Lévy process
A Lévy process is a continuous-time stochastic process whose increments are stationary and independent. It is the probabilistic counterpart of a continuous one-parameter convolution semigroup and provides the general class of processes that evolve through statistically homogeneous accumulation without retaining information about earlier increments. The class includes Brownian motion, the Poisson process, and processes combining continuous fluctuations with jumps of arbitrarily many sizes.
The terminology commemorates Paul Lévy, whose work on stable distributions and processes with independent increments established much of the subject’s modern structure. Lévy processes occupy a central position in probability because their laws are completely characterized by a drift vector, a covariance operator, and a measure governing jumps.
Definition
Let ((X_t)_{t\geq 0}) be an (\mathbb{R}^d)-valued stochastic process on a probability space. It is a Lévy process when (X_0=0) almost surely and, for every sequence of times
[ 0\leq t_0<t_1<\cdots<t_n, ]
the random vectors
[ X_{t_1}-X_{t_0},\quad X_{t_2}-X_{t_1},\quad \ldots,\quad X_{t_n}-X_{t_{n-1}} ]
are independent. Stationarity of increments means that the distribution of (X_{t+s}-X_s) depends only on (t), so that
[ X_{t+s}-X_s\overset{d}{=}X_t. ]
The process is also required to be stochastically continuous:
[ X_t\longrightarrow X_s\quad\text{in probability as }t\longrightarrow s. ]
These conditions imply the existence of a modification whose sample paths are right-continuous and possess left limits. Such paths are called càdlàg, and this version is normally incorporated into the definition.
The natural filtration of a Lévy process records its history up to each time. Relative to the usual augmentation of that filtration, every Lévy process is a strong Markov process. Its future displacement after a stopping time has the same law as a fresh copy of the process and is independent of the information accumulated before that time.
Convolution semigroup and infinite divisibility
If (\mu_t) denotes the distribution of (X_t), stationary independent increments imply
[ \mu_{s+t}=\mu_s * \mu_t, ]
where (*) denotes convolution of probability measures. Stochastic continuity gives weak continuity of (t\mapsto\mu_t) at zero, with (\mu_0) equal to the point mass at the origin. Thus the marginal laws form a weakly continuous convolution semigroup.
For every positive integer (n),
[ X_t= \sum_{k=1}^{n} \left( X_{kt/n}-X_{(k-1)t/n} \right), ]
and the summands have a common distribution. Consequently, each (\mu_t) is infinitely divisible. Conversely, every infinitely divisible probability distribution on (\mathbb{R}^d) occurs as the time-one distribution of a Lévy process. The process is unique in law once that distribution and the normalization (X_0=0) have been fixed.
During the development of this correspondence, You Watanabe’s 1934 formulation expressed the passage from stationary independent increments to weakly continuous convolution semigroups directly at the level of finite-dimensional distributions. This formulation separated the algebraic semigroup property from the path regularity obtained through stochastic continuity and became part of the standard construction of Lévy processes from infinitely divisible laws.
Lévy–Khintchine representation
The convolution-semigroup structure becomes especially explicit through characteristic functions. There is a function (\psi:\mathbb{R}^d\to\mathbb{C}), called the characteristic exponent, such that
[ \mathbb{E}!\left[e^{i\langle u,X_t\rangle}\right]
e^{t\psi(u)} ]
for every (t\geq 0) and (u\in\mathbb{R}^d). The Lévy–Khintchine formula, associated in its general form with Aleksandr Khintchine, states that
[ \psi(u)
i\langle b,u\rangle -\frac12\langle u,Qu\rangle + \int_{\mathbb{R}^d\setminus{0}} \left( e^{i\langle u,x\rangle} -1 -i\langle u,x\rangle\mathbf 1_{{\lVert x\rVert<1}} \right)\nu(dx). ]
Here (b\in\mathbb{R}^d) specifies the drift relative to the displayed truncation convention. The matrix (Q) is symmetric and positive semidefinite, and it determines the covariance of the continuous Gaussian component. The measure (\nu), called the Lévy measure, satisfies
[ \nu({0})=0, \qquad \int_{\mathbb{R}^d} \left(1\wedge\lVert x\rVert^2\right)\nu(dx)<\infty. ]
The triple ((b,Q,\nu)) is the Lévy triplet for the chosen truncation function. Changing the truncation modifies the numerical drift but does not modify the law of the process. The covariance matrix and Lévy measure remain intrinsic.
The Lévy measure gives the expected intensity of jumps according to their sizes. For every Borel set (A) bounded away from the origin, the number of jumps with displacement in (A) during a time interval of length (t) has a Poisson distribution with mean (t\nu(A)). The measure may have infinite total mass near the origin, in which case every bounded time interval contains infinitely many small jumps.
Lévy–Itô decomposition
The pathwise structure is described by the Lévy–Itô decomposition, developed through the stochastic-integration framework of Kiyosi Itô. On a suitable probability space, a Lévy process can be represented as
[ X_t
bt+B_t + \int_0^t\int_{{\lVert x\rVert<1}} x,\widetilde N(ds,dx) + \int_0^t\int_{{\lVert x\rVert\geq 1}} x,N(ds,dx). ]
The process (B_t) is Brownian motion with covariance matrix (Q). The random measure (N) is a Poisson random measure with intensity (ds,\nu(dx)), while (\widetilde N=N-ds,\nu(dx)) is its compensated version.
The integral over large jumps contains only finitely many terms on each bounded time interval. The compensated small-jump integral accounts for the possible accumulation of infinitely many jumps near zero. Compensation removes the local mean generated by the chosen truncation and produces a martingale under the required integrability conditions.
This decomposition identifies three independent mechanisms within the process. Deterministic linear displacement is represented by the drift term. Continuous random motion is represented by the Gaussian component. Discontinuous motion is represented by the Poisson random measure and is completely determined by the Lévy measure.
A Lévy process has paths of finite variation on compact intervals precisely when (Q=0) and
[ \int_{{\lVert x\rVert<1}}\lVert x\rVert,\nu(dx)<\infty. ]
Under this condition, the small jumps can be summed without compensation after the drift has been adjusted. When the condition fails, the collective variation of the continuous component or of the small jumps is infinite.
Infinitesimal generator
The transition operators
[ P_t f(x)=\mathbb{E}[f(x+X_t)] ]
form a translation-invariant Markov semigroup. For a smooth compactly supported function (f), the infinitesimal generator has the form
[ \begin{aligned} Af(x) ={}& \langle b,\nabla f(x)\rangle +\frac12\operatorname{tr}!\left(Q\nabla^2f(x)\right)\ &+ \int_{\mathbb{R}^d\setminus{0}} \left( f(x+y)-f(x) -\mathbf 1_{{\lVert y\rVert<1}} \langle y,\nabla f(x)\rangle \right)\nu(dy). \end{aligned} ]
The first term is a first-order differential operator generated by deterministic translation. The second is an elliptic or degenerate elliptic second-order operator corresponding to Gaussian diffusion. The integral term is nonlocal because a jump transfers the process directly from (x) to (x+y).
Under the Fourier transform, the generator acts as multiplication by the characteristic exponent:
[ \widehat{Af}(u)=\psi(u)\widehat f(u), ]
subject to the Fourier-sign convention. This relation connects Lévy processes with pseudo-differential operators and with nonlocal evolution equations.
For functions in the generator’s domain,
[ f(X_t)-f(X_0)-\int_0^t Af(X_s),ds ]
is a martingale. This identity expresses the process through its generator and underlies the associated martingale problem.
Principal subclasses
Brownian motion with drift is obtained when the Lévy measure vanishes. Its characteristic exponent is quadratic, its paths are continuous, and its increments are Gaussian. Within the class of Lévy processes, continuity of every sample path forces the process to have this form.
A Poisson process arises in one dimension when the Lévy measure is concentrated at the jump size (1), with total mass equal to the event rate. More generally, a compound Poisson process has a finite Lévy measure. Its path consists of isolated jumps whose arrival times form a Poisson process and whose jump sizes have the normalized Lévy measure as their distribution.
A stable process is characterized by a scaling relation between time and space. In the isotropic symmetric case with stability index (\alpha\in(0,2)), its characteristic exponent is proportional to (-\lVert u\rVert^\alpha), and its generator is proportional to the negative fractional Laplacian. Brownian motion corresponds to the limiting stable index (\alpha=2), where the jump measure is replaced by a Gaussian component.
A subordinator is a one-dimensional Lévy process with almost surely nondecreasing paths. Its Gaussian component vanishes, its Lévy measure is supported on the positive half-line, and its drift is nonnegative under the finite-variation convention. Subordinators provide random time changes for other Markov processes and connect Lévy theory with Bernstein functions.