Lévy Colored Noise

• Lévy colored noise is a non-Gaussian random field that embeds correlation into Lévy white noise using spatial convolution kernels or temporal processes.
• It leverages methods like Fourier transforms, tempered spectral measures, and convolution operators to manage heavy-tailed jump statistics and infinite divisibility.
• The framework finds applications in SPDEs and fluctuation theory, unifying Gaussian colored noise limits with systems exhibiting memory and nonlocal jumps.

Lévy colored noise denotes a class of non-Gaussian random fields obtained by introducing correlation structure into Lévy white noise. In one line of work, the coloring is spatial: the noise is white in time, spatially homogeneous, and correlated in space through a deterministic kernel or spectral measure. In another line of work, the coloring is temporal: the driving signal is an Ornstein–Uhlenbeck–Lévy process with finite correlation time. These usages share a common core—infinitely divisible jump statistics, compensated Poisson random measures, and heavy-tailed or stable laws—but they differ in whether correlation is imposed in space or in time (Balan, 2013, Srokowski, 2010).

1. Construction from Lévy white noise

A standard SPDE construction starts from a space-time Lévy white noise $L$ on $\mathbb{R}_+\times\mathbb{R}^d$, built from a compensated Poisson random measure with Lévy measure $v$ and finite second moment

$\int_{\mathbb{R}} z^2\,v(dz)<\infty.$

For bounded Borel $B\subset \mathbb{R}_+\times\mathbb{R}^d$,

$$L(B)=\int_{B\times(\mathbb{R}\setminus\{0\})} z\,\tilde N(dt,dx,dz),$$

and, for $\varphi\in L^2(\mathbb{R}_+\times\mathbb{R}^d)$,

$$E|L(\varphi)|^2=v_0\int_0^\infty\int_{\mathbb{R}^d}|\varphi(t,x)|^2\,dx\,dt, \qquad v_0=\int_{\mathbb{R}} z^2\,v(dz).$$

Balan’s construction of Lévy colored noise introduces a tempered spectral measure

$\mu(d\xi)=(2\pi)^{-d}|h(\xi)|^2\,d\xi$

and defines

$X_t(\varphi)=L_t(\mathcal{F}\varphi\cdot h).$

Its covariance is

$\mathbb{R}_+\times\mathbb{R}^d$0

so the field is white in time and colored in space (Balan, 2013).

In the one-dimensional formulation used for recent fluctuation theory, the same object is written through a spatial convolution kernel $\mathbb{R}_+\times\mathbb{R}^d$1: $\mathbb{R}_+\times\mathbb{R}^d$2 Here the coloration is purely spatial: time has independent increments, while space is correlated through $\mathbb{R}_+\times\mathbb{R}^d$3. The covariance of the Gaussian part is

$\mathbb{R}_+\times\mathbb{R}^d$4

with

$\mathbb{R}_+\times\mathbb{R}^d$5

This is identical in form to Dalang’s Gaussian colored noise, with the variance constant $\mathbb{R}_+\times\mathbb{R}^d$6 coming from the Lévy measure (Balan et al., 26 Feb 2026).

2. Temporally correlated Lévy noise

A different, older usage of the same expression appears in linear dynamical systems, where “colored” refers to time rather than space. The basic model is the Ornstein–Uhlenbeck–Lévy process

$\mathbb{R}_+\times\mathbb{R}^d$7

where $\mathbb{R}_+\times\mathbb{R}^d$8 is a symmetric $\mathbb{R}_+\times\mathbb{R}^d$9-stable Lévy process with characteristic function

$v$0

The solution

$v$1

has a finite relaxation time $v$2, so it is colored in time. Its one-point distribution remains $v$3-stable, with characteristic function

$v$4

The corresponding fractional Fokker–Planck equation is

$v$5

For $v$6, the autocorrelation is exponential,

$v$7

with Lorentzian spectrum

$v$8

For $v$9, the variance diverges, so the usual autocorrelation is not well defined; nevertheless, the normalized spectrum keeps the Lorentzian shape, and truncated covariance calculations decay as $\int_{\mathbb{R}} z^2\,v(dz)<\infty.$0. In this literature, Lévy colored noise therefore means heavy-tailed forcing with finite memory time rather than spatially colored martingale noise (Srokowski, 2010).

3. Generalized-function and white-noise frameworks

The distributional foundation of Lévy colored noise is supplied by the theory of Lévy white noise as a generalized random field. Dalang and Humeau define the Lévy white noise $\int_{\mathbb{R}} z^2\,v(dz)<\infty.$1 as the distributional derivative of a Lévy process or Lévy field and prove a sharp criterion for when $\int_{\mathbb{R}} z^2\,v(dz)<\infty.$2 belongs almost surely to the space of tempered distributions: this holds if and only if the underlying Lévy measure has a positive absolute moment,

$\int_{\mathbb{R}} z^2\,v(dz)<\infty.$3

If no such moment exists, then

$\int_{\mathbb{R}} z^2\,v(dz)<\infty.$4

Under this condition, colored noise can be defined by any continuous linear operator $\int_{\mathbb{R}} z^2\,v(dz)<\infty.$5,

$\int_{\mathbb{R}} z^2\,v(dz)<\infty.$6

which covers convolution kernels, Fourier multipliers, Green operators, and fractional operators (Dalang et al., 2015).

Fageot, Humeau, and MacKinlay extend Lévy white noise from a generalized random process to an independently scattered random measure and characterize its domain of definition through the Rajput–Rosiński exponent $\int_{\mathbb{R}} z^2\,v(dz)<\infty.$7. For measurable $\int_{\mathbb{R}} z^2\,v(dz)<\infty.$8,

$\int_{\mathbb{R}} z^2\,v(dz)<\infty.$9

This yields explicit admissibility criteria for kernels used to filter white noise into colored noise and provides concrete domains such as $B\subset \mathbb{R}_+\times\mathbb{R}^d$0 for symmetric $B\subset \mathbb{R}_+\times\mathbb{R}^d$1-stable noise and $B\subset \mathbb{R}_+\times\mathbb{R}^d$2 or $B\subset \mathbb{R}_+\times\mathbb{R}^d$3 for other Lévy families (Fageot et al., 2017).

A complementary approach is provided by the multiparameter Lévy sheet theory of Hida type. In that setting, a pure jump Lévy sheet on $B\subset \mathbb{R}_+\times\mathbb{R}^d$4 is written

$B\subset \mathbb{R}_+\times\mathbb{R}^d$5

and its white noise is the generalized derivative

$B\subset \mathbb{R}_+\times\mathbb{R}^d$6

The theory constructs Lévy chaos expansions, stochastic test function spaces $B\subset \mathbb{R}_+\times\mathbb{R}^d$7, stochastic distribution spaces $B\subset \mathbb{R}_+\times\mathbb{R}^d$8, and the white noise of the compensated Poisson random measure

$B\subset \mathbb{R}_+\times\mathbb{R}^d$9

This furnishes a full white-noise calculus for linear filtering and for stochastic convolutions that produce Lévy colored fields (Draouil et al., 29 Oct 2025).

4. Stochastic integration and SPDE realizations

The SPDE theory of Lévy colored noise follows the Gaussian pattern but with Poissonian rather than Wiener calculus. Balan constructs the stochastic integral first for smooth elementary processes and then extends it by closure in the norm

$$L(B)=\int_{B\times(\mathbb{R}\setminus\{0\})} z\,\tilde N(dt,dx,dz),$$0

For a predictable $$L(B)=\int_{B\times(\mathbb{R}\setminus\{0\})} z\,\tilde N(dt,dx,dz),$$1-valued process $$L(B)=\int_{B\times(\mathbb{R}\setminus\{0\})} z\,\tilde N(dt,dx,dz),$$2 whose Fourier transform is a function and satisfies

$$L(B)=\int_{B\times(\mathbb{R}\setminus\{0\})} z\,\tilde N(dt,dx,dz),$$3

the integral

$$L(B)=\int_{B\times(\mathbb{R}\setminus\{0\})} z\,\tilde N(dt,dx,dz),$$4

is a càdlàg square-integrable martingale with predictable variation

$$L(B)=\int_{B\times(\mathbb{R}\setminus\{0\})} z\,\tilde N(dt,dx,dz),$$5

For linear stochastic heat and wave equations, existence of a random field solution is equivalent to

$$L(B)=\int_{B\times(\mathbb{R}\setminus\{0\})} z\,\tilde N(dt,dx,dz),$$6

and, in the heat and wave examples, this reduces to the Dalang-type condition

$$L(B)=\int_{B\times(\mathbb{R}\setminus\{0\})} z\,\tilde N(dt,dx,dz),$$7

(Balan, 2013).

A recent one-dimensional application is the hyperbolic Anderson model

$$L(B)=\int_{B\times(\mathbb{R}\setminus\{0\})} z\,\tilde N(dt,dx,dz),$$8

with mild solution

$$L(B)=\int_{B\times(\mathbb{R}\setminus\{0\})} z\,\tilde N(dt,dx,dz),$$9

Under Assumption A on the coloration kernel and $\varphi\in L^2(\mathbb{R}_+\times\mathbb{R}^d)$0, there exists a unique mild solution. Under the moment condition $\varphi\in L^2(\mathbb{R}_+\times\mathbb{R}^d)$1 and

$\varphi\in L^2(\mathbb{R}_+\times\mathbb{R}^d)$2

the solution satisfies uniform $\varphi\in L^2(\mathbb{R}_+\times\mathbb{R}^d)$3 bounds, and for each fixed $\varphi\in L^2(\mathbb{R}_+\times\mathbb{R}^d)$4 the field $\varphi\in L^2(\mathbb{R}_+\times\mathbb{R}^d)$5 is strictly stationary in space (Balan et al., 26 Feb 2026).

5. Fluctuation theory and large-scale limits

For the hyperbolic Anderson model, the central object is the spatial integral

$\varphi\in L^2(\mathbb{R}_+\times\mathbb{R}^d)$6

Under Assumption B, two kernel classes are distinguished. If $\varphi\in L^2(\mathbb{R}_+\times\mathbb{R}^d)$7, then

$\varphi\in L^2(\mathbb{R}_+\times\mathbb{R}^d)$8

If

$\varphi\in L^2(\mathbb{R}_+\times\mathbb{R}^d)$9

then

$$E|L(\varphi)|^2=v_0\int_0^\infty\int_{\mathbb{R}^d}|\varphi(t,x)|^2\,dx\,dt, \qquad v_0=\int_{\mathbb{R}} z^2\,v(dz).$$0

The limiting covariance theorem states that

$$E|L(\varphi)|^2=v_0\int_0^\infty\int_{\mathbb{R}^d}|\varphi(t,x)|^2\,dx\,dt, \qquad v_0=\int_{\mathbb{R}} z^2\,v(dz).$$1

and in particular

$$E|L(\varphi)|^2=v_0\int_0^\infty\int_{\mathbb{R}^d}|\varphi(t,x)|^2\,dx\,dt, \qquad v_0=\int_{\mathbb{R}} z^2\,v(dz).$$2

The covariance structure coincides with that of the solution to the same SPDE driven by a Gaussian colored noise $$E|L(\varphi)|^2=v_0\int_0^\infty\int_{\mathbb{R}^d}|\varphi(t,x)|^2\,dx\,dt, \qquad v_0=\int_{\mathbb{R}} z^2\,v(dz).$$3 with the same spatial covariance kernel $$E|L(\varphi)|^2=v_0\int_0^\infty\int_{\mathbb{R}^d}|\varphi(t,x)|^2\,dx\,dt, \qquad v_0=\int_{\mathbb{R}} z^2\,v(dz).$$4, so the second-order scaling is the same as in the Gaussian colored case (Balan et al., 26 Feb 2026).

The same paper proves a quantitative central limit theorem. If there exists $$E|L(\varphi)|^2=v_0\int_0^\infty\int_{\mathbb{R}^d}|\varphi(t,x)|^2\,dx\,dt, \qquad v_0=\int_{\mathbb{R}} z^2\,v(dz).$$5 such that

$$E|L(\varphi)|^2=v_0\int_0^\infty\int_{\mathbb{R}^d}|\varphi(t,x)|^2\,dx\,dt, \qquad v_0=\int_{\mathbb{R}} z^2\,v(dz).$$6

then for fixed $$E|L(\varphi)|^2=v_0\int_0^\infty\int_{\mathbb{R}^d}|\varphi(t,x)|^2\,dx\,dt, \qquad v_0=\int_{\mathbb{R}} z^2\,v(dz).$$7,

$$E|L(\varphi)|^2=v_0\int_0^\infty\int_{\mathbb{R}^d}|\varphi(t,x)|^2\,dx\,dt, \qquad v_0=\int_{\mathbb{R}} z^2\,v(dz).$$8

For $$E|L(\varphi)|^2=v_0\int_0^\infty\int_{\mathbb{R}^d}|\varphi(t,x)|^2\,dx\,dt, \qquad v_0=\int_{\mathbb{R}} z^2\,v(dz).$$9,

$\mu(d\xi)=(2\pi)^{-d}|h(\xi)|^2\,d\xi$0

with $\mu(d\xi)=(2\pi)^{-d}|h(\xi)|^2\,d\xi$1 and $\mu(d\xi)=(2\pi)^{-d}|h(\xi)|^2\,d\xi$2 equal to the Fortet–Mourier, $\mu(d\xi)=(2\pi)^{-d}|h(\xi)|^2\,d\xi$3-Wasserstein, or Kolmogorov distance. For $\mu(d\xi)=(2\pi)^{-d}|h(\xi)|^2\,d\xi$4, $\mu(d\xi)=(2\pi)^{-d}|h(\xi)|^2\,d\xi$5, and $\mu(d\xi)=(2\pi)^{-d}|h(\xi)|^2\,d\xi$6,

$\mu(d\xi)=(2\pi)^{-d}|h(\xi)|^2\,d\xi$7

for any

$\mu(d\xi)=(2\pi)^{-d}|h(\xi)|^2\,d\xi$8

A functional CLT also holds: $\mu(d\xi)=(2\pi)^{-d}|h(\xi)|^2\,d\xi$9 with zero-mean Gaussian limit $X_t(\varphi)=L_t(\mathcal{F}\varphi\cdot h).$0 and covariance $X_t(\varphi)=L_t(\mathcal{F}\varphi\cdot h).$1. The field is ergodic in space, and

$X_t(\varphi)=L_t(\mathcal{F}\varphi\cdot h).$2

An almost sure central limit theorem for the normalized spatial integral was then established under the same two kernel classes, using the CLT and a Malliavin derivative estimate for the solution (Balan et al., 26 Feb 2026, Balan et al., 27 Feb 2026).

6. White-noise baselines, terminology, and scope

A persistent source of confusion is that several influential papers concern Lévy-driven dynamics that are directly relevant to colored-noise intuition but do not study colored noise in a strict sense. For example, the bistable system analyzed in “Stationary States in Bistable System Driven by Lévy Noise” is driven by heavy-tailed white symmetric Lévy noise. Its central result is that, in contrast with the Gaussian case, the maxima of the stationary PDF do not coincide with the minima of the bistable potential; for $X_t(\varphi)=L_t(\mathcal{F}\varphi\cdot h).$3 an explicit stationary density is obtained, and for general $X_t(\varphi)=L_t(\mathcal{F}\varphi\cdot h).$4 the system is treated by Langevin simulation and a space-fractional kinetic equation. This is a white-noise result, not a colored-noise construction (Sliusarenko et al., 2012).

The same is true for multiplicative Lévy-noise problems and simple confining potentials. In Srokowski’s nonlinear Langevin model, the white Lévy interpretation determines whether ordinary rules of calculus are valid, whether stationary algebraic tails occur, and whether the variance can be finite in the Stratonovich interpretation. In the V-shaped potential model $X_t(\varphi)=L_t(\mathcal{F}\varphi\cdot h).$5, the noise is again white in time; the stationary density is expressed through the Mittag–Leffler function, is normalizable for $X_t(\varphi)=L_t(\mathcal{F}\varphi\cdot h).$6, and ceases to be normalizable in the Cauchy-like limit $X_t(\varphi)=L_t(\mathcal{F}\varphi\cdot h).$7 (Srokowski, 2010, MacKay, 2024).

These white-noise baselines are not extraneous to Lévy colored noise. They supply the limiting cases that spatially colored SPDEs recover when the coloring kernel approaches a delta profile, and that temporally colored models recover when the correlation time tends to zero. This suggests that heavy tails, nonlocal jumps, and non-Boltzmann stationary structures are structural features of Lévy forcing that coloring modifies rather than removes.