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In the leading massless collinear parton approximation, $P^2=0$ and $q^2=-Q^2$ imply $(\xi P+q{)}^2=2\xi P\cdot q-Q^2$. With Bjorken x $x=Q^2/(2P\cdot q)$ and $P\cdot q>0$, the Dirac delta change-of-variable formula yields the displayed identity. The positive-energy branch of the final parton also gives $\delta (E_q+\xi E_P-E_k)/E_k=2\delta ((E_q+\xi E_P{)}^2-E_k^2)$ locally on physical support. Keeping target mass instead introduces ${\xi }^2{M}^2$ and changes the root and Jacobian.

For a surface thermostat, the planes containing $F$ form the projective bundle of $T(SM)/\mathbb{R}F$. The vertical Maslov cycle is the section selecting the image of the vertical vector field of a surface unit tangent bundle. It is a smooth hypersurface. In the local projective chart $\mathbb{R}F+\mathbb{R}(V+rH)$, it is $r=0$; the lifted flow satisfies $\dot{r}=1$ on the cycle, hence crosses it transversely.

For discrete-time Markov chains run at time step $1/n$, suppose the displayed conditional moments converge uniformly on compact sets to continuous coefficients, the local conditional large-jump second moments vanish, initial laws converge, and compact containment holds on each finite horizon. If the limiting diffusion martingale problem is well-posed and nonexplosive, the chain laws converge to its solution. Vanishing jumps give a continuous limit; linearly interpolated paths converge in the uniform path topology and step paths in Skorokhod J1 topology. Local moment bounds give tightness after exit-time localization. Taylor expansion identifies the limiting test-function martingales, and uniqueness of the martingale problem makes every subsequential limit the same. The diffusion approximation of a birth-death process is an important example.

A continuous adapted process started at a fixed point is Brownian motion exactly when $f(X_t)-f(X_0)-\frac{1}{2}{\int }_0^t\Delta f(X_s)ds$ is a martingale for every $f\in C_c^{\infty }({\mathbb{R}}^d)$. Cutoff coordinate and coordinate-product tests give continuous local martingales with quadratic covariations $[X^i,X^j{]}_t={\delta }_{ij}t$. The Itô formula makes $\exp (iu\cdot (X_t-X_0)+|u{|}^{2}t/2)$ a local martingale, whose deterministic modulus bound on bounded horizons makes it a true martingale. Its conditional expectation identifies every increment as centered Gaussian with covariance $(t-s)I$, independent of the past. This proves the converse via the Lévy characterization of multidimensional Brownian motion. The forward implication is the Itô formula.

For a positive function $g$ in the domain of a Markov jump-process generator, the generator martingale for $g(X_t)$ and semimartingale integration by parts give the displayed local martingale. Suitable finite-horizon deterministic bounds on $g$ and $Lg/g$ make it a true martingale. For a single marked exponential jump process, take $g(x)=e^{\theta x}$. Then the exponent's compensating process is $r_{\theta }(t\wedge T)$, where $r_{\theta }=(e^{\theta }-1)/\theta -1$ for $\theta \ne 0$ and $r_0=0$; it is continuous adapted and hence predictable.

With scale $N$, such a process has finitely many possible jump vectors $\ell /N$ occurring at rates $N{\beta }_{\ell }(x)$ depending on the current scaled state. Its drift is ${\sum }_{\ell }\ell {\beta }_{\ell }(x)$, while the predictable covariance rate of its martingale fluctuations is $N^{-1}{\sum }_{\ell }\ell {\ell }^T{\beta }_{\ell }(x)$.

// Target: probability-and-statistics.bigb

For a Markov jump process with off-diagonal rate kernel $q(x,dy)$, the jump measure records the jump time and the destination state. Relative to a filtration for which the process has the Markov jump property, its predictable compensator is the displayed rate measure, up to explosion. The corresponding generator is $Qf(x)=\int [f(y)-f(x)]q(x,dy)$. An enlarged filtration revealing future jumps need not preserve this compensator, so the filtration condition is essential.

A kernel specifying the jump rate from state $x$ for each displacement $z$. With the displacement convention the infinitesimal generator is $L\phi (x)=\int (\phi (x+z)-\phi (x))K(x,dz)$. The destination convention instead records a rate measure on the new state, translating each displacement by $x$. For finite total rate, that total is the exponential holding rate and the normalized measure is the next-jump distribution.

Let $T$ be a rate-one exponential random variable independent of a uniform random variable $U$ on $(0,1)$. In its natural filtration, this Markov jump process jumps once from zero to its mark and then stays there. Its destination rate kernel is $q(x,dy)=1_{\{x=0\}}{1}_{(0,1)}(y)dy$. For a Borel mark set $B$, the martingale $1_{\{T\le t,U\in B\}}-|B|(t\wedge T)$ follows directly from the memoryless residual lifetime. Information about the future mark or lifetime must not be inserted into the initial filtration.

This finite-state Markov jump process models two opposing populations, each suffering losses at a rate proportional to the surviving opposing population. Both coordinate axes are absorbing. Its embedded jump chain removes one individual at each step, with probabilities $b/(a+b)$ and $a/(a+b)$. The corresponding deterministic equations $\dot{a}=-\kappa b$, $\dot{b}=-\kappa a$ conserve $a^2-b^2$; finite stochastic populations satisfy a near-invariant instead.

A diffusion limit rescales a sequence of stochastic processes so that its limit is a diffusion process. For suitably rescaled independent small jumps, the first two jump moments determine the limiting drift and diffusion coefficient. Brownian motion arises when the limiting drift is zero and the variance grows linearly in time.

A driftless square-root diffusion process is the nonnegative solution of $d{X}_t=\sqrt{X_t}d{W}_t$, with zero an absorbing boundary. For elapsed time $h>0$, starting from $x\ge 0$, its Laplace transform isTo prove this, fix the final time $h$ and set $F(t,x)=\exp [-xu/(1+u(h-t)/2)]$. Differentiation gives $F_t+\frac{1}{2}x{F}_{xx}=0$, while $F(h,x)=e^{-ux}$. By Itô formula, $F(t,X_t)$ is a local martingale; it takes values in $[0,1]$, so localization and bounded convergence make it a true martingale. Taking expectations at the two endpoints proves the formula. The Markov property gives the same formula conditionally at any starting time.

In a one-Brownian-factor arbitrage-free market, every traded derivative with nonzero diffusion exposure $s$ has the same excess-drift/exposure ratio. Otherwise holdings $s_2$ in the first and $-s_1$ in the second, financed in the bank account, cancel all diffusion risk while leaving a nonzero excess drift. At zero exposure, the correct statement is $m-rV=\theta s=0$, not division by zero. The short rate itself need not be a traded asset, so its physical drift does not alone fix this risk price.

A positive state-price density prices a productive stock through a conditional dividend sum plus a future deflated-price term. The displayed condition removes that terminal term, yielding transversality and fundamental dividend prices. An individual consumption budget instead has terminal term $\mathbb{E}[{\zeta }_N{S}_N{\theta }_{N+1}]$, so its vanishing must be imposed or proved for the particular holdings as well.

For nested compact H-hulls whose closures avoid $x\in \mathbb{R}$, the boundary derivative is an excursion avoidance probability. Avoiding the larger hull is a subset of avoiding the smaller one, which proves the derivative inequality. For a hull contained in $\{|z|\le R\}$ and $|x|>R$, comparison with the filled half-disc gives the sharp bound $1-R^2/x^2\le g_A^{\prime }(x)\le 1$, since that half-disc has mapping-out function $z+R^2/z$. The empty hull and filled half-disc attain the two bounds.

Decreasing strictly positive summable sequences with equal totals and $x\prec y$ admit an infinite doubly stochastic matrix realizing the displayed relation. Choose $k$ with $y_k\ge x_1>y_{k+1}$ and average the pair to obtain $x_1$ and $y_k+y_{k+1}-x_1$. The remaining decreasing sequence majorizes the remaining target sequence: short initial sums contain values at least $x_1$, and longer sums are the old sums minus $x_1$. Repeat on the tail, freezing one normalized finite-support row at each stage. Limit column sums are at most one; Tonelli's theorem, equal finite totals and strict positivity of every $y_j$ force every column sum to equal one.

For anticommuting components and an antisymmetric numerical matrix $A$, $\delta ({\psi }^{T}A\psi )=2\delta {\psi }^{T}A\psi$: move the second varied field past the first and use $A^T=-A$. Applying this to $C^{-1}$ and to $C$ cancels the factor $1/2$ in $\mathcal{L}=\bar{\psi }i/\partial \psi +\frac{1}{2}(m{\psi }^T{C}^{-1}\psi -m^{\ast }\bar{\psi }C{\bar{\psi }}^T)$. Varying the Dirac adjoint gives the first displayed equation. Integration by parts in the spinor variation gives $i{\gamma }^{\mu T}{\partial }_{\mu }{\bar{\psi }}^T+m{C}^{-1}\psi =0$; multiplication by $C$ gives the second. Squaring the coupled equations yields $(\square +|m{|}^2)\psi =0$, so the physical Majorana fermion mass is $|m|$.

For one-dimensional flow obeying an isothermal equation of state with $u_z=B_z=0$, the primitive variables $(\rho ,u_x,u_y,B_y)$ form a four-variable hyperbolic system and $B_x$ is constant. The speeds relative to $u_x$ satisfy $c^4-(c_s^2+v_a^2)c^2+c_s^2{v}_{ax}^2=0$. These are the fast and slow magnetosonic waves. The independent out-of-plane Alfvén wave polarization and a freely advected entropy mode are absent from this reduced isothermal system.

For sound speed $v_s$ much larger than Alfvén speed $v_a$, and angle $\theta$ between the wavevector and magnetic field, the fast magnetosonic wave has $v_f^2=v_s^2+v_a^2{\sin }^2\theta +O(v_a^4/v_s^2)$. The slow magnetosonic wave has $v_{\mathrm{slow}}^2=v_a^2{\cos }^2\theta [1-(v_a^2/v_s^2){\sin }^2\theta +O(v_a^4/v_s^4)]$. The fast mode is predominantly longitudinal sound. The slow mode becomes almost incompressible, polarized in the plane of the wavevector and magnetic field; the Alfvén wave is polarized perpendicular to that plane. The latter two are predominantly magnetic-tension waves with equal leading phase speeds.

For two incompressible streams of equal density with a common tangential magnetic field parallel to their velocities, decaying normal modes and interface displacement/total-pressure continuity give $\rho (\sigma +k_x{U}_1{)}^2+\rho (\sigma +k_x{U}_2{)}^2=2k_x^2{B}^2$ in units ${\mu }_0=1$. The displayed dispersion relation follows, with $v_A^2=B^2/\rho$. Thus $|\Delta U|<2v_A$ excludes exponential Kelvin-Helmholtz instability; equality is marginal. Above threshold the growth rate is $|k_x|\sqrt{(\Delta U{)}^2/4-v_A^2}$. The transverse horizontal wavenumber affects the decay length through $k=(k_x^2+k_y^2{)}^{1/2}$ but not this growth threshold. Magnetic tension supplies the stabilizing restoring force.