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If a Lévy process has mean zero and $\mathrm{Var}(X_1)={\sigma }^2<\infty$, then $\mathrm{Var}(X_t)=t{\sigma }^2$ and $X_t^2-t{\sigma }^2$ is a martingale.

The intersection $L^1\cap L^2$ is Banach for $\Vert f{\Vert }_1+\Vert f{\Vert }_2$, but it is generally incomplete when equipped with the $L^1$ norm alone. Truncations of an $L^1$ function outside $L^2$ provide an $L^1$-Cauchy sequence with no limit in the intersection.

For $1\le p\le \infty$, the Lebesgue space $L^p$ is complete in its usual norm.

For a small-noise diffusion $dX=a(X)dt+\sqrt{ϵ}\sigma (X)dW$, the Freidlin--Wentzell action of a smooth path is

A rate function is a lower-semicontinuous map $I$ into $[0,\infty ]$. It is good when every sublevel set $\{x:I(x)\le c\}$ is compact.

Axisymmetric harmonic functions separate into radial powers times Legendre polynomials.

A Dirichlet Laplacian eigenfunction vanishes on the boundary and satisfies $-\Delta u=\lambda u$.

The normalized order-zero function has the representation

For the Laguerre equationthe ansatz $y={\int }_{\gamma }{e}^{zt}f(t)dt$ givesThe contour must make the endpoint contributionvanish.

For $x>0$, substituting $y(x)={\int }_{\gamma }{e}^{xt}f(t)dt$ into $x{y}^{(3)}+2y=0$ and integrating by parts givesThe amplitude equation $(t^{3}f{)}^{\prime }=2f$ has solution $f(t)=C{t}^{-3}{e}^{-1/t^2}$. On $\gamma =(-\infty ,0)$ both endpoint terms vanish, and hence

In plane polar coordinates, Laplace's equation isSeparation of variables gives radial powers $r^n,r^{-n}$ for positive angular modes and $1,\log r$ for the zero mode.

In three dimensions,Consequently the decaying solution of $-{\nabla }^{2}u=f$ in free space is

This is the free-space Green function for the negative Laplace operator.

At the energy-critical exponent $p=(d+2)/(d-2)$, the normalized Aubin-Talenti bubblesolves $\Delta W+W^p=0$ and optimizes the critical Sobolev embedding theorem.

When $a=2/(p-1)$ and $a<d-2$, the Lane-Emden equation has the singular scale-invariant solution

For a quadratic kernel $K(t,k)\sim t+k^2$, the singular heat-capacity correction is proportional to

For zero initial velocity and Fourier amplitude $a(k)$, observation along $x=Vt$ with $0\le V<c$ selects the two stationary wavenumbers $\pm k_0$, whereWriting $s=\sqrt{c^2-V^2}$, the leading oscillation has angular frequency $As$ and amplitude proportional to $t^{-1/2}$.

In a flat fundamental square, straight horizontal, vertical and rational-slope lines project to closed geodesics. The central horizontal glide axis is one-sided, a vertical line is two-sided, and the line $(2t,t)$ closes with one transverse self-intersection.

Forthe transformations with even $c$ form an index-two translation subgroup. Its quotient is a torus, giving the orientation double cover $T^2\to K$.

If spatial reflection leaves a Stokes configuration and forcing equivalent to flow reversal while preserving one candidate velocity component, uniqueness forces that component to vanish.

The Penrose process extracts rotational energy by allowing a fragment with negative stationary Killing energy to enter the black hole while another fragment escapes with more energy than the original body carried.