the characteristic exponent
Where a single random variable has a characteristic function, a Lévy process has a characteristic exponent: the single function psi that captures the entire law of the process through time. It is the natural unknown of the theory because, by stationary independent increments, the whole multi-time structure is generated by one exponent acting linearly in time.
If (X_t) is a Lévy process with X_0 = 0, then E[e^(i theta X_t)] = e^(t psi(theta)) for all t >= 0, where psi is the characteristic exponent. The factor t in the exponent is the signature of stationary independent increments: the increment over [0, t] is the sum of n increments over intervals of length t/n, each with exponent (t/n) psi, and the characteristic functions multiply. By Lévy-Khintchine, psi has the canonical form psi(theta) = i b theta - (1/2) sigma^2 theta^2 + integral (e^(i theta x) - 1 - i theta x 1_{|x|<1}) nu(dx). The map t -> psi determines the process in law completely, and psi is also the symbol of the process's generator: the generator acts on smooth f as a pseudo-differential operator whose Fourier multiplier is psi.
The characteristic exponent is the working coordinate for almost every computation: subordination composes exponents, the stable processes are exactly those with psi homogeneous in theta, and moments of X_t (when they exist) come from derivatives of psi at 0. A caveat: e^(t psi) being a characteristic function for all t requires psi to be of Lévy-Khintchine type; an arbitrary continuous psi with psi(0) = 0 need not exponentiate to a valid family — for instance psi(theta) = -|theta|^alpha is admissible only for 0 < alpha <= 2 (the stable range).
For Brownian motion with drift mu and variance sigma^2, psi(theta) = i mu theta - (1/2) sigma^2 theta^2, so E[e^(i theta X_t)] = e^(t(i mu theta - sigma^2 theta^2/2)) = the characteristic function of N(mu t, sigma^2 t), as expected. For the symmetric alpha-stable process psi(theta) = -c |theta|^alpha, giving the self-similar scaling X_t = t^(1/alpha) X_1 in law.
The exponent times t recovers the marginal law at each time.
psi(theta) and the cumulant generating function of X_1 differ only by the imaginary unit: psi(theta) = log E[e^(i theta X_1)]. Do not confuse psi with the Laplace exponent used for subordinators, which is defined on the real argument and has the opposite sign convention.