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If a nondecreasing oscillation quantity satisfiesfor $0<\rho \le R$, with $\gamma >\beta$, then iteration gives $\Phi (\rho )\le C(\rho /R{)}^{\beta }\Phi (R)+CB{\rho }^{\beta }$.

An operator $U$ on a Hilbert space is a partial isometry when it is isometric on $(\ker U{)}^{\perp }$. Equivalently, $U^{\ast }U$ is the orthogonal projection onto this initial space.

An element $a$ of a C-star algebra is positive when $a=b^{\ast }b$ for some $b$, equivalently when $a=a^{\ast }$ and $\sigma (a)\subseteq [0,\infty )$. Continuous functional calculus gives it a unique positive square root $a^{1/2}$.

Let $U,V,X,Y$ take values in the vector space ${\mathbb{F}}_2^n$ over the two-element finite field. In this additive group, subtraction equals addition. Suppose $(U,V)$ is a C-relevant pair of random variables relative to $(X,Y)$, and take mutually independent copies $U_1,U_2,V_1,V_2$. Then $(U_1+U_2,V_1+V_2)$ is $2C$-relevant to $(X,Y)$.

For a proof, let $S=U_1+U_2$ and take an independent representative of $X$. The entropy submodularity for three independent sums givesso the definition of Entropic Ruzsa distance yieldsThe last step is the Entropic Ruzsa triangle inequality. Apply the same argument to $V_1+V_2,Y$ and add the inequalities.^[ref]

The binary finite field hypothesis makes the displayed self-sum argument agree with the difference convention in Entropic Ruzsa distance. On a general abelian group, signs must be retained explicitly.

For distinct accessible subset states, choose a state in their symmetric difference. A word taking it to the unique final state accepts from one subset; uniqueness of the starting state prevents acceptance from the other. Thus the accessible part of the subset automaton is irreducible.

Let $W$ be continuous with $W_0=0$. Ifis a martingale for every real $c$, then conditional expectations solveThus $W_t-W_s$ is independent of ${\mathcal{F}}_s$ and distributed as $N(0,t-s)$, so $W$ is Brownian motion.

For suitable $f$,defines the Brownian transition semigroup. Its generator identity is

Each Gaussian-convolution operator is a rescaled Weierstrass transform; the semigroup combines all times.

For the Cameron-Martin space of Wiener measure path $h(s)=\mu s$, the Cameron-Martin theorem says that translation of Brownian motion by this linear function changes its law through time $t$ by the density

A Brownian motion with drift $\mu$ is $X_t=W_t+\mu t$ for a standard Brownian motion $W$.

If $W$ is Brownian motion, thenis also Brownian motion. Its covariance is $\min (s,t)$, and continuity at zero follows from $W_u/u\to 0$ almost surely.

A centered Gaussian process with almost surely continuous sample paths and covariance $\mathbb{E}[B_s{B}_t]=\min (s,t)$ is standard Brownian motion. Here Gaussian process means that every finite-dimensional distribution is a multivariate normal distribution; having separate normal distributions at each time would not suffice. This characterization is stated in Pitman, Basic Properties of Brownian Motion, Definition 15.2.

To compute the Brownian covariance kernel directly, suppose $0\le s\le t$. Since $B_t=B_s+(B_t-B_s)$ and the added Brownian increment is centered and independent of $B_s$,Symmetry gives $\min (s,t)$ for arbitrary nonnegative times. For instance, $\mathrm{Cov}(B_1,B_4)=1$ and their correlation is $1/\sqrt{1\cdot 4}=1/2$. Positions are therefore related even though disjoint Brownian increments are independent: both positions contain the displacement accumulated during the first unit of time. Conversely, taking differences of $\min (s,t)$ gives zero covariance for disjoint increments; their joint multivariate normal distribution turns this into independence.

For every real $c$,is a martingale. Conditional expectation factors at time $s$ because the Gaussian increment $W_t-W_s$ is independent of the past and has exponential moment $e^{c^2(t-s)/2}$.

A standard Brownian bridge is the centered Gaussian process $B(t)=W(t)-tW(1)$ on $[0,1]$. Its covariance is $\min (s,t)-st$ and it satisfies $B(0)=B(1)=0$.

For a domain $D$, the Brownian exit time is $T_D=\inf \{t\ge 0:B_t\notin D\}$. It is a stopping time, and it has finite expectation when $D$ is bounded.

Standard Brownian motion $B$ in ${\mathbb{R}}^d$ is transient for $d\ge 3$: $|B_t|\to \infty$ almost surely. In dimension three, start away from zero and use the positive local martingale $1/|B_t|$. It is a supermartingale, hence converges to a finite limit. Brownian scaling and the normal distribution of $B_t/\sqrt{t}$ show that $1/|B_t|\to 0$ in probability, so its almost-sure limit must be zero. Thus $|B_t|\to \infty$.

For a start at zero, apply the same argument from any positive deterministic time; the position there is nonzero almost surely. In higher dimensions, project onto three coordinates: the full radius dominates a radius already tending to infinity. This differs from recurrence of planar Brownian motion, where every neighbourhood is visited repeatedly.

For every $c>0$, the stochastic process ${\tilde{B}}_t=c^{-1/2}{B}_{ct}$ is standard Brownian motion. Its Brownian increments remain independent and have variance $c^{-1}c(t-s)=t-s$; continuity and the zero starting value are preserved. Equivalently,Equality here is of whole-process distributions, including every finite-dimensional distribution. It is not an equality of the two expressions along a particular realization. With $c=4$, speeding time up fourfold requires dividing displacement by two to recover the standard law. The figure applies exactly these coordinate changes to four finite samples; the shared colors indicate reused samples, while the dashed lines show pointwise standard deviations, not bounds on entire sample paths.

The same square-root normalization explains how a simple random walk approaches Brownian motion. Let $S_k={\sum }_{j=1}^k{\xi }_j$, where the independent steps take values $+1$ and $-1$ with equal probability. Then $R_n(t)=S_{\lfloor nt\rfloor }/\sqrt{n}$ has variance $\lfloor nt\rfloor /n$. Its linear interpolation isand ${\sup }_{0\le t\le 1}|Y_n(t)-R_n(t)|\le 1/\sqrt{n}$. Donsker's theorem says $Y_n\Rightarrow B$ in $C[0,1]$ with the uniform norm: this is convergence in distribution of random paths. The interpolation bound vanishes, but the theorem does not assert pointwise convergence of an arbitrarily fixed walk to an independently chosen Brownian motion. The usefulness is that many-step discrete motion has a universal continuous limit.^[ref]

A continuous local martingale $M$ with $M_0=0$ is standard Brownian motion in its given filtration if and only if its quadratic variation satisfies $[M{]}_t=t$ for every $t\ge 0$, simultaneously almost surely.

For the nontrivial direction, the Itô formula makes $\exp (i\theta M_t+{\theta }^{2}t/2)$ a complex local martingale. Its absolute value is deterministically bounded on each finite horizon, so it is a true martingale. Conditioning at $s<t$ givesThe conditional characteristic-function criterion for Brownian increments identifies a centered normal distribution independent of ${\mathcal{F}}_s$. This turns a bracket computation into a proof of the complete Brownian motion law.

Brownian paths are almost surely nowhere differentiable. An increment over duration $h$ is typically of order $\sqrt{h}$, so its difference quotient is of order $h^{-1/2}$ rather than approaching a finite limit.

A real- or complex-valued function is unbounded when no finite constant bounds the moduli of all its values. Equivalently, for every $M>0$ there is an allowed input $x$ with $|f(x)|>M$.

Unboundedness does not say that all sufficiently large inputs produce large values. For example, $f(x)=x\sin x$ on the nonnegative real numbers is zero at $x=n\pi$, yet $f(\pi /2+2n\pi )=\pi /2+2n\pi$ grows without bound. Thus this function is unbounded but has no limit $+\infty$ as $x\to \infty$.

A far-field boundary condition prescribes the limiting behavior of a solution as one or more spatial coordinates tend to infinity.