Quantitative Laplace Principle in SPDEs

• The Quantitative Laplace Principle is a rigorous framework that characterizes rare events in stochastic partial differential equations through variational control and large deviation principles.
• It applies to SPDEs like the three-dimensional stochastic wave equation under small noise scaling, employing weak convergence and continuity techniques to assess exponential decay rates.
• The principle informs practical analysis in mathematical physics and signal processing, offering precise asymptotics and a transferable methodology to other infinite-dimensional systems.

The quantitative Laplace principle provides a rigorous asymptotic characterization of the behavior of functionals of stochastic partial differential equations (SPDEs), formulating the exponential decay rate of rare event probabilities via a variational control problem. For the stochastic wave equation in spatial dimension three driven by Gaussian noise, the principle characterizes the limiting exponential moment of path functionals in terms of a variational cost over deterministic controls. This framework both refines and extends classical large deviation theory to the regime of non-Gaussian, infinite-dimensional systems by employing a weak convergence methodology and precise continuity/tightness arguments on spaces of Hölder continuous functions.

1. Core Principle and Large Deviations Context

The Laplace principle, in this context, is equivalent to proving a large deviation principle (LDP) for the solution of an SPDE under small noise scaling. Concretely, consider the stochastic wave equation in $\mathbb{R}^{3}$: $$u(t, x) = w(t, x) + \int_{0}^{t} \int_{\mathbb{R}^{3}} G(t-s, x-y) \sigma(u(s, y)) M(ds, dy) + \int_{0}^{t}[G(t-s) * b(u(s, \cdot))](x) ds$$ where $w(t,x)$ is derived from the initial data, $G(\cdot, \cdot)$ is the fundamental solution (kernel) of the wave operator, $\sigma$ and $b$ are Lipschitz nonlinearities, and $M(ds, dy)$ is a martingale measure associated with a spatially correlated Gaussian noise. The noise may be multiplicatively perturbed by a parameter $\epsilon \in (0,1]$: $$u^{\epsilon}(t, x) = w(t, x) + \sqrt{\epsilon}\;\text{(stochastic integral)} + \text{(deterministic convolution term)},$$ which yields a family of laws on the Banach space of Hölder continuous paths.

The Laplace principle, for a continuous bounded function $f$ on the solution space, states: $$-\epsilon \log \mathbb{E}\left[\exp(-f(u^{\epsilon}))\right] \to \inf_{\phi} \{ f(\phi) + I(\phi) \}$$ as $\epsilon \to 0$, where $I(\cdot)$ is a rate function determined by the pathwise control problem. This embodies the LDP by specifying the exponential rate of decay for rare fluctuations away from typical behavior in terms of a variational minimization.

2. Mathematical Framework for the SPDE

Several structural and probabilistic ingredients are central:

• Noise structure: The noise is Gaussian, white in time, and admits a stationary spatial covariance characterized by a tempered measure on $\mathbb{R}^3$ with density proportional to $|x|^{-8}$ up to a positive function.
• Solution space: Solutions $u^{\epsilon}$ are constructed as random fields with jointly Hölder continuous sample paths in space-time. The ambient solution space is a Polish space of functions with Hölder regularity.
• Cylindrical Wiener process: The source of randomness is represented as a cylindrical Wiener process $B(t) = \sum_{k} B_{k}(t) e_{k}$ with $(e_{k})$ an orthonormal basis of $L^2(\mathbb{R}^3)$, and $B_k$ independent standard Brownian motions.

3. Main Theorem and Rate Function

The key result is that, under appropriate conditions on initial data, coefficients, and covariance structure, the family $\{u^{\epsilon}\}_{\epsilon \in (0,1]}$ satisfies a large deviation principle on the solution path space. The rate function $I$ is variational: $$I(f) = \inf_{h \in H_T : G^{0}(h) = f} \left\{ \frac{1}{2} \|h\|^2 \right\}$$ where:

• $H_T$ is the Cameron-Martin space associated with the driving Wiener process,
• $G^{0}(h)$ solves the deterministic "controlled" wave equation (obtained by turning off the stochastic term and introducing a deterministic control $h$ in its place),
• The infimum is taken over controls $h$ which steer the deterministic system to the path $f$.

Thus, the cost of a rare event (a large deviation) is governed by the minimal quadratic cost needed to steer the system to it via deterministic controls.

4. Weak Convergence Approach and Continuity/Tightness

The establishment of the Laplace principle is achieved by the weak convergence method, in which the stochastic control representation of exponential functionals is pivotal. Specifically, for a bounded measurable functional $g$: $$-\log \mathbb{E}\left[ \exp(-g(B)) \right] = \inf_{u \in L^2} \mathbb{E} \left[ \frac{1}{2} \|u\|^2_{L^2} + g\left(B + \int u ds\right) \right],$$ where $B$ is the infinite-dimensional Brownian motion.

The proof uses:

• Continuity: The mapping from control $h$ to the controlled solution $V^h$ is continuous in the weak topology on $H_T$.
• Compactness: The collection $\{V^h : h \in H_A\}$ is compact in the solution space.
• Convergence: The perturbed solutions $u^{\epsilon, v^{\epsilon}}$ converge to $V^h$ as $\epsilon \to 0$, quantifying the "closeness" of stochastic and controlled deterministic evolutions.

Burkholder’s inequality, Hölder regularity estimates, and Gronwall-type arguments control the stochastic and deterministic integrals.

5. Applications and Theoretical Implications

The Laplace principle for this SPDE allows for:

• Quantification of rare events: The exponential probability of observing atypical path functionals is governed by the rate function. This is crucial for mathematical physics, wave propagation in random media, and signal processing, where probabilistic bounds for deviations in hyperbolic SPDEs are required.
• Asymptotics of small noise: The result precisely describes the probability law's limiting behavior as noise amplitude $\epsilon$ vanishes, including the exponential scaling and the shape of fluctuations determined by deterministic control problems.
• Transferability to other SPDEs: The methodology is robust and, under similar structural and regularity conditions, extends to other infinite-dimensional stochastic equations.

6. Comparison with Prior Literature

Relative to prior work:

• Existence, uniqueness, and joint Hölder continuity for solutions to the stochastic wave equation in three dimensions had been established by Dalang and Sanz-Solé ("Memoirs of the AMS, Vol 199, 2009").
• This work supplements those foundations by including a small-noise scaling parameter and develops the large deviation (Laplace) principle using the weak convergence strategy of Dupuis-Ellis.
• Refinements over Dalang and Quer-Sardanyons are achieved, especially in functional analytic regularity.
• The analysis adapts variational representations and tightness/compactness arguments from the theory of infinite-dimensional Brownian motions, as in Budhiraja, Dupuis, and Maroulas.

7. Technical Summary and Conclusions

By perturbing the stochastic wave equation with a small parameter $\epsilon$ in the noise and analyzing the induced family of solution laws, the quantitative Laplace principle provides the exact exponential rate—via the minimization over deterministic controls—of large deviations in Hölder norm for the solution. All critical arguments rest on the continuity, compactness, and convergence properties of the control-to-solution map and precise pathwise and moment estimates for the SPDE dynamics.

This quantitative framework now underpins the rigorous understanding of rare event probabilities, asymptotics of SPDEs in small noise, and forms a prototype for future work in the area of stochastic analysis and infinite-dimensional variational problems.