Symmetric Lévy Basis in SPDE Analysis

• Symmetric Lévy basis is a stochastic measure characterized by pure-jump, symmetric, and heavy-tailed behavior without Gaussian components, constructed via Poisson random measures.
• It underpins rigorous SPDE analysis by enabling both mild and generalized solutions through measurable integration, stable chaos expansions, and specific integrability criteria.
• Key applications include modeling infinite-variance dynamics in heat and wave equations, advancing research in non-Gaussian stochastic calculus and multiplicative SPDE models.

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 (Dalang et al., 2018, Balan et al., 18 Sep 2024).

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 $\tilde J(ds,dz) = J(ds,dz) - ds\, v(dz)$ (Dalang et al., 2018).

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 (Dalang et al., 2018).

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 (Balan et al., 18 Sep 2024): $$Z(B) = \sum_{i=1}^\infty \varepsilon_i \Gamma_i^{-1/\alpha} \psi(T_i, X_i)^{-1} 1_B(T_i, X_i),$$ 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 (Balan et al., 18 Sep 2024).

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 $\alpha > d/(d-2)$ (Dalang et al., 2018, Balan et al., 18 Sep 2024).

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 (Balan et al., 18 Sep 2024). This architecture enables path regularity and intermittency studies for heavy-tailed SPDEs, highlighting an infinite-variance extension of the classical chaos approach.