Fractional Stochastic Heat Equations

• Fractional stochastic heat equations are SPDEs characterized by a fractional Laplacian that models nonlocal anomalous diffusion with power-law decay.
• They integrate heavy-tailed Lévy noise to capture random fluctuations and long-memory effects, essential for understanding super- and subdiffusive behaviors.
• The framework rigorously quantifies moment growth via Lyapunov exponents and exponential indices, highlighting intermittent high peaks and spatial propagation.

Fractional stochastic heat equations are stochastic partial differential equations (SPDEs) in which anomalous, nonlocal diffusion—expressed via the fractional Laplacian or other nonlocal operators—interacts with stochastic forcing such as space-time white noise, Lévy noise, or spatially/temporally correlated Gaussian noise. These equations offer a rigorous mathematical framework for modeling physical, biological, or financial systems exhibiting superdiffusive or subdiffusive behavior and random fluctuations with heavy tails or long memory. The theory covers existence and uniqueness of mild solutions, moment growth (including Lyapunov exponents and intermittency), spatial/temporal regularity, integral tests for extremes, and statistical inference for model parameters.

1. Formulation and Operators

The prototypical form of a fractional stochastic heat equation is

$$\frac{\partial X}{\partial t}(t,x) = {\cal L}_{\alpha} X(t,x) + \sigma(X(t,x))\dot{\Lambda}(t,x),$$

where ${\cal L}_\alpha = -(-\Delta)^{\alpha/2}$ is the fractional Laplacian in $\mathbb{R}^d$ for $\alpha \in (0,2)$, and $\dot{\Lambda}(t,x)$ represents a space-time Lévy white noise. The operator ${\cal L}_\alpha$ is the infinitesimal generator of an isotropic rotationally symmetric $\alpha$-stable process, and is nonlocal: its Green kernel decays polynomially,

$$q_t(x)\asymp \frac{t}{(t^{2/\alpha}+|x|^2)^{(d+\alpha)/2}},$$

demonstrating slower (algebraic) spatial decay and stronger long-range interactions compared to the rapid exponential decay in the Gaussian ($\alpha = 2$) case. The choice of $\alpha$ tunes the strength of the nonlocality and the spatial tails of the process, which in turn impacts moment estimates and intermittency.

2. Lévy White Noise: Structure and Impact

The driving noise $\dot{\Lambda}(t,x)$ is space-time Lévy white noise, specified via a Lévy–Itô decomposition as a sum of small jump (compensated) and large jump (Poissonian) terms,

$$\Lambda(t,x)= \rho\,W(t,x) + \left(\int_{|z|<1} z\, (\mu-\nu)(t,x,dz) + \int_{|z|\ge1} z\,\mu(t,x,dz)\right),$$

where $W(t,x)$ is a Gaussian term, and $\mu$ is a Poisson random measure with compensator $\nu$. In most of the regime analyzed, one sets $\rho = 0$ to emphasize the pure jump (non-Gaussian) mechanism. Unlike Gaussian noise, Lévy noise typically features heavy tails and infinite variance, so higher moments may not exist. The resultant long-range space-time dependencies and the absence of Gaussian isometries require fundamentally different analytic tools, including new inequalities for Poisson integrals.

3. Lyapunov Exponents and Exponential Growth Indices

The temporal and spatial growth of $p$th moments of the mild solution $X(t,x)$ is measured by Lyapunov exponents and exponential growth indices: $$\overline{\gamma}(p) = \limsup_{t\to\infty} \frac{1}{t}\,\sup_{x\in\mathbb{R}^d}\log \mathbb{E}|X(t,x)|^p, \qquad \underline{\gamma}(p) = \liminf_{t\to\infty} \frac{1}{t}\,\inf_{x\in\mathbb{R}^d}\log \mathbb{E}|X(t,x)|^p,$$ which quantify the long-time exponential growth rate of the $p$th moments. For spatial propagation of peaks, the exponential growth indices are defined as

$$\overline{\lambda}(p) = \inf\left\{\eta>0: \limsup_{t\to\infty} \frac{1}{t} \sup_{|x|\geq e^{\eta t}} \log \mathbb{E}|X(t,x)|^p < 0 \right\},$$

$$\underline{\lambda}(p) = \sup\left\{\eta>0: \limsup_{t\to\infty} \frac{1}{t} \sup_{|x|\geq e^{\eta t}} \log \mathbb{E}|X(t,x)|^p > 0 \right\},$$

measuring the rate at which the spatial intermittency front propagates. The presence of polynomial kernel decay necessitates these indices over the classical regime, as exponential kernel weights cannot control the polynomial spatial tails.

4. Intermittency, Moment Growth, and High Peaks

A central phenomenon in these SPDEs is weak intermittency, where solutions develop randomly located but extremely high peaks, even if lower moments remain finite. The haLLMark is positive Lyapunov exponents for some range $p\in (1,1+\alpha)$: $$0 < \underline{\gamma}(p) \leq \overline{\gamma}(p) < \infty,$$ and exponential moment growth away from the origin, but with spatial decay governed by the fractional kernel,

$$q_t(x) \asymp |x|^{-(d+\alpha)} \quad \text{for large } |x|.$$

Unlike the Gaussian setting (with exponential spatial decay), the fractional Laplacian induces persistent influence of large jumps. High peaks of the solution propagate outward at exponential rates dictated by the above growth index, reflecting the balance between rare but strong spatial displacements and polynomial spatial averaging.

5. Methodological Innovations: Heavy-Tailed Kernels and Poisson Stochastic Integrals

The polynomial decay of the fractional heat kernel precludes classical Fourier analytic approaches and requires new convolution inequalities adapted to the heavy-tailed kernel: $$q_t(x)\asymp \frac{t}{(t^{2/\alpha}+|x|^2)^{(d+\alpha)/2}},$$ which do not admit exponential weights. The analysis leverages weighted polynomial norms and develops lower-bound inequalities for Poisson stochastic integrals, compensating for the absence of Itô's isometry. These are crucial for establishing existence, uniqueness, and moment bounds. The stochastic integral for the mild solution is represented by a Poisson sum, and the novel lower bounds (see Lemma 3.4 in (Shiozawa et al., 28 Sep 2025)) are essential for sharp estimates.

6. Comparison and Extensions

The results for fractional Laplacian with space-time Lévy noise (Shiozawa et al., 28 Sep 2025) both generalize and differentiate previous work:

• In [CD15-1], the fractional heat equation with Gaussian noise was treated, with growth indices reflecting the heavy-tailed fractional kernel.
• In [CK19], stochastic heat equations with standard Laplacian and Lévy noise were studied, but the kernel was exponentially decaying.

The present setting treats both the nonlocal operator and heavy-tailed noise, requiring spatially polynomial weights and a new analytic framework. The theory entails:

• Existence and uniqueness via suitable Banach spaces adapted to the kernel,
• Quantitative moment bounds and sharp (strictly positive and finite) Lyapunov exponents in the non-Gaussian, heavy-tailed regime,
• Explicit characterizations of how intermittent islands (peaks) form and spatially propagate, governed by the new exponential indices.

7. Summary Table

Quantity	Definition & Role	Features in Fractional Lévy Setting
${\cal L}_\alpha$	$-(-\Delta)^{\alpha/2}$	Nonlocal, heavy-tailed heat kernel
$\dot{\Lambda}$	Lévy space-time white noise	Heavy tail, infinite variance possible
Moment Growth	Lyapunov exponents $\overline\gamma(p), \underline\gamma(p)$	$p \in (1,1+\alpha)$, strictly positive
Growth Indices	$\overline\lambda(p), \underline\lambda(p)$	Exponential rate; polynomial weights
Intermittency	$0 < \underline\gamma(p) \leq \overline\gamma(p)$	Peaks at random, sparse, explosive

8. Concluding Remarks

Fractional stochastic heat equations driven by space-time Lévy white noise display intricate interplay between nonlocal jump-driven diffusion and heavy-tailed stochastic forcing. The main theoretical constructs—Lyapunov exponents and exponential growth indices—capture both temporal explosion of moments (indicative of intermittency) and spatial propagation of rare, large peaks. The proofs leverage kernel-weighted functional spaces and novel inequalities suited to Poissonian stochastic integrals, extending methods beyond the reach of classical Gaussian/Fourier techniques. These results both complement and generalize previous analyses for Gaussian and non-fractional settings, providing a rigorous toolkit for studying the high-dimensional and heavy-tailed stochastic dynamics intrinsic to many modern applied models (Shiozawa et al., 28 Sep 2025).