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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.

Brownian motion in ${\mathbb{R}}^d$ is transient for $d\ge 3$: its distance from the origin tends to infinity almost surely. In dimension three, the positive local martingale $|B_t{|}^{-1}$ and Brownian scaling give a short proof.

It is a dimension-dependent long-time property of Brownian motion.

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 exactly when its quadratic variation satisfies $[M{]}_t=t$.

The characterization asserts that a continuous local martingale starting at zero with quadratic variation $t$ is Brownian motion.

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.

An unbounded function has no finite bound on the moduli of its values.

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

A mixed boundary condition imposes different kinds of boundary conditions on different portions of the boundary. A common elliptic example combines a Dirichlet boundary condition on one portion with a Neumann boundary condition on the remainder.

A Neumann boundary condition prescribes the outward normal derivative of the unknown function on the boundary.

A Dirichlet boundary condition prescribes the value of the unknown function on the boundary.

An ideal energy measurement returns an eigenvalue of the Hamiltonian with probability equal to the squared norm of the state's projection onto its eigenspace. After obtaining a nondegenerate energy, the state is the corresponding energy eigenstate.

A Boolean function $f:\{0,1{\}}^n\to \{0,1\}$ is $(\epsilon ,p,r)$-quasirandom when conditioning any set of at most $r$ coordinates to any values changes its ${\mu }_p$-expectation by at most $\epsilon$.

A $J$-junta is a function whose value depends only on coordinates indexed by $J$. A $k$-junta depends on at most $k$ coordinates.