---
title: Symmetric Lévy Basis in SPDE Analysis
url: https://www.emergentmind.com/topics/symmetric-levy-basis
type: topic
---

# Symmetric Lévy Basis in SPDE Analysis

A symmetric Lévy basis is a fundamental construct in stochastic analysis, explicitly realized as a symmetric pure-jump Lévy space–time white noise. This independently scattered random measure acts on Borel subsets of a Euclidean space, typically representing space-time windows in SPDE modeling. It is defined via a Poisson random measure parameterized by a symmetric Lévy measure, with the absence of Gaussian and drift components ensuring its pure-jump character. In the case of symmetric α-stable noise ($0<\alpha<2$), the Lévy measure takes the form $v(dz) = C_\alpha |z|^{-1-\alpha} dz$, rendering the noise strictly infinite-variance and heavy-tailed. Such bases underpin the stochastic integration theory necessary for the rigorous analysis of SPDEs with jump-driven inputs and have been central in recent developments in multiplicative SPDE models and non-Gaussian stochastic calculus [1809.09999], [2409.12286].

## 1. Construction and Defining Properties

Let $S$ be a Borel subset of $\mathbb{R}^n$, commonly space–time such as $\mathbb{R}_+ \times \mathbb{R}^d$. A symmetric pure-jump Lévy basis $X$ is given as
\[
X(ds) = \int_{|z|<1} z\,\tilde J(ds, dz) + \int_{|z| \geq 1} z\,J(ds, dz),
\]
where $J(ds,dz)$ is a Poisson random measure on $S \times \mathbb{R}$ with intensity $\mu(ds,dz) = ds\, v(dz)$ and $v$ is a symmetric Lévy measure ($v(-B) = v(B)$, $\int (1\wedge z^2) v(dz) < \infty$). The compensated measure is $\tilde J(ds,dz) = J(ds,dz) - ds\, v(dz)$ [1809.09999].

Given a test function $\varphi \in \mathcal{D}(S)$,
\[
X(\varphi) = \int_S \varphi(s) X(ds)
\]
is infinitely divisible with characteristic functional
\[
E[e^{i X(\varphi)}] = \exp\left\{ \int_S \int_\mathbb{R} \left(e^{i z \varphi(s)} - 1 - i z \varphi(s) 1_{|z|<1}\right) v(dz) ds \right\}.
\]
In the symmetric α-stable case ($v(dz) = C_\alpha |z|^{-1-\alpha} dz$), for $A$ measurable and $|A| < \infty$,
\[
E[e^{i u X(A)}] = \exp(-\sigma_\alpha |A| |u|^\alpha).
\]

## 2. Measure Structure, Symmetry, and Integrability

The intensity measure for the location variable is Lebesgue; the Lévy measure $v$ is symmetric and satisfies integrability $\int(1 \wedge z^2)v(dz) < \infty$.

Symmetry ensures $E[X(\varphi)] = 0$ and is essential for isomorphic properties of stochastic integrals, as in Rajput–Rosiński's theory. Integrable deterministic functions $f: S \to \mathbb{R}$ belong to $L(X, S)$ if and only if
\[
\int_S \int_{\mathbb{R}} (|f(s)z|^2 \wedge 1) v(dz) ds < \infty.
\]
In the symmetric α-stable regime, this reduces to $f \in L^\alpha(S)$, i.e.,
\[
\int_S |f(s)|^\alpha ds < \infty.
\]
This classification underpins the distinction between random-field (mild) and generalized (distribution-valued) SPDE solutions.

## 3. Mild and Generalized Solutions to SPDEs

Given a linear SPDE $\mathcal{L} u = X$ with fundamental solution $p$, mild solutions exist given the integrability hypothesis (H2): $p(t-\cdot) \in L(X, S)$ for each $t$, leading to
\[
u_{\mathrm{mild}}(t) = \int_S p(t-s) X(ds).
\]
For symmetric α-stable noise, the criterion becomes $\int_S |p(t-s)|^\alpha ds < \infty$.

Generalized solutions require the weaker hypothesis (H1): for $\varphi \in \mathcal{D}(\mathbb{R}^m)$, $\varphi * p \in L(X, S)$, enabling solution via
\[
(u_{\mathrm{gen}},\varphi) = X(\varphi * p).
\]
Theoretical equivalences and distinctions between these solutions follow from measure and integrability criteria and stochastic Fubini arguments [1809.09999].

## 4. LePage Series Representation and Stable Chaos Expansions

The symmetric α-stable Lévy basis admits almost sure representation via the LePage series as detailed in [2409.12286]:
\[
Z(B) = \sum_{i=1}^\infty \varepsilon_i \Gamma_i^{-1/\alpha} \psi(T_i, X_i)^{-1} 1_B(T_i, X_i),
\]
where $\varepsilon_i$ are i.i.d. Rademacher variables, $\Gamma_i$ are Poisson arrival times, and $(T_i, X_i)$ are i.i.d. locations with density $\psi^\alpha$.

For linear multiplicative SPDEs (Anderson model),
\[
u(t,x) = 1 + \int_0^t \int_{\mathbb{R}^d} G_{t-s}(x-y) u(s,y) Z(ds, dy).
\]
The solution admits a chaos expansion, replacing Gaussian tools with multiple stable integrals:
\[
u(t,x) = 1 + \sum_{n=1}^\infty \int_{([0,t]\times \mathbb{R}^d)^n} f_n(\cdots; t, x) Z(dt_1, dx_1)\cdots Z(dt_n, dx_n),
\]
where $f_n$ is the iterated kernel derived from convolutions of the fundamental solution [2409.12286].

## 5. Existence and Uniqueness Criteria

In infinite-variance settings, analysis proceeds in $L^0$ (convergence in probability). Existence requires kernel integrability:
\[
\int_0^T \int_{\mathbb{R}^d} G_t(x)^\alpha dx dt < \infty.
\]
Further summability assumptions guarantee almost sure convergence of chaos expansions and solution identification. For heat kernels ($G_t(x) \sim t^{-d/2}e^{-|x|^2/(2t)}$), existence holds for $\alpha < 1 + 2/d$; for wave kernels in $d \le 2$, no further restriction on $\alpha$ arises. Uniqueness is established in some cases via light-cone arguments (hyperbolic SPDEs).

## 6. Applications: Classical SPDEs Driven by Symmetric Lévy Bases

Three canonical equations exemplify the theory:

| Equation           | Mild Solution Exists       | Gen./Random-field Solution Exists    |
|--------------------|---------------------------|--------------------------------------|
| Heat ($d$)         | $\alpha < 1+2/d$          | Same                                 |
| Wave ($d=1$)       | All $\alpha < 2$          | Same                                 |
| Wave ($d=2$)       | $\alpha < 2$              | Same                                 |
| Wave ($d \ge 3$)   | None                      | Only generalized                     |
| Poisson ($d$)      | None                      | $d>4$, $\alpha > d/(d-2)$            |

For the heat equation $\partial_t u - \Delta u = X$, the random-field solution demands $\alpha < 1 + 2/d$. The wave equation exhibits mild solutions only for $d \le 2$ and specific $\alpha$. The Poisson equation admits no mild solution for any $0<\alpha<2$ but admits generalized solutions only for $d > 4$ and $\alpha > d/(d-2)$ [1809.09999], [2409.12286].

## 7. Infinite-Variance Chaos vs. Gaussian Chaos

Unlike Wiener chaos expansions applicable to Gaussian noise ($\alpha=2$), symmetric α-stable bases necessitate series in multiple stable integrals (Samorodnitsky–Taqqu), constructed via the LePage representation. $L^2$-based, Hilbert-space, and Malliavin-calculus techniques cannot be directly employed; analysis relies on probabilistic convergence and combinatorial techniques developed for stable chaos [2409.12286]. This architecture enables path regularity and intermittency studies for heavy-tailed SPDEs, highlighting an infinite-variance extension of the classical chaos approach.

Source: https://www.emergentmind.com/topics/symmetric-levy-basis