Gaussian heat kernel ($p_t(x)=(2\pi t{)}^{-d/2}{e}^{-|x{|}^2/(2t)}$)

For standard Brownian motion in ${\mathbb{R}}^d$, $p_t(y-x)$ is the transition density from $x$ to $y$. It is the heat kernel for generator $\frac{1}{2}\Delta$. For every unit vector $e$, $\int |{\partial }_e{p}_t(x)|dx=\sqrt{2/\pi }/\sqrt{t}$, which bounds the change in this density under a spatial translation and proves the Gaussian heat-kernel proof of the harmonic Liouville theorem.

For $D,\gamma >0$, the displayed function solves the heat equation $u_t=D{u}_{xx}$ and has initial condition $u(x,0)=A{e}^{-\gamma (x-x_0{)}^2}$. Set $s=1+4\gamma Dt$. Direct differentiation givesThese identities prove the equation. The total integral is $A\sqrt{\pi /\gamma }$, independent of time. The real smooth formula is valid for $t>-1/(4\gamma D)$, including a finite interval of negative time for this particular initial profile. For nonzero $A$, its value at $x_0$ diverges as the lower time endpoint is approached. The superposition principle combines finitely many such profiles with different centers or widths.

For $D,\gamma ,a>0$, apply the superposition principle to two translated solutions for Gaussian initial data for the heat equation. The displayed solution of $u_t=D{u}_{xx}$ has odd initial profile $A(e^{-\gamma (x-a{)}^2}-e^{-\gamma (x+a{)}^2})$. Its total spatial integral is zero. For $A>0$, it is positive on $x>0$ and negative on $x<0$ at every $t\ge 0$.

Writing it as $2A{s}^{-1/2}{e}^{-\gamma (x^2+a^2)/s}\sinh (2\gamma ax/s)$ shows how cancellation affects its late-time shape. On the diffusive scale $x=O(\sqrt{s})$, its leading form is $4A\gamma ax{s}^{-3/2}{e}^{-\gamma x^2/s}$. Thus the two lobe maxima asymptotically lie at $x=\pm \sqrt{s/(2\gamma )}$, while their absolute heights decay as $s^{-1}$. Their centers are not fixed at the initial Gaussian centers. The first spatial moment remains ${\int }_{\mathbb{R}}xu(x,t)dx=2Aa\sqrt{\pi /\gamma }$, as follows directly from the Gaussian integral and translation.

For $d\ge 3$, $A_d=\Gamma (d/2-1)/(2{\pi }^{d/2})$. Substitute $u=|x{|}^2/(2s)$ to obtain the displayed identity for $x\ne 0$. The semigroup property and Tonelli theorem then give $P_t(|\cdot {|}^{2-d})(x)=A_d^{-1}{\int }_t^{\infty }{p}_s(x)ds$. The coefficient multiplying the time integral is the reciprocal of the coefficient in the full Green integral, a distinction important in Brownian motion hitting asymptotics.