An increment of a real- or vector-valued stochastic process is the difference $X_t-X_s$ between its values at two times $s<t$. A random walk accumulates its one-step increments; a Poisson process has nonnegative integer increments counting arrivals in $(s,t]$. A process can have independent increments even though its values at different times are dependent.

A locally bounded process can be made bounded by stopping it on an increasing sequence of random time intervals. For stochastic integration on positive times, one useful convention is that there exist stopping times ${\tau }_n\uparrow \infty$ almost surely and finite deterministic constants $K_n$ such thatThe interval convention matters at time zero: including zero additionally requires a deterministic bound on $H_0$, whereas the positive-time Stochastic integral does not use that value.

Every continuous adapted process with finite initial value has this positive-time localization. Set ${\tau }_n=\inf \{t\ge 0:|H_t|\ge n\}\wedge n$. If its initial magnitude already exceeds $n$, this Stopping time is zero and the indicated interval is empty. Otherwise continuity prevents overshoot at the first exit, so $K_n=n$ works. continuity on each compact time interval makes ${\tau }_n\to \infty$ pathwise.

A predictable locally bounded process is integrable against a semimartingale: the bounded stopped integrands define compatible integrals, which are joined by localization. Predictability supplies the required information condition; a magnitude bound alone does not replace it. For example, Brownian motion is not bounded on the whole infinite time axis but its continuous adapted paths give the stated localization.

// Target: probability-and-statistics.bigb

A Random field is a family of random variables indexed by points of a spatial domain, or more generally by a multi-dimensional parameter set. When second moments are finite, its componentwise covariance function and spectral tensor describe second-order dependence. A Gaussian random field has jointly Gaussian finite-dimensional distributions; a general Random field need not be Gaussian.

A process $Z$ is self-similar with exponent $H$ if $Z(a\cdot )$ and $a^{H}Z$ have the same finite-dimensional distributions for every $a>0$, or the same path law when the process is considered as a random path. Fractional Brownian motion has this property with its Hurst exponent. It converts dilation of time to deterministic scaling of amplitude: $Z(N\cdot )/N\stackrel{d}{=}{N}^{H-1}Z$. For $H<1$, so that $N^{H-1}\to 0$, a small-noise large deviation principle with inverse-variance speed produces the large-deviation speed $N^{2(1-H)}$ for those rescaled paths.

Find a constrained path and a boundary regulator which together modify a given driving path. In the half-line case, write $x=g+k\ge 0$, require $k(0)=0$ and $k$ nondecreasing, and require its increases to occur only where $x=0$. For multidimensional orthants the regulator acts through a specified reflection matrix. The Skorokhod reflection map on the half-line solves the one-dimensional continuous-path case explicitly.

A Bernoulli process is a sequence of independent and identically distributed random variables with a Bernoulli distribution: $\mathbb{P}(B_n=1)=p$ and $\mathbb{P}(B_n=0)=1-p$. It models repeated Bernoulli trials with a fixed success probability and no dependence between trials. The count $K_n={\sum }_{j=1}^n{B}_j$ has a binomial distribution, expected value $np$, and variance $np(1-p)$; the first success time has a geometric distribution on $\{1,2,\dots \}$. The observations $B_n$ and the accumulated count $K_n$ are different stochastic processes: the observations are independent, while consecutive counts share all earlier observations. ^[ref]

mean reversion is a tendency for a stochastic process to move back towards a reference level. For a mean-reverting Ornstein-Uhlenbeck process with $\theta >0$, the conditional expected value of a future deviation is the present deviation multiplied by $e^{-\theta h}$ after time $h$. Individual sample paths can move away from that level: mean reversion describes a conditional tendency, not monotone motion or a guaranteed return by a deadline. It also does not by itself imply a stationary process; the initial probability distribution matters.

A controlled Markov process has transition probabilities or rates depending on its current state and the chosen control action. A Markov policy selects an action from the current state and possibly the time. Once such a policy is fixed, the induced state evolution is a Markov process. For general history-dependent controls, the state alone need not remain Markov. A dynamic programming operator optimizes over the available actions at each state.

At stationarity, the mean number ${\mu }_i$ divided by the molecular removal flux $v_i$ is the mean time a molecule remains in the system. Birth or death event rates must be multiplied by the numbers of molecules changed in each event before computing this lifetime.

// Target: mathematical-biology.bigb

This noise-relevant event size weights absolute stoichiometric jump sizes by their molecular turnover flux. At stationarity the denominator is twice the balanced production or removal flux $v_i$, so the normalized diffusion diagonal is $D_{ii}=2⟨s_i⟩/({\tau }_i{\mu }_i)$. Fixed birth bursts of size $b$ and unit deaths give $⟨s⟩=(b+1)/2$. This quantity is not the event-count-weighted average absolute jump size.

// Target: mathematical-biology.bigb

The Bell number $B_n$ counts all set partitions of an $n$-element finite set, with no restriction on the number of blocks of a set partition. Summing the Stirling numbers of the second kind gives $B_n={\sum }_{k=0}^{n}S(n,k)$. The first five values, for $n=0,1,2,3,4$, are $1,1,2,5,15$. In particular, $B_4=1+7+6+1=15$. ^[ref]

A population inversion puts more emitters in the upper than in the lower quantum state of a transition in the nondegenerate two-level model. Let $N_2,N_1$ be their number densities. Equal stimulated-transition strengths make the emission rate proportional to $N_2$ and the competing photon absorption rate proportional to $N_1$, so net optical gain is proportional to $N_2-N_1$. For $N_2=6\times 10^{20}{\mathrm{m}}^{-3}$ and $N_1=4\times 10^{20}{\mathrm{m}}^{-3}$, the difference is positive; swapping them makes the medium absorb. Equal populations give zero net stimulated contribution. With degenerate levels, the relevant comparison uses populations per statistical weight of an atomic level, rather than simply $N_2>N_1$.

Why pump through extra levels? Ordinary steady resonant pumping of a nondegenerate two-level system drives both photon absorption and stimulated emission. With equal induced rate $W$ and upper-state decay rate $\Gamma$, its elementary population equation isat a stationary population. Increasing this pump saturates towards $N_2=N_1$; it cannot sustain population inversion in this model. This is a steady rate-equation conclusion, not a universal prohibition on all driven two-level dynamics or all possible laser gain mechanisms. ^[ref].

In a familiar three-level path, pumping excites a higher level which relaxes into the upper lasing level; the lower lasing level is the ground state, so substantial ground-state depletion is needed. A four-level path also provides rapid emptying of the lower lasing level into a separate ground state. Suitable lifetimes then allow accumulation above and depletion below the lasing transition. Pumping still supplies the energy: extra levels provide a route to population inversion, not free energy. ^[ref].

Uniform solid-body rotation adds the centrifugal potential $-{\Omega }^2{\varpi }^2/2$ to the gravitational potential. In the corotating frame, hydrostatic equilibrium becomes $\nabla P=-\rho \nabla \varphi$. The Poisson equation gives ${\nabla }^2\varphi =4\pi G\rho -2{\Omega }^2$. On regular connected equipotential surfaces, $P$ is constant; taking the curl shows that $\rho$ is constant too. Uniform composition and an invertible equation of state then make $T$ constant on each such surface. These statements do not require $|\nabla \varphi |$ to be constant there.

A stellar core is the central region distinguished from the surrounding envelope by its composition, energy generation or thermodynamic state. A core boundary at radius $R_c$ encloses mass $M_c=m(R_c)=4\pi {\int }_0^{R_c}\rho (r)r^{2}dr$. In a hydrogen-shell-burning red giant, a helium-rich core can be inert while a narrow surrounding layer supplies the luminosity. Core mass and core radius are separate variables: the relation between them must come from a structural model, such as a polytropic mass-radius relation, rather than from the word core alone.

The temperature at a star's centre. In gas-pressure-supported stellar homology, the stellar hydrostatic equation and ideal gas law give $T_c\propto \mu GM/(\mathcal{R}R)$, with a coefficient set by the dimensionless density and pressure profiles. It controls the strongly temperature-dependent stellar energy-generation rate.

Taking the curl of axisymmetric hydrostatic equilibrium with centrifugal acceleration gives $\varpi {\partial }_z{\Omega }^2=(\nabla p\times \nabla \rho {)}_{\varphi }/{\rho }^2$. For an approximately spherical ideal gas with constant composition, use $\nabla \rho =\nabla p/c_s^2-\rho \nabla S/c_p$ to obtain the displayed relation. Latitudinal entropy differences can support noncylindrical differential rotation. A barotropic fluid gives cylindrical rotation in this leading balance; the relation does not determine the transport stresses or circulation rate.

In a steady axisymmetric star, a torque due to Reynolds stress, Maxwell stress tensor or viscosity must balance advection of specific angular momentum by stellar meridional circulation. Writing the nonadvective angular momentum transport flux as ${\mathbf{F}}_J$ gives the displayed balance. Thus stresses that maintain differential rotation also drive a meridional response. The stellar thermal-wind balance supplies a separate leading force constraint, not a replacement for this torque equation.

Integrated spectra can confuse older stellar populations with more metal-rich ones because both can redden the light and change absorption lines. Line strengths depend on stellar metallicity, temperatures, ages, and abundance ratios. Weak metal lines in a young population dominated by hot bright stars need not indicate low stellar metallicity.

Helioseismology infers the Sun's structure and dynamics from its stellar oscillations. Spatial projection of resolved surface observations onto spherical harmonics identifies angular degree and azimuthal order; a temporal Fourier transform measures mode frequencies. Stellar acoustic modes constrain the adiabatic sound speed, while rotational splitting constrains stellar rotation. Their finite wavelength, surface reflection phases and finite observational coverage limit inversion resolution.

The slow neutron-capture process builds nuclei heavier than iron when neutron captures occur more slowly than typical beta decays. Thermal pulses in asymptotic-giant-branch stars produce characteristic enhancements of elements such as strontium, zirconium, and barium.