Carleman Estimate in Backward Stochastic Parabolic PDEs

• The paper introduces a weighted Carleman estimate that secures stability and uniqueness for backward stochastic parabolic equations.
• The methodology employs tailored exponential weight functions to counteract ill-posedness and manage stochastic perturbations.
• The approach underpins observability, null controllability, and unique continuation, thereby advancing inverse problem resolution in SPDEs.

A Carleman estimate for backward stochastic parabolic equations is a powerful weighted energy inequality that provides essential quantitative control over weak solutions of stochastic parabolic partial differential equations (SPDEs) posed in the backward time direction. This analytical tool enables stability and uniqueness results for inverse problems, unique continuation properties, and observability inequalities underpinning controllability theory in stochastic parabolic settings. Carleman estimates are a central mechanism to overcome ill-posedness, low regularity, and lack of compactness that are intrinsic to stochastic parabolic problems, with broad implications for mathematical analysis, control, and inverse identification of SPDEs.

1. Global Carleman Estimates: Weight Functions, Structure, and Stochastic Calculus

Carleman estimates for backward stochastic parabolic equations typically concern equations of the form

$$dy - \Bigg(\sum_{i,j=1}^{n} (b_{ij} y_{x_j})_{x_i}\Bigg)dt = [(a_1, \nabla y) + a_2 y + f]dt + (a_3 y + g)dB(t)$$

posed in a space-time cylinder $Q = (0,T) \times G$ with Dirichlet or general boundary conditions, where $B(t)$ is a Brownian motion and $f$, $g$ may be random and adapted processes.

A central step in constructing a Carleman estimate is the selection of weight functions $\varphi = e^{\lambda v}$ and $\theta = e^{s\varphi}$, where $v$ is typically independent of spatial variables (for global-in-space estimates), and $\lambda>0$, $s>0$ are large parameters. The weight “blows up” or is highly damped near selected space-time boundaries, enabling the absorption of lower-order terms and control of stochastic perturbations. In the stochastic setting, Itô's formula introduces additional quadratic variation terms and necessitates manipulations to maintain adaptedness.

A prototypical Carleman estimate takes the form

$$\lambda\,\mathbb{E}\int_\delta^T\int_G \theta^2 |\nabla y|^2 dxdt + s^2\lambda^4\,\mathbb{E}\int_\delta^T\int_G \theta^2 y^2 dxdt \leq C\Big\{ \theta^2(T)\|\nabla y(T)\|^2_{L^2(G)} + \theta^2(0)\|\nabla y(0)\|^2_{L^2(G)} + \cdots \Big\}$$

for large $\lambda$ and $s$ (Lu, 2011). This weighted energy inequality controls high-norm quantities of the solution via the prescribed data and source terms. The introduction of two large parameters (typically $\lambda$ for spatial and $s$ for temporal amplification) enhances flexibility and sharpness.

2. Inverse Problems: Conditional Stability and Source Identification

Carleman estimates enable rigorous stability and uniqueness statements in inverse stochastic parabolic problems, notably for:

Backward historical determination: Given $y(T)$, one aims to recover $y(t_0)$ for $t_0 < T$, a severely ill-posed backward problem. Carleman estimates yield interpolation (conditional stability) inequalities of the type

$$\|y(t_0)\|_{L^2(G)} \leq C \|y\|_{L^2(0,T;L^2(G))}^{1-\delta} \|y(T)\|_{H^1(G)}^{\delta}$$

where $\delta\in(0,1)$ and $C>0$ depend on a priori bounds (Lu, 2011). This quantifies the extent of ill-posedness and allows a backward uniqueness principle: if $y(T)=0$, then $y\equiv 0$ on $[0,T]$.

Inverse source problems: When seeking uniqueness/stability for an unknown source $h(t,x')$ from lateral Neumann data, the Carleman estimate (typically after a cutoff and localization procedure) yields that the vanishing of an appropriate measurement implies $h\equiv 0$ (Lu, 2011). The tool is robust for both deterministic and stochastic perturbations and extends to more general data and multiple types of observations.

For degenerate and stochastic degenerate cases, singular weight functions that 'blow up' near the degeneracy are employed, e.g., for equations like $$dz + ((x+\varepsilon)^{\alpha}z_x)_xdt = f dt + F dB(t)$$, with tailored parameter regimes and boundary functions (Wu et al., 2019, Baroun et al., 2022).

3. Null Controllability, Observability, and Duality

Carleman estimates are foundational in deriving observability inequalities for backward stochastic parabolic equations, which, via duality (Hilbert Uniqueness Method or penalized minimization), establish null controllability for the corresponding forward (or backward) system.

Given a backward equation as the adjoint system,

$$dz + \mathrm{div}(A\nabla z)dt = (F_0 + \mathrm{div}F)dt + Z\,dW(t)$$

a global Carleman estimate yields an observability inequality of the form

$$\|z(0)\|_{L^2(G)}^2 \leq C\,\mathbb{E}\int_{0}^{T}\int_{G_0} \theta^2 z^2 dxdt$$

enabling the construction of a control which steers the forward state to zero at the designated terminal time (Baroun et al., 9 Jan 2024, Boulite et al., 14 Oct 2025, Boulite et al., 13 Nov 2024).

The explicit dependence of observability (and thus control cost) constants on the final time $T$, coefficients, and weights is often computed, providing quantitative insight on short-time blow-up and parameter sensitivity (Baroun et al., 2023, Boulite et al., 14 Oct 2025). For systems with dynamic boundary conditions, Robin conditions, or singular coefficients, the methodology is adapted with suitable modifications of weight functions and careful tracking of trace and surface terms (Boulite et al., 12 Jun 2024, Boulite et al., 14 Oct 2025).

4. Unique Continuation and Quantitative Stability

Carleman estimates for stochastic backward parabolic equations have been shown to yield strong unique continuation properties. If a solution vanishes to infinite order at a point (in a probabilistic, mean-square, or almost sure sense), then it must vanish everywhere (pathwise or in expectation). This is established via iterated three-cylinder or two-cylinder inequalities: $$E \int_{(0,T)\times B(x_0, r)} |y(t,x)|^2 dx dt \leq C_N r^{2N} \ \forall N\in\mathbb{N}, r>0 \implies y\equiv 0$$ demonstrating SUCP (Strong Unique Continuation Property) (Liao et al., 2017).

These results propagate to quantitative unique continuation and stability in inverse boundary problems, including logarithmic-type stability for identifying time-varying (or moving) boundaries from partial interior measurements (Liao et al., 2023): $$d_H(G_1(t), G_2(t)) \leq C \exp(e\cdot\gamma(t)) |\log \tilde{\epsilon}|^{-1}$$ which precisely quantifies the ill-posedness and sensitivity to measurement error.

5. Extensions: Degeneracy, High Order, and Semi-Discrete Systems

Recent developments extend Carleman estimates for backward stochastic parabolic equations to degenerate, semi-linear, and high-order (fourth-order) SPDEs as well as to spatially semi-discretized or semi-discrete settings:

• Degenerate parabolic SPDEs require carefully designed singular weights, which accommodate vanishing diffusion coefficients at the boundary (Wu et al., 2019, Baroun et al., 2022).
• High-order (e.g., Cahn-Hilliard or bi-Laplacian) backward stochastic parabolic equations are handled via new weighted identities and suitable time-space weights, ensuring non-degeneration at $t=0$ (Lü et al., 2021, Zhang et al., 7 Aug 2024, Wang, 2023).
• In semi-discrete (finite-difference in space) frameworks, the Carleman estimate must balance parameter regimes and capture mesh-dependent error terms, producing "relaxed" or $\phi$-null controllability in the limit as the mesh is refined (Zhao, 20 Feb 2024, Wang et al., 6 May 2024, Lecaros et al., 5 Mar 2025).

The essential features preserved are: the exponential amplification of solutions via weight functions, localization to control problematic components, and techniques for handling discrete difference operators under stochastic calculus.

6. Concluding Significance and Technical Advances

Carleman estimates for backward stochastic parabolic equations have become central tools in the analysis of stochastic inverse and control problems. The core methodological advance is the coupling of exponential weights, stochastic integration (notably Itô’s formula adaptations), and duality. These lead to the following canonical outcomes:

• Amplified observability and stability estimates for ill-posed inverse and backward-time problems, overcoming both deterministically and stochastically driven instability.
• Unique continuation and backwards (and forward) uniqueness even for solutions lacking high regularity or possessing only weak adaptedness.
• Null controllability and cost quantification via duality, with explicit dependence on all relevant PDE, stochastic, and geometric parameters.
• Extension to broad classes of boundary conditions (Dirichlet, Neumann, Robin, and dynamic), spatial degeneracy, high-order operators, and spatial semi-discretizations.

Collectively, these results form the analytic underpinning for rigorous inverse problem stability, control design (continuous and discrete), and unique identification in stochastic parabolic systems. The field continues to evolve, with challenges remaining in extending such estimates to general nonlinearities, weak regularity settings, and coupled forward–backward or nonlocal systems.