forward uniqueness. It separates the existence and continuous dependence of the forward evolution from the very different inverse task of recovering information that diffusion has nearly erased.