a diffusion process
A diffusion process is a continuous-path strong Markov process whose local motion is described by a drift and a diffusion coefficient — informally, a process that 'moves smoothly but jiggles randomly', with no jumps. Brownian motion is the prototype, and the solution of a well-posed SDE dX = b(X) dt + sigma(X) dB is the canonical example. Diffusions are the continuous-state, continuous-time members of the Markov family, and they are the bridge between probability and second-order partial differential equations.
There are two equivalent ways to characterize a diffusion. Analytically, it is the Markov process whose transition semigroup P_t f(x) = E[f(X_t) given X_0 = x] is generated by a second-order elliptic operator L f = sum_i b_i(x) partial_i f + (1/2) sum_{i,j} a_ij(x) partial_i partial_j f, with a = sigma sigma^T nonnegative; this L is the infinitesimal generator and it is the limit (1/t)(P_t f - f) as t -> 0. Probabilistically, a diffusion is exactly a solution of the SDE (strong or weak) with those coefficients, equivalently a solution of the martingale problem for L: f(X_t) - integral_0^t L f(X_s) ds is a martingale for every smooth compactly supported f. The continuity of paths is built in, and the strong Markov property lets you restart at stopping times. The drift b(x) is the infinitesimal mean velocity and a(x) the infinitesimal covariance rate, as in the drift-and-diffusion-coefficients entry.
Diffusions are the workhorses of stochastic modelling — finance (asset prices, interest rates), physics (Langevin dynamics), biology (population and gene-frequency models), and the probabilistic side of PDE theory. The link to PDEs is the whole point: the generator's parabolic equation (backward Kolmogorov) governs E[f(X_t)], the forward equation (Fokker-Planck) governs the density, and Feynman-Kac solves boundary-value problems by averaging over diffusion paths. Honest cautions: a diffusion is a continuous process by definition (jump processes are NOT diffusions — they are the realm of Levy processes); and the generator is only elliptic, not strictly elliptic, when a = sigma sigma^T can degenerate, in which case the diffusion may be confined to lower-dimensional manifolds and smoothing of the density can fail (Hormander's condition is then what restores hypoellipticity).
The Ornstein-Uhlenbeck process dX = -theta X dt + sigma dB is a diffusion with generator L f = -theta x f'(x) + (sigma^2/2) f''(x). Its transition density is Gaussian for every t, and as t -> infinity it relaxes to its invariant N(0, sigma^2/(2 theta)) distribution — the diffusion's stationary law.
The OU process: a mean-reverting diffusion with Gaussian transitions and a Gaussian invariant law.
A diffusion has continuous paths by definition — jump processes are not diffusions. When a = sigma sigma^T degenerates the generator is only elliptic, not uniformly elliptic, and the density may concentrate on a lower-dimensional set unless Hormander's bracket condition holds.