Well-Posedness of Hyperbolic SPDEs

• The paper demonstrates global existence and uniqueness in nonlinear hyperbolic SPDEs using kinetic formulation and stochastic averaging lemmas.
• It employs truncation, vanishing viscosity, and approximation techniques to manage degenerate diffusion and non-Lipschitz noise coefficients.
• The results show robust L1 and Hᴿ stability with both Gaussian and Lévy noise, ensuring contraction properties and ergodic behavior.

Global well-posedness of hyperbolic stochastic partial differential equations (SPDEs) concerns the existence, uniqueness, and stability of solutions for nonlinear hyperbolic systems under random forcing, with particular focus on mixed or degenerate parabolic-hyperbolic phenomena and non-classical noise. The foundational approaches—kinetic solution theory, stochastic averaging lemmas, and functional analytic techniques for infinite variance—enable precise treatment of these systems under minimal regularity and growth conditions for the nonlinearities and the driving noise.

1. Principal SPDE Models: Structure and Noise

Hyperbolic SPDEs are defined on spatial domains such as the torus $\mathbb{T}^N$ or a bounded region $D \subset \mathbb{R}^d$, with time horizons $[0,T]$. Typical equations include both conservation law-type hyperbolic fluxes and possibly degenerate parabolic diffusion: $$du + \operatorname{div}(B(u))\,dt = \operatorname{div}(A(u)\nabla u)\,dt + \Phi(x,u)\,dW_t$$ where $B: \mathbb{R} \to \mathbb{R}^N$, $A(\xi)$ symmetric and nonnegative, and $\Phi(x,u)$ encodes spatially distributed multiplicative noise (Gess et al., 2016). In other instances, the hyperbolic operator may be fractional: $$\partial_{tt} u(t,x) = -(-\Delta)^{\gamma} u(t,x) + b(u(t,x)) + \sigma(u(t,x))\dot{L}(t,x)$$ with $\dot{L}$ denoting space-time Lévy white noise, either of finite or infinite variance. Solutions are constructed in Sobolev spaces $H^r(D)$ for $r > d/2$ (Balan et al., 28 Nov 2025).

Multiplicative noise is modeled via:

• Series expansions with independent Brownian motions: $W(t) = \sum_{k \ge 1} \beta_k(t) e_k$
• Lévy bases, either finite-variance (Itô integral) or infinite-variance symmetric $\alpha$-stable measures, necessitating construction via Daniell mean and L$^0$ topology The regularity of noise coefficients $g_k$ or $\sigma$ is typically only locally Lipschitz with linear growth, supporting highly non-Lipschitz regimes.

2. Structural Hypotheses and Non-degeneracy Requirements

The nonlinearity and noise coefficients satisfy:

• $B \in C^2$; $A \in C^1$; $g_k$ Lipschitz in $(x, \xi)$
• Growth conditions: Linear in $\xi$, i.e., $|b(\xi)| \leq D_b(1+|\xi|)$, $|g_k(x,\xi)| \leq C(1+|\xi|)$
• Diffusion $A(\xi)$ may be degenerate, i.e., vanish for some $\xi$ Well-posedness hinges on non-degeneracy in the kinetic sense: The symbol $$\mathcal{L}(i u, i n, \xi) = i(u + b(\xi)\cdot n) + n^T A(\xi) n$$ must not vanish excessively on sets of $\xi$, quantified by stochastic velocity-averaging hypotheses involving exponents $(\alpha, \beta)$ and frequency scaling.

This suggests that classical uniform ellipticity is replaced by localized structural control via the kinetic symbol, ensuring sufficient dissipation or propagation in averaged senses.

3. Kinetic Solution Theory and Functional Formulations

Solutions are defined via kinetic formulations. For $u(t,x)$, the indicator $f(t,x,\xi) = 1_{u(t,x) > \xi}$ and the kinetic measure $m$ encode the fine structure of dissipation/concentration. The following random-parabolic-hyperbolic equation is satisfied in distributions: $$\partial_t f + b(\xi) \cdot \nabla_x f - A(\xi): D^2_x f = \partial_\xi\left(m-\tfrac{1}{2}G^2(x,\xi)\delta_{u=\xi}\right) - \sum_k g_k(x,\xi)\partial_\xi f\,\dot{\beta}_k + \sum_k \delta_{u=\xi}g_k(x,\xi)\,\dot{\beta}_k$$ A kinetic solution is an $L^1_{t,x}$-adapted process satisfying chain rules, $L^2$ composition properties, and kinetic measure decay conditions.

For wave equations in bounded domains driven by Lévy noise, solutions are constructed in the mild sense using the Green kernel $G_t(x,y)$ and spectral decompositions in $H^r$ spaces, with noise integrals defined as Itô or Daniell depending on the variance regime.

This framework is robust under minimal regularity and does not require uniform growth conditions or global Lipschitz regularity for coefficients.

4. Global Existence, Uniqueness, and Stability Results

Main theorems include:

• Existence and Uniqueness in $L^1$: For degenerate parabolic-hyperbolic SPDEs, under local Hölder continuity for $\sigma$, initial $u_0 \in L^1$, and mild non-degeneracy, there exists a unique kinetic solution $u \in C([0,T]; L^1)$ almost surely. For $p,q \geq 1$,

$$\mathbb{E}\left[\sup_{0 \leq t \leq T} \|u(t)\|_{L^p}^{pq}\right] \leq C(1 + \mathbb{E}\|u_0\|_{L^p}^{pq})$$

with $L^1$-contractivity and comparison property

$$\|u_1(t) - u_2(t)\|_{L^1} \leq \|u_{1,0} - u_{2,0}\|_{L^1}$$

(Gess et al., 2016).

• Well-posedness for Fractional Hyperbolic SPDEs: For $\gamma > d$, and locally Lipschitz/linear growth drift and noise, unique global solutions exist in $C(\mathbb{R}_+; H^r(D))$; for all $T > 0$

$$\mathbb{E}\left[\sup_{t \leq T} \|u(t)\|_{H^r(D)}^2\right] < \infty$$

Both finite-variance Lévy noise and symmetric infinite-variance cases (e.g., $\alpha$-stable) are covered. The infinite-variance regime uses truncation and pasting, with the stochastic integral defined in $L^0$ (Balan et al., 28 Nov 2025).

• Comparison and Contraction: Markov semigroups associated to the SPDE solutions are Feller and have the $L^1$ contraction (the “e-property”), enabling stability under initial data perturbations and uniqueness.

5. Core Analytical Instruments: A Priori Estimates and Stochastic Averaging

Central to the construction and stability analysis are:

• A Priori Energy Estimates: Itô formula applied to powers of $(1+u^2)$ yield for all $p, q \ge 1$,

$$\mathbb{E} \sup_{t \leq T} \|u(t)\|_{L^p}^{pq} \leq C(1 + \mathbb{E}\|u_0\|_{L^p}^{pq})$$

• Decay of Kinetic Measure and Equi-integrability: For trimming regions $A_R = \{(t,x,\xi): R \leq |\xi| \leq 2R\}$,

$$\frac{1}{R}\, \mathbb{E}\, m(A_R) \to 0 \text{ as } R \to \infty,\quad \operatorname{ess\,sup}_t \mathbb{E} \|(u(t) - R)^+\|_{L^1_x} \to 0$$

providing replacement for BV (bounded variation) bounds.

• Stochastic Averaging Lemmas: These enable fractional Sobolev regularity for velocity averages $$\bar{\eta}(u(t,x)) = \int_\mathbb{R} \chi_{u(t,x)}(\xi)\eta(\xi)\,d\xi$$, with $\bar{\eta}(u) \in L^r(\Omega; L^1_t W^{s,r}_x)$ for some $s > 0$, $r > 1$. The proof combines frequency decomposition (Littlewood-Paley), kinetic non-degeneracy, and stochastic integral inequalities (Burkholder–Davis–Gundy).
• Spectral and Sobolev Embedding Bounds: In wave equations, embedding $H^r \hookrightarrow L^\infty$ for $r > d/2$ and convergence of spectral sums are used alongside Grönwall-type inequalities for moment bounds.

6. Methodological Details: Truncations, Approximation, and Pasting

The existence proofs are constructed via approximation and truncation techniques:

• Vanishing Viscosity and Cutoff Approximation: Solutions are initially obtained for globally regularized coefficients. Compactness arguments and uniqueness by doubling variables extend the results to original equations.
• Stopping Times and Consistency: Solutions with truncated coefficients are patched over stopping times $\tau_n$ where $H^r$ norm exceeds $n$, establishing global existence as $\tau_n \to \infty$ with high probability.
• Infinite-Variance Noise Handling: Truncation of jumps larger than $K$ yields finite-variance approximations $L^K$. Solutions $u^K$ and $u^{K+1}$ are shown to agree before first big jump $\tau^K$, and pasting yields the global solution. The stochastic integral is defined in L$^0$ using the Daniell mean, and maximal inequalities replace moment estimates.

This methodology allows for rigorous treatment of non-Lipschitz and heavy-tailed noise within the hyperbolic SPDE framework.

7. Generalizations and Plausible Implications

Current results demonstrate that global well-posedness holds for broad classes of hyperbolic SPDEs under minimal regularity and non-degeneracy conditions, accommodating degenerate parabolic effects, locally non-Lipschitz nonlinearities, and both Gaussian and highly non-Gaussian Lévy-driven noise (Gess et al., 2016, Balan et al., 28 Nov 2025). A plausible implication is the potential for analogous kinetic and averaging approach extension to more general mixed-type systems, including systems on manifolds or with boundary dynamics, provided velocity-averaging non-degeneracy can be ensured.

Further, the contractive Feller semigroup structures and stochastic smoothing suggest robust ergodic and regularity properties, opening avenues for probabilistic analysis of long-time behavior, invariant measure existence, and stability under parameter perturbation.