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Processes $X,Y$ on the same probability space and index set are modifications of each other if $\mathbb{P}(X_t=Y_t)=1$ for each fixed index $t$. This differs from indistinguishability of stochastic processes, which requires equality at all indices on a single event of probability one. Equality at a countable dense set upgrades to indistinguishability when both processes have continuous paths.

For a stopping time $T$, the stochastic interval $[0,T)$ means the subset $\{(\omega ,t):0\le t<T(\omega )\}$ of sample-time space. Its endpoint varies with the sample. A locally defined stochastic process on it has no prescribed value at the endpoint. This notation is distinct from an ordinary deterministic interval.

A continuous locally defined process is a pair $(X,T)$ specified on the stochastic interval $[0,T)$, with a lifetime stopping time $T$ approached by an announcing sequence for a stopping time $T_n$. Each stopped process is an ordinary continuous adapted process. This is the convention used for continuous local differential equations; no value at $T$ is required. Restricting a globally defined process gives examples, but finite lifetimes also allow approach to a domain boundary or explosion.

A stellar composition profile gives chemical mass fractions against radius or enclosed mass. Flat portions can indicate mixing in a convective core or convective envelope, while burning shells and moving convective boundaries leave gradients and discontinuities.

The opacity mechanism drives stellar pulsation when compression increases opacity in a layer, trapping heat at a phase that feeds mechanical oscillation. Helium ionization zones provide this driving in Cepheid variables.

The helium mass fraction is the fraction of a material's mass carried by helium nuclei. Hydrogen burning raises it, whereas core helium burning lowers it while producing carbon and oxygen.

A helium-burning shell converts helium into carbon and oxygen outside a helium-exhausted stellar core. Thin-shell burning on the Asymptotic giant branch can develop a thermal pulse of an asymptotic-giant-branch star.

A helium star has lost most of its hydrogen-rich envelope and is made predominantly of helium. If massive enough, its core can ignite helium; a lower-mass stripped core instead becomes a helium white dwarf.

Stellar rejuvenation occurs when mixing brings fresh hydrogen into the core of a mass-gaining or merged main sequence star, making its remaining nuclear lifetime resemble that of a younger star. The amount of rejuvenation depends on the accretor's internal mixing and prior composition.

A red giant is an evolved star with a large radius and low effective temperature. On the first giant branch, a low- or intermediate-mass star is powered primarily by shell hydrogen burning around a helium-rich core.

A subgiant has begun evolving away from the main sequence after central hydrogen exhaustion, increasing its radius before the extended red giant stage.

The Hertzsprung gap is a sparsely populated region of the Hertzsprung-Russell diagram traversed rapidly by many stars evolving away from the main sequence toward the red giant region.

A blue loop is an excursion to higher effective temperature, followed by a return redward, during core helium burning in some evolved stars. Its extent depends on core and envelope structure, composition gradients and mixing. Only loops reaching the Cepheid instability strip produce the associated additional Cepheid variable crossings.

A carbon-oxygen core is the product of core helium burning. After central helium exhaustion it has little hydrogen or helium, while helium-rich and hydrogen-rich layers surround it.

For equal phase mass densities, equal sensible heat capacities per unit volume, and negligible thermal diffusion, a porous melting front driven by hot Darcy flux $U$ has speedHere $\Delta \varphi$ is the newly melted volume fraction. Mass conservation preserves $U$ on both sides: the liquid gained by melting fills the newly available pores. The enthalpy increase per unit bulk volume is $\rho c_p\Delta T+\rho L\Delta \varphi$, so its rate of production is balanced by incoming heat flux $\rho c_{p}U\Delta T$.

Directional solidification advances a freezing region preferentially along an imposed temperature gradient. Pulling material through fixed heat exchangers can make the mushy layer stationary in the apparatus frame even while solid and liquid pass through it.

A mushy layer contains solid crystals and liquid in local phase equilibrium, rather than a single sharp phase boundary. Its solid fraction changes as heat and solute are transported. A concentration-dependent liquidus couples temperature to liquid composition; a connected crystal framework makes the residual liquid behave as a porous-media flow.

A nuisance parameter is a statistical parameter required to specify the distribution of observations but not itself the target of inference. A profile likelihood maximizes over it; Bayesian model evidence integrates over it using a proper prior distribution. These operations differ and need not give the same inference. A baseline hazard in a Cox proportional-hazards model is an infinite-dimensional example.

An estimator is a function of observed data used to estimate a parameter or a specified function of parameters. Its definition cannot involve the unknown parameter. Its bias, variance, and expected loss describe its performance under repeated sampling.