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For a bounded continuous adapted process $H$ and a Brownian motion $B$, ${\int }_t^{t+h}{H}_{s}d{B}_s/(B_{t+h}-B_t)$ converges in probability to $H_t$ as $h\downarrow 0$. The Itô isometry makes the error numerator small relative to $\sqrt{h}$; the normal distribution of the denominator controls its chance of being too close to zero.

One-dimensional Brownian motion satisfiesalmost surely. Its continuous path therefore visits every point, including zero, infinitely often.

For the unit-rate continuous-time symmetric simple random walk, the processes $n^{-\alpha }{X}_{nt}$ have three finite-dimensional regimes. For $0<\alpha <1/2$ no probability limit is possible, because the one-time characteristic functions tend to zero off the origin and are discontinuous there. At $\alpha =1/2$ the limit is standard Brownian motion. For $\alpha >1/2$ it is the identically zero Lévy process. This follows from $n(1-\cos (n^{-\alpha }\theta ))\sim ({\theta }^2/2)n^{1-2\alpha }$ and independent increments. Thus a nonzero limit uniquely determines the exponent, whereas merely asking for a Lévy limit does not.

Chordal Schramm–Loewner evolution in the complex upper half-plane from $0$ to $\infty$ is the random Loewner chain driven by $U_t=\sqrt{\kappa }{B}_t$, where $B$ is standard Brownian motion and $\kappa \ge 0$.

A sharp-k filter includes every Fourier mode below a cutoff with unit weight and excludes all modes above it. For a Gaussian random field, increasing the cutoff adds independent Fourier shells, so the smoothed density contrast is a Brownian motion when parameterized by its variance. The real-space window is not a localized spherical top-hat, so its halo mass assignment needs a specified convention.

The Skorokhod embedding theorem represents a centered probability distribution $\mu$ on the real numbers with finite variance as $B_T\sim \mu$ for a stopping time $T$ of Brownian motion. One may choose $T$ so that the stopped process $(B_{t\wedge T})$ is uniformly integrable and $\mathbb{E}T=\int x^2\mu (dx)$.

The Skorokhod space consists of real paths that are right-continuous and have a finite left limit at every positive time. Its $J_1$ topology allows small changes of time: $x_n\to x$ when there are increasing homeomorphisms ${\lambda }_n$ of $[0,1]$ such that $\Vert {\lambda }_n-\mathrm{id}{\Vert }_{\infty }\to 0$ and $\Vert x_n\circ {\lambda }_n-x{\Vert }_{\infty }\to 0$, where $\mathrm{id}$ denotes the identity time map. This topology aligns nearby jumps rather than insisting on uniform closeness at identical times. The step-path version of the Donsker invariance principle converges here to continuous Brownian motion; its polygonal version uses the ordinary supremum norm on continuous paths.

The space-time Probabilists' Hermite polynomial $H_n(x,t)=t^{n/2}{\mathrm{He}}_n(x/\sqrt{t})$ extends polynomially to $t=0$ and solves ${\partial }_t{H}_n+\frac{1}{2}{\partial }_x^2{H}_n=0$. Therefore $H_n(B_t,t)$ is a martingale for a standard Brownian motion, by the Itô formula and Gaussian moments.

Stochastic calculus extends integration and differential calculus to stochastic processes such as Brownian motion and semimartingales.

For an unbounded nondecreasing function $f:[0,\infty )\to \mathbb{R}$, its strict generalized inverse is $G(a)=\inf \{t\ge 0:f(t)>a\}$. It is nondecreasing and right-continuous in $a$. If $a_n\downarrow a$, then for any $t>G(a)$ monotonicity gives $f(t)>a$, so eventually $f(t)>a_n$ and $G(a_n)\le t$; also $G(a_n)\ge G(a)$. Local finiteness gives finite left limits. Flat parts of $f$ create jumps of $G$. For the continuous running maximum of Brownian motion, this gives the càdlàg Brownian first-passage subordinator.

Two scalar SDEs driven by the same Brownian motion, with the same globally Lipschitz diffusion coefficient and ordered globally Lipschitz drifts, preserve a strict initial order at every finite time. If $Z=Y-X$ starts positive, apply a reciprocal barrier to the first hit of zero. A Gronwall bound on $(Z+ϵ{)}^{-1}$ makes the hitting probability at most $ϵe^{(L+K^2)t}/(Z_0+ϵ)$, which tends to zero. Common noise and the Lipschitz bound on the diffusion difference are essential to this argument.

A stochastic differential equation has strong existence for specified coefficients and initial data if a strong stochastic solution can be constructed on every prescribed stochastic basis carrying the required initial variable and driving Brownian motion. This existence assertion is distinct from pathwise uniqueness and from a definition of the solution itself.

A strong solution is adapted to the filtration of a prescribed driving Brownian motion on a prescribed probability space and satisfies the stochastic integral equation almost surely.

If $X$ is a Lévy process and $T$ an independent subordinator, then $Y_t=X_{T_t}$ is a Lévy process. Conditional on the clock, increments of $X$ over disjoint clock intervals are independent; averaging over the independent stationary increments of the clock gives the same properties for $Y$. If $\mathbb{E}{e}^{iu{X}_s}=e^{-s\Psi (u)}$, its characteristic function is $\mathbb{E}{e}^{-T_t\Psi (u)}$. For standard Brownian motion, $\Psi (u)=u^2/2$, so this is the clock's Laplace transform at $u^2/2$.

Start traps at a unit-intensity Poisson random measure in Euclidean space and attach mutually independent Brownian motions, independent of the initial measure. For a deterministic target path $f$, independently thin initial positions according to whether their marked path comes within radius one of the target by time $t$. The retention probability $p_t(x)$ integrates, by the Tonelli theorem, to the expected volume of the sausage along $f-B$. Symmetry of Brownian motion makes this the expected volume of the sausage along $f+B$. The Poisson zero-count probability proves the displayed survival formula. This derives the formula directly from independent thinning, without assuming a separate marking or displacement theorem.

With $s(x)=1$ for $x\ge 0$ and $s(x)=-1$ for $x<0$, the equation starting at zero has uniqueness in law but not pathwise uniqueness. For existence, take a Brownian motion $W$ and set $B=\int s(W)dW$; the Lévy characterization of Brownian motion and associativity of stochastic integration give $W=\int s(W)dB$. Every solution has quadratic variation $t$ and hence the Brownian motion law. Both $W$ and $-W$ are solutions driven by the same $B$, since the time spent at zero is zero and $s(-W)=-s(W)$ away from zero. Using $s(0)=0$ instead would allow the constant solution and invalidate this uniqueness-in-law example.

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.

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.