Brownian motion is often thought of as an infinitesimal random walk. That works nicely when dealing with distributions. However, the 1-dimensional Brownian motion cannot drive (innovate) the 2-dimensional Brownian motion, while for random walks it can happen readily. In general, some combinatorial (discrete-time) constructions have (continuous-time) scaling limits, others have not. The key to the distinction is stability (versus sensitivity), an important idea emerged independently in probability and computer science.
Some processes are, in a sense, (1+)-dimensional Brownian motions. I mean a diffusion process in a 1-dimensional topological space with branching points (say, a graph, or just 3 rays with a common origin). It was conjectured [Barlow, Pitman, Yor 1989] that such a process (Walsh's Brownian motion, spider martingale) cannot be driven by the 1-dimensional Brownian motion (since its splitting multiplicity exceeds 2). The conjecture was proven by me [1997]. Stability was the key idea (in the form of cosiness).
A Brownian motion in a finite-dimensional Lie group corresponds naturally to a Brownian motion in the tangent space to the Lie group at the unit. The same holds for the infinite-dimensional Lie group of smooth diffeomorphisms of Rn (or a manifold): stochastic flows are driven via stochastic differential equations (which is classical). However, singularities (such as splitting, stickiness, coalescence; and maybe turbulence in a future) lead to non-smooth stochastic flows, or Brownian motions in semigroups of non-smooth transformations (of Rn, say).
Such semigroups are sometimes quite elementary and finite-dimensional (even 2-dimensional) but very different from Lie groups (typically they have no one-parameter sub-semigroups). Brownian motions in such semigroups (singular stochastic flows) cannot be driven by classical Brownian motions of any dimension (even infinite). Classical Brownian motions are stable, nonclassical are not.
Brownian motion (or rather its derivative, the white noise) may be roughly thought of as an infinitely divisible collection of independent random variables, or rather, sigma-fields. Every interval on the time axis determines its sigma-field, and disjoint intervals determine independent sigma-fields. The same holds for elementary sets (I mean, finite unions of intervals) on the time axis. Does it hold for non-elementary measurable sets? For classical Brownian motions the positive answer is classical. In full generality, the question was asked in [Feldman 1971]. The negative answer was given by me [1999]. Stability was the key idea, again.
A finite or countable collection of independent random variables corresponds to a product of probability spaces. Its continuous-time counterpart is a continuous product of probability spaces. Corresponding Hilbert spaces L2 form a continuous tensor product of Hilbert spaces, called also a product system. Such objects are investigated by functional analysts, motivated especially by quantum theory. Classical Brownian motions correspond to classical product systems (Fock spaces); these are type I product systems. The theory of nonclassical (type II and III) product systems suffered from lack of rich sources of examples. Nowadays, probability theory provides such a source. Using it, several questions of Arveson are answered. Nonclassical Brownian motions lead to type II product systems. Type III product systems emerge via Vershik's idea of continuous products of measure type spaces.