Parabolic Anderson Model of Skorokhod Type

• Parabolic Anderson Model of Skorokhod Type is a stochastic PDE framework that uses Malliavin calculus and Wick products to handle singular multiplicative noise.
• It employs Wiener–Itô chaos expansions and Feynman–Kac representations to derive precise moment bounds and intermittency properties.
• The model bridges continuous and discrete analyses, enabling numerical schemes that connect to directed polymers in Gaussian environments.

The Parabolic Anderson Model (PAM) of Skorokhod Type refers to a class of stochastic partial differential equations (SPDEs) of reaction-diffusion type, where the multiplicative noise term is interpreted via the Skorokhod integral, i.e., the Wick or divergence sense in Malliavin calculus. The primary analytical and probabilistic techniques employed for these models include Wiener chaos expansions, Feynman–Kac representations, and moment and intermittency analyses under various regimes of spatial and/or temporal noise roughness. The Skorokhod interpretation is essential in handling singular noises, subcritical and critical regimes, and distinguishing from alternative stochastic calculus conventions (notably the Stratonovich sense).

1. Model Definition and Skorokhod Interpretation

The general parabolic Anderson model of Skorokhod type, in $d$ spatial dimensions, is formulated as

$$\frac{\partial u}{\partial t}(t,x) = \frac{1}{2}\Delta u(t,x) + u(t,x) \,\diamond\, \dot{W}(t,x),$$

where %%%%1%%%% and $\diamond$ denotes the Wick (Skorokhod) product, and $\dot{W}(t,x)$ is a generalized Gaussian noise, possibly rough in time and/or space (Kim et al., 2018, Ma et al., 2020, Chen et al., 2020).

The Skorokhod product is defined via Malliavin calculus. For suitable $\mathcal{H}$-valued processes $u$, the Skorokhod integral $\delta(u)$ is the adjoint of the Malliavin derivative $D$, i.e., for $F\in \mathbb{D}^{1,2}$

$$\mathbb{E}[F\, \delta(u)] = \mathbb{E}[\langle D F, u\rangle_{\mathcal{H}}].$$

This framework accommodates distribution-valued noises beyond the scope of Itô or classical stochastic calculus.

In one dimension with purely spatial Brownian noise, the model becomes

$$\partial_t u^S(\varepsilon; t, x) = \partial_{xx} u^S(\varepsilon; t, x) + \varepsilon u^S(\varepsilon; t, x) \, \diamond \, dW(x),$$

for $x\in[0,\pi]$ with Dirichlet boundary conditions, without further need for renormalization (Kim et al., 2018).

2. Chaos Expansion, Regularity, and Analyticity

Solutions to the PAM of Skorokhod type admit Wiener–Itô chaos expansions: $u(t,x) = \sum_{n=0}^\infty I_n(f_n(\cdot, t, x)),$ where $I_n$ denotes the $n$-th order multiple Wiener–Itô integral, and $f_n\in \mathcal{H}^{\otimes n}$ are explicitly constructed in terms of heat kernels and initial data (Hu et al., 2016, Balan et al., 2018, Kim et al., 2017).

Convergence of the chaos series in $L^2(\Omega)$ (and higher $L^p$) is guaranteed under Dalang-type conditions on the spectral measure of the driving noise: $$\int_{\mathbb{R}^d} \frac{1}{1 + |\xi|^2} \, \mu(d\xi) < \infty,$$ which become sharp in the presence of spatial roughness (Balan et al., 2018, Hu et al., 2016).

The parameter $\varepsilon$ (noise intensity) enters the expansion as a real-analytic parameter; both Skorokhod and Stratonovich solutions can be expanded in powers of $\varepsilon$, and the series is absolutely convergent for all real $\varepsilon$ (Kim et al., 2018). Each coefficient is explicitly constructed as a Picard/Duhamel iteration over heat kernel convolutions and repeated Wick products.

Sample path regularity (space and time) matches that of the additive (non-multiplicative) models for one-dimensional spatial noise. For instance, with Dirichlet conditions and $u_0\in C^{3/2}$, the solution is $C^{3/4-\epsilon}$ in time and $C^{3/2-\epsilon}$ in space for any $\epsilon>0$; these are optimal (Kim et al., 2017).

3. Moment Formulas, Feynman–Kac Representation, and Intermittency

A central tool is the Feynman–Kac representation of moments: $$\mathbb{E}[u(t,x)^p] = \mathbb{E}^B \exp\left\{ \sum_{1\le i<j\le p} \iint_{[0,t]^2} K(s,r) Q(B^i_s, B^j_r) \, ds \, dr \right\},$$ where $B^i$ are independent copies of the underlying Markov process (Brownian motion, $\alpha$-stable, etc.), and $K(s,r)$, $Q(x,y)$ encode the space-time covariance structure (Ma et al., 2020, Chen et al., 2020, Hu et al., 2016).

This allows derivation of sharp upper and lower moment bounds. In rough settings ($H<1/2$ for temporal fractional noise, or rough space as $H\downarrow 1/4$), moments exhibit super-exponential growth in $n$—the hallmark of intermittency: $$\log \mathbb{E}[u(t,x)^n] \asymp n^{1+H} t \quad \text{(space roughness } H\in(1/4,1/2)),$$ or more generally via exponents determined by model parameters and spatial regularity (Hu et al., 2016, Ma et al., 2020).

Moment Lyapunov exponents

$$\lambda_p = \lim_{t\to\infty} \frac{1}{t} \log \mathbb{E}[u(t,x)^p]$$

are strictly convex functions of $p$, reflecting multifractal (fully intermittent) behavior in all considered parameter regimes (Ma et al., 2020, Chen et al., 2016).

4. Comparison with Stratonovich Interpretation and Renormalization

In dimensions $d=1$, the Skorokhod and Stratonovich solutions are asymptotically close as $\varepsilon\to 0$, differing only by an explicit deterministic $O(\varepsilon^2)$ term (Kim et al., 2018): $$u^{\mathrm{Strat}}(\varepsilon; t,x) - u^S(\varepsilon; t,x) = \varepsilon^2 C(t,x) + o(\varepsilon^2),$$ where $C(t,x)$ is an explicit integral over heat kernels and the initial profile.

The first $O(\varepsilon)$ corrections coincide, and all randomness in the correction cancels in one dimension. This deterministic shift can be seen as a finite “renormalization” effect. In higher spatial dimensions ($d\geq 2$), such corrections diverge, necessitating infinite renormalization and invoking the theory of regularity structures to even make sense of the model, as the Skorokhod product alone is not sufficient (Chen et al., 2020).

5. Criticality, Blow-Up, and Regularity Structures

The Skorokhod-type PAM exhibits a rich phase diagram:

• Subcritical regime: Under explicit conditions on Hurst exponents and dimension (e.g., $d-H<1$ for fractional settings), all moments remain finite and explicit chaos/Feynman–Kac analysis applies (Chen et al., 2020).
• Critical regime: When the criticality threshold is reached ($d-H=1$ or explicit combinations of Hurst indices and spatial dimension), only local-in-time solutions exist and moments of order $p\geq2$ blow up beyond a finite time $t_0(p)$, given by sharp inequalities involving the best Sobolev constant (e.g., Gagliardo–Nirenberg for $L^4$) (Chen et al., 2020).
• Supercritical regime: No meaningful solution exists in the Skorokhod framework; regularity structures or renormalized models must be invoked (Chen et al., 2020).

In the critical case, the transition time for moment blow-up is precisely characterized: $t_0(p) = \kappa^{-4/(2H_0-1)} (p-1)^{-1/(2H_0-1)},$ where $\kappa$ is the best constant in a functional inequality tied to the noise covariance (Chen et al., 2020).

When the noise is more singular (e.g., $H_j<1/2$ in space), $\gamma$ becomes a distribution, and for Stratonovich models, explicit renormalization via counterterms is necessary, as constructed by regularity structures; in the Skorokhod case, finite moments persist up to the subcritical threshold without infinite counterterms (Chen et al., 2020, Chen et al., 2016).

6. Numerical Schemes and Connections to Discrete Models

The Feynman–Kac representation enables discrete numerical schemes via random walks, providing accurate approximations for the one-dimensional Skorokhod-type PAM (Xia et al., 28 Dec 2025). For discretization mesh $h$, with driving fractional Brownian sheet ($H,H_*\ge 1/2$), the $L^p(\Omega)$ error rate matches the solution's time-Hölder regularity up to an arbitrarily small $\epsilon$: $$\| u_h(m,n) - u(t,x) \|_{L^p} = O\left(h^{\frac12[(2H+H_*-1)\wedge 1] - \epsilon}\right).$$ The discrete model corresponds exactly to the partition function of directed polymers in Gaussian environments, thus unifying continuum intermittency properties with their discrete analogues in statistical mechanics (Xia et al., 28 Dec 2025).

7. Spatial Averages, Central Limit Theorems, and Advanced Phenomena

Spatial average fluctuations over large scales for Skorokhod-type PAMs driven by rough noise exhibit Gaussian CLT-type behavior: $$\frac{1}{\sqrt{R}} \int_{-R/2}^{R/2} [u(t,x) - \mathbb{E}u(t,x)] dx \xrightarrow[R\to\infty]{\mathrm{law}} \mathcal{G}(t),$$ where the limiting process is centered Gaussian with covariance determined by explicit functionals of the Feynman–Kac representation (Nualart et al., 2020).

Rigorous control of spatial averages and chaos norms provides the foundation for functional CLTs and elucidates the mesoscopic fluctuation structure of PAMs, even in highly singular (fractional, colored) noise settings.

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References:

(Kim et al., 2018, Kim et al., 2017, Hu et al., 2016, Balan et al., 2018, Ma et al., 2020, Chen et al., 2020, Xia et al., 28 Dec 2025, Nualart et al., 2020, Chen et al., 2016)