Linear Stochastic Fractional Heat Equation

Updated 20 November 2025

The topic is defined by replacing the Laplacian with its fractional counterpart and incorporating white or fractional noise to model anomalous diffusion and memory effects.
The methodology involves adapting classical PDE techniques through mild solution theory and spectral summability conditions to ensure well-posedness under non-standard noise.
The framework has significant implications for modeling complex systems in physics, biology, and finance, and guides statistical inference and numerical discretization strategies.

The linear stochastic fractional heat equation (SFHE) generalizes the classical stochastic heat equation by replacing the Laplacian with a fractional Laplacian operator and introducing randomness with white or fractional noise, often acting in both temporal and spatial dimensions. This framework captures anomalous diffusion and memory effects observed in various physical, biological, and financial systems. Rigorous analysis of such equations encompasses mild solution theory, regularity, moment and sample path properties, and parameter estimation.

1. Mathematical Formulation

Let $D\subset\mathbb{R}^d$ be a $C^{2+\beta}$ domain, $T>0$ , and consider a probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq0},\mathbb{P})$ . The prototypical linear stochastic fractional heat equation with both white and fractional noise may be written as

$\begin{aligned} du(x,t) &= [\mathrm{Div}(k(x,t)\nabla u(x,t)) + f(u(x,t))]\,dt \ &\quad+ g(u(x,t))\,dW(x,t) + h(u(x,t))\,dW^H(x,t),\quad (x,t)\in D\times(0,T], \ u(x,0)&=\varphi(x),\; x\in D, \ \partial u/\partial n(k)(x,t) &= 0, \; (x,t)\in\partial D\times(0,T], \end{aligned}$

where $k$ is a symmetric, uniformly elliptic matrix field of regularity $C^{\beta,\beta'}$ , $f,g$ are Lipschitz, $h$ is affine, $W$ is an $L^2(D)$ -valued cylindrical Wiener process, and $W^H$ is an $L^2(D)$ -valued fractional Brownian motion with Hurst parameter $H\in(1/2,1)$ (Mishura et al., 2018).

In unbounded domains and for the space-fractional model, the equation is often formulated as

$\partial_t u(t,x) = -(-\Delta)^{\alpha/2}u(t,x) + \dot{W}(t,x)$

with $\alpha\in(0,2]$ , and $\dot{W}$ as generalized Gaussian noise, which may possess non-trivial spatial correlation structure, e.g., fractional Brownian covariance (Song et al., 2023, Chen et al., 2012, Li et al., 2015, Mishura et al., 2019, Liu et al., 30 Jul 2025).

2. Existence, Uniqueness, and Mild Solution Theory

Inhomogeneous, Non-autonomous Setting

Under global Lipschitz conditions on the coefficients, affine structure of the fractional noise, spectral summability, and regularity/ellipticity of $k(x,t)$ , one constructs a mild solution in $W^{\alpha,2}(0,T;L^2(D))$ for any $\alpha\in(1-H,1/(d+2))$ , provided $H>(d+1)/(d+2)$ and $\varphi$ is compatible with the conormal boundary condition. The mild solution takes the form (Mishura et al., 2018): $\begin{aligned} u(t) &= U(t,0)\varphi + \int_0^t U(t,s)f(u(s))\,ds \ &\quad + \int_0^t U(t,s)g(u(s))\,dW(s) + \int_0^t U(t,s)h(u(s))\,dW^H(s), \end{aligned}$ where $U(t,s)$ is the parabolic evolution operator generated by the deterministic part.

Space-Fractional and Infinite-Dimensional Fractional Noise

For equations with the fractional Laplacian, white or colored in time and space, existence and uniqueness often rest on spectral integrability (Dalang-type conditions) for the noise and appropriate regularity of the initial data. In particular, under the spectral/space-time integrability constraint: $\int_{\mathbb{R}^d} \frac{\mu(d\xi)}{(1+|\xi|^\alpha)} < \infty$ for spatial spectral measure $\mu$ , well-posedness holds for the Walsh-Dalang mild formulation (Assaad et al., 2020, Zhang et al., 2023). For equations on $\mathbb{R}$ with infinite-dimensional fBm input and piecewise-constant coefficients, a pathwise mild solution can be constructed using Riemann-Liouville fractional calculus and fixed-point arguments in Besov-type spaces, whenever $H>1/2$ and structural summability/regularity of the basis is met (Mishura et al., 2019).

Time-Fractional Caputo Derivative

In equations with Caputo time-fractional derivative $D_t^\alpha$ (with $\alpha\in(0,2)$ ) and additive fractional time-space noise, a unique solution in the sense of distributions exists for all $d$ ; mild $L^2$ -solutions exist only for $\alpha\geq 1$ and $d\leq 2$ (Hachemi et al., 24 Feb 2024).

3. Probabilistic Representations and Moment Formulas

For linear and multiplicative models with the fractional Laplacian, fundamental results include:

Feynman–Kac-type representations: For both Stratonovich and Skorohod solutions in multiplicative noise settings, pathwise solutions can be represented as exponential functionals over stable Lévy processes, e.g.,

$u(t,x) = \mathbb{E}^X\left[ f(X_t^x) \exp\left(\int_0^t \int_{\mathbb{R}^d} \delta(X_{t-s}^x - y)W(ds,dy)\right)\right],$

under critical integrability conditions linking dimension, stability index, and Hurst indices (Song et al., 2023, Chen et al., 2012).

Moment formula: For the $p$ -th moment,

$\mathbb{E}[u(t,x)^p] = \mathbb{E}^{X^{(1)},\dots,X^{(p)}}\left[\prod_{j=1}^pf(X_t^{(j),x}) \exp \left(\sum_{j<k} I_{jk} \right)\right],$

where $I_{jk}$ is a singular double integral depending on the noise's covariance structure (Song et al., 2023, Chen et al., 2012, Hu et al., 2015). The form and asymptotics of moment Lyapunov exponents elucidate intermittency phenomena.

4. Regularity, Path Properties, and Scaling

Analyses yield sharp regularity and sample path results:

Hölder regularity: For mild solutions on $\mathbb{R}^d$ with fractional noise, the optimal Hölder exponents in time and space are governed by the kernels' regularity and the precise conditions $2H_0 + \frac{1}{\alpha}\sum_{i=1}^d (2H_i - 2)>1$ for Stratonovich models (Chen et al., 2012), with exponents less than $\kappa/2$ in time and $\kappa$ in space with $\kappa = 2H_0 + \tfrac{1}{\alpha}\sum_{i=1}^d (2H_i-2) -1$ .
Sample path moduli, LIL, and suprema: For spatially rough noise (white in time, fBm in space), the sharp spatial increment behavior and the almost-sure moduli of continuity/supremum growth are captured by parameters such as $\kappa = (2H+\alpha-2)/\alpha$ (Liu et al., 30 Jul 2025, Chang et al., 19 Nov 2025, Tudor et al., 2015). Chung-type laws of the iterated logarithm at $x=0$ and for local/pointwise process increments are available (Chang et al., 19 Nov 2025, Tudor et al., 2015).
Strong local nondeterminism and Slepian models: Solutions exhibit Gaussian field structures with explicit covariance representation, which support the application of majorizing measure and Sudakov's minoration tools for almost-sure laws and modulus of continuity (Liu et al., 30 Jul 2025).

5. Statistical Inference and Fluctuations

Recent developments include statistical and probabilistic analysis of the SFHE:

Parameter estimation: Maximum likelihood estimators (MLE) and power variation estimators for the drift parameter (e.g., in $-\theta(-\Delta)^{\alpha/2}$ ) exhibit strong consistency and CLTs with explicit rates in Wasserstein distance, under observation of finitely many Fourier modes or over time/space grids. Berry–Esséen bounds for central limit theorem rates are established, depending on $(T,N)$ and mesh size/scaling (Douissi et al., 9 Sep 2024, Khalil et al., 2019).
Spatial averaging and central limit theorems: Quantitative normal approximations (with explicit error rates in total variation and density) are proved for spatial averages over large domains, both for additive and multiplicative noise models, covering the parabolic Anderson model and spatially colored noise (Zhang et al., 2023, Assaad et al., 2020).
Moderate deviation principles: Moderate deviation results for centered fluctuations around deterministic centering, at scaling regimes between strong law and large deviations, are available under sufficient regularity on the operator's symbol and the noise's spectral measure (Li et al., 2015).

6. Numerical Approximation and Discretization

A full discretization methodology is available for SFHEs with rough space-time noise, based on:

Noise mollification by spectral cutoff and piecewise-constant representation as a finite Gaussian sum on space-time grids.
Spatial Galerkin finite-element discretization and time-implicit Euler stepping.
Convergence proofs in distributional Sobolev spaces, with explicit convergence rates depending on discretization parameters (Deya et al., 2021).

Such schemes accommodate the highly singular nature of some space-time fractional noise settings and demonstrate empirical convergence toward the theoretical limit.

7. Connections, Thresholds, and Parameter Regimes

Critical integrability thresholds: The existence and uniqueness of SFHE mild solutions (especially in the multiplicative case) are tightly linked to dimensionality $d$ , the stability index $\alpha$ , and Hurst/exponent parameters of the noise. For Skorohod (Wick) solutions, the key threshold is $d<2+\alpha$ (Song et al., 2023), while for fractional-white-noise driven time-fractional equations, mild $L^2$ -solutions exist only in dimension $d\le2$ when $\alpha\ge1$ (Hachemi et al., 24 Feb 2024).
Noise roughness and regularity: The spatial Hurst index $H$ for fractional-colored noise (in the white-in-time, $H\in((2-\alpha)/2,1/2)$ , $\alpha\in(1,2)$ ) controls the global regularity and growth exponents. As $H$ decreases (the noise becomes rougher), regularity and solution existence are constrained (Liu et al., 30 Jul 2025).
Boundary conditions and non-autonomous/nonlinear coefficients: The approach adapts to Neumann, Dirichlet, and conormal boundary conditions; both time-dependent coefficients and divergence-model operators with discontinuity are tractable under ellipticity and summability constraints (Mishura et al., 2018, Mishura et al., 2019).