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Classical heat equation solutions with identical continuous initial data and a uniform Gaussian spatial growth bound on each finite time interval are unique. For the zero-data difference, choose $b>a$ and compare on a short slab with $\epsilon {\Phi }_b$, where ${\Phi }_b=(1-4bt{)}^{-n/2}\exp (b|x{|}^2/(1-4bt))$ solves the equation for $t<1/(4b)$. Its faster spatial growth controls the lateral boundary of large cylinders. The heat equation maximum principle bounds both signs of the difference by $\epsilon {\Phi }_b$ inside. Increasing the cylinder radius and then sending $\epsilon$ to zero proves uniqueness on that slab; repeating slabs covers the finite interval.

Let $M$ be a closed manifold with a Riemannian metric and use $\Delta =-\mathrm{div}\mathrm{grad}$, the positive Laplace-Beltrami operator. This is the setting in which the final discrete eigenfunction expansion applies without additional boundary or noncompact spectral hypotheses. A Riemannian heat kernel is a smooth $H(t,x,y)$ for $t>0$ such thatfor every smooth initial function $f$. Here $\nu$ solves the heat equation. The limit is the statement that the initial kernel is the Dirac delta distribution on the diagonal. In the closed setting this uniquely determines the heat kernel; it is symmetric, preserves constants, is nonnegative, and satisfies the semigroup property.

A heat parametrix $P(t,x,y)$ is an approximate version with the same delta initial limit and with errorregular enough at $t=0$ to correct by a convergent integral series. One may construct it by the local Gaussian ansatzwhere $\chi =1$ near the diagonal and is supported in a convex normal neighborhood, and the smooth coefficients solve the usual radial transport equations. The leading coefficient accounts for the Riemannian volume form. Taking $q$ sufficiently large makes the residual extend continuously, with any prescribed finite number of derivatives, to $t=0$. Alternatively a smooth asymptotic summation of all transport coefficients gives a residual vanishing to every order. These local construction, extension and differentiability facts are used here as subsidiary results, as permitted.

For the correction proof take a heat parametrix whose residual is smooth up to $t=0$ on a short interval $[0,T]$. We also use its uniform integral bound ${\sup }_{0<t\le T,x}{\int }_M|P(t,x,y)|d{V}_y<\infty$, its approximate-identity limit, and the following standard differentiability property: convolution with a smooth time-dependent kernel gives a smooth kernel for $t>0$, and differentiating it yields the identity below. These properties follow from the Gaussian estimates and the delta initial limit; no conclusion about the final exact heat kernel is assumed.

Define the Volterra convolution of kernels byIt is associative wherever these integrals converge. The delta term at the upper endpoint givesSeek $H=P+P\ast Q$. Then its error vanishes exactly when $Q+R+R\ast Q=0$. The solution is the Volterra parametrix correctionThe signs matter: the first correction is $-P\ast R$.

To prove convergence, let $|R|\le C$ and $V=\mathrm{vol}(M)$. The time variables in $R^{\ast k}$ range over a simplex of volume $t^{k-1}/(k-1)!$, soThe series for $Q$ therefore converges uniformly on $[0,T]\times M\times M$. Derivatives obey analogous bounds, with polynomial factors in $k$, by the quoted residual extension and differentiation properties. Thus the series can be convolved and differentiated as above. Associativity and absolute convergence give $Q+R+R\ast Q=0$, proving $({\partial }_t+{\Delta }_x)H=0$. The correction $P\ast Q$ has integral norm $O(t)$ by the integral bound on $P$ and boundedness of $Q$. It has zero initial limit, so $H$ has precisely the required delta initial data.

For uniqueness, a smooth solution $w$ with zero initial data satisfies the heat equation energy identityIts initial norm is zero, hence $w=0$. Applying uniqueness to consecutive evolutions proves the semigroup property. Choose $s_0<T$ and, for arbitrary $t>0$, compose enough kernels $H(t/r)$ that $t/r<s_0$. The resulting kernel is independent of the subdivision by uniqueness, is smooth for positive time by the quoted differentiability property, and extends the construction to every $t>0$. This proves a parametrix determines the global heat kernel in the closed setting. The heat equation maximum principle gives nonnegativity and applying uniqueness to the constant initial function gives conservation of total mass. Self-adjointness of the positive Laplace-Beltrami operator $\Delta$ gives symmetry.

Finally let $\{{\varphi }_j\}$ be a complete complex orthonormal eigenbasis of $\Delta$, with eigenvalues ${\lambda }_j\ge 0$, repeated by multiplicity. As subsidiary analytic facts we use the compact elliptic compact elliptic spectral theorem, elliptic regularity bounds making each fixed derivative of ${\varphi }_j$ grow at most polynomially in $1+{\lambda }_j$, and a polynomial eigenvalue-counting bound. The spectral expansion of the Riemannian heat kernel converges because exponential decay makes the following series converge with every derivative when $t\ge \epsilon >0$:Indeed, evolving initial data ${\varphi }_j$ gives $e^{-t{\lambda }_j}{\varphi }_j$ by direct substitution into the heat equation and uniqueness. For general $f\in L^2(M)$, completeness and the heat equation energy identity giveThe smoothly convergent kernel series represents this same operator, and equality for all smooth $f$ identifies it pointwise with the constructed heat kernel. The complex conjugate is required by the inner product; for a real eigenbasis it may be omitted. A noncompact Riemannian manifold may instead require a spectral integral, and the unqualified discrete formula should not be asserted there.

The bounded classical solution is the heat-kernel convolutionThe two-dimensional heat kernel has integral one. Boundedness of $g$ permits differentiation under the integral for $t>0$, proving $u_t=\Delta u$ and $|u|\le \Vert g{\Vert }_{\infty }$. Its approximate identity property gives $u(x,t)\to g(x)$, locally uniformly since $g$ is continuous. This bounded solution satisfies the required Gaussian-growth bound.

For heat-equation uniqueness under Gaussian growth, let $w$ be the difference of two solutions, so $w(x,0)=0$ and $|w|\le M_0{e}^{a|x{|}^2}$ on a finite time interval. Choose $b>a$ and a slab $0\le t\le \tau <1/(4b)$. The positive comparison solutionsatisfies $({\Phi }_b{)}_t=\Delta {\Phi }_b$. For any $\epsilon >0$, its faster spatial growth makes $|w|\le \epsilon {\Phi }_b$ on a sufficiently large lateral cylinder boundary. At the initial boundary, $w=0$. The heat equation maximum principle applied to $\pm w-\epsilon {\Phi }_b$ proves the same inequality inside. First allow the cylinder radius to increase, then let $\epsilon \downarrow 0$. Repeating these slabs proves uniqueness on every finite interval with the stated uniform growth bound. No spatial integrability of $g$ is required.

At time $T$, this is Gaussian filtering with covariance $2TI$, hence $\sigma =\sqrt{2T}$ per coordinate. With the angular-frequency Fourier transform convention $\hat{f}(\xi )=\int e^{-ix\cdot \xi }f(x)dx$, the Gaussian filtering Fourier multiplier isFor merely bounded $g$, this identity is understood through tempered distributions. The multiplier leaves the zero frequency unchanged and attenuates large frequencies exponentially: it suppresses rapid noise fluctuations but blurs image edges. In cycles-per-length frequency $k$, the multiplier is $e^{-4{\pi }^{2}T|k{|}^2}$.

For the printed Perona-Malik equation, retain the supplied conductance $c(s)=s{e}^{-s^2/(2{\lambda }^2)}$, which includes an extra factor $s$. In one dimension write $p=u_x$ and $F(p)=c(|p|)p=p|p|e^{-p^2/(2{\lambda }^2)}$. Then $u_t=F^{\prime }(u_x)u_{xx}$, withThe forward-backward threshold for gradient-weighted exponential diffusion is thereforeAt $|u_x|=0$ and $\sqrt{2}\lambda$ the coefficient vanishes, giving degenerate diffusion. Increasing $\lambda$ increases the forward-diffusion range. Small nonzero slopes smooth, while large slopes formally sharpen; a negative coefficient produces short-wave growth and ill-posedness, so this is a dynamics explanation, not a general existence theorem for arbitrary data. Replacing the printed conductance by $e^{-s^2/(2{\lambda }^2)}$ would give threshold $\lambda$, but that is a different equation.

The four-neighbour mean expansion follows by Taylor expansion: opposing first and third derivatives cancel, soIf $u(x_0)={\mathrm{mean}}_h(u)(x_0)$ for every sufficiently small $h$, division by $h^2$ and passage to the limit give $\Delta u(x_0)=0$. This is a pointwise conclusion; it does not assert harmonicity in a whole neighbourhood.

For the area median over the disk, the disk-median curvature expansion, at a point where $|\nabla u|\ne 0$, isConsequently the analogous median fixed-point condition yieldsThis is the vanishing of the level-line curvature at $x_0$: to second order the level line is straight there. It need not be a straight segment. For example $u(x,y)=y+x^3$ has zero disk median at the origin for every radius, by odd symmetry, but its zero level line is the cubic $y=-x^3$. Vanishing curvature on an entire connected regular level arc would force that arc to be straight. The factor $1/6$ is for the disk's uniform area measure; a circle-boundary median has a different scale factor, while producing the same zero-curvature equation.