Kolmogorov equations for stochastic convective Brinkman-Forchheimer equations forced by Lévy Noise and its application to infinite horizon problems

Abstract: This article examines the Kolmogorov equation corresponding to the following stochastic two- and three-dimensional incompressible ($\nabla\cdot\boldsymbol{u}=0$) convective Brinkman-Forchheimer equations, also known as the damped Navier-Stokes equations, driven by Lévy noise on the torus: \begin{align*} \mathrm{d}\boldsymbol{u}+[-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p]\mathrm{d} t =\sqrt{\mathrm{Q}}\mathrm{d}\mathrm{W}+\int_{Z}σ(t,z)\widetildeπ(\mathrm{d} t,\mathrm{d} z), \end{align*} where $μ,α,β&gt;0$ are physical constants; $\mathrm{Q}$ is a non-negative, trace-class operator; $\mathrm{W}$ is a cylindrical Wiener process on a Hilbert space; $σ$ represents the jump-noise coefficient; $(Z,\mathscr{B}(Z))$ is a measurable space; $π$ is a time-homogeneous Poisson random measure; and $\widetildeπ$ denotes its compensator. The main contribution of this work is the establishment of the essential $m$-dissipativity of the corresponding Kolmogorov operator, a property that has received limited attention in the existing literature for systems driven by jump-type noise. \emph{Our main innovation is that, in contrast to traditional techniques which crucially depend on exponential moment estimates, we utilize the intrinsic structure of the absorption term $$α\boldsymbol{u}+β|\boldsymbol{u}|<sup>{r-1}\boldsymbol{u}$$ to dispense with these requirements. This allows us to establish the essential $m$-dissipativity of the Kolmogorov operator without the need for exponential moments.} We apply the developed framework to an infinite-horizon stochastic optimal control problem, demonstrating the solvability of the associated infinite-dimensional Hamilton-Jacobi-Bellman (integro-differential) equation.

• The paper proves essential m-dissipativity of the SCBF Kolmogorov operator in L²(H,η) using invariant-measure moment bounds and Forchheimer coercivity instead of exponential moments.
• The analysis derives a carré du champ identity with both Wiener-gradient and Lévy-jump dissipation, along with resolvent estimates and bounded-drift perturbation results.
• The paper establishes a unique mild solution to the infinite-horizon HJB equation under a discount-dependent Lipschitz condition and identifies an optimal feedback control for quadratic costs.

This paper studies the Kolmogorov equation associated with the stochastic convective Brinkman–Forchheimer (SCBF) equations — the damped Navier–Stokes system on the two- and three-dimensional torus $\mathbb{T}^d$ — driven by additive Lévy noise. The central result is the essential $m$-dissipativity of the Kolmogorov operator in $\mathrm{L}^2(H,\eta)$, where $H$ is the solenoidal $\mathrm{L}^2$ space and $\eta$ the unique invariant measure of the transition semigroup. The analysis is then applied to an infinite-horizon stochastic optimal control problem, yielding existence and uniqueness of a mild solution to the associated infinite-dimensional Hamilton–Jacobi–Bellman (HJB) integro-differential equation.

Setting and model

The SCBF system under consideration reads, after projection onto divergence-free fields,

$$dY(t)+[\mu\mathcal{A}Y(t)+\mathfrak{B}(Y(t))+\alpha Y(t)+\beta\mathfrak{C}(Y(t))]\,dt=\sqrt{Q}\,dW(t)+\int_Z G(z)\,\pi(dt,dz),$$

where $\mathcal{A}$ is the Stokes operator, $\mathfrak{B}(y)=(y\cdot\nabla)y$, and $\mathfrak{C}(y)=\mathscr{P}(|y|^{r-1}y)$ is the Forchheimer absorption term with exponent $m$0. The noise consists of a trace-class Wiener perturbation satisfying $m$1, plus a compensated Poisson random measure with intensity $m$2. A notable technical point is that, unlike the Navier–Stokes case, zero-mean velocity fields are not assumed, since the absorption term does not preserve this property; consequently the Poincaré inequality is unavailable and the full $m$3-norm must be used throughout.

The well-posedness framework is taken from prior work: for $m$4 (and for $m$5 under $m$6), a pathwise unique strong solution exists with paths in $m$7, together with an energy equality.

Invariant measure and key a-priori bounds

Existence of an invariant measure follows from energy estimates combined with the Krylov–Bogoliubov theorem via tightness of time-averaged laws. Uniqueness follows from exponential stability of solutions: for $m$8 with $m$9 (where $\mathrm{L}^2(H,\eta)$0), or $\mathrm{L}^2(H,\eta)$1 with $\mathrm{L}^2(H,\eta)$2, any two solutions satisfy $\mathrm{L}^2(H,\eta)$3 almost surely. This yields exponential mixing of the semigroup toward equilibrium.

The technically decisive estimate concerns moments of the invariant measure. For test functions $\mathrm{L}^2(H,\eta)$4, applying Itô's formula, using invariance ($\mathrm{L}^2(H,\eta)$5), and exploiting the identity $\mathrm{L}^2(H,\eta)$6 on the torus, the authors obtain, for all $\mathrm{L}^2(H,\eta)$7,

$\mathrm{L}^2(H,\eta)$8

with $\mathrm{L}^2(H,\eta)$9 depending only on $H$0, the jump-moment quantity $H$1, and physical parameters. This bound on $H$2 is precisely what earlier treatments of stochastic Navier–Stokes obtained through exponential moment estimates of the form $H$3 and mixed exponential-interpolation integrals.

Essential $H$4-dissipativity without exponential moments

The main theorem states that the Kolmogorov operator $H$5, defined on the algebra of exponential functions $H$6 by

$H$7

is dissipative in $H$8 and its closure coincides with the infinitesimal generator $H$9 of the transition semigroup. The proof proceeds in three steps:

Extension. Via the infinite-dimensional Itô formula applied to $\mathrm{L}^2$0, one shows $\mathrm{L}^2$1 pointwise, and the invariant-measure bound above guarantees that difference quotients are equibounded in $\mathrm{L}^2$2, so $\mathrm{L}^2$3 extends $\mathrm{L}^2$4.

Approximation. The nonlinearities are truncated at scale $\mathrm{L}^2$5 in the $\mathrm{L}^2$6-norm, producing smooth bounded coefficients $\mathrm{L}^2$7, $\mathrm{L}^2$8 whose resolvent problems are classically solvable. Derivative estimates for the resolvent follow from the exponential decay of the linearized flow, $\mathrm{L}^2$9, which itself rests on the coercivity of the absorption term rather than on exponential integrability.

Perturbation vanishing. The truncation error terms satisfy, e.g., $\eta$0, which requires $\eta$1 and $\eta$2. These are established case-by-case: in dimension three for $\eta$3 via a weighted interpolation of $\eta$4 norms with exponents chosen so that both factors fall within the available moment bounds (with $\eta$5); for $\eta$6 via an $\eta$7-$\eta$8 interpolation with $\eta$9; and in dimension two via Gagliardo–Nirenberg. The Lumer–Phillips theorem then gives $$dY(t)+[\mu\mathcal{A}Y(t)+\mathfrak{B}(Y(t))+\alpha Y(t)+\beta\mathfrak{C}(Y(t))]\,dt=\sqrt{Q}\,dW(t)+\int_Z G(z)\,\pi(dt,dz),$$0-dissipativity of the closure, and uniqueness of $$dY(t)+[\mu\mathcal{A}Y(t)+\mathfrak{B}(Y(t))+\alpha Y(t)+\beta\mathfrak{C}(Y(t))]\,dt=\sqrt{Q}\,dW(t)+\int_Z G(z)\,\pi(dt,dz),$$1-dissipative extensions identifies it with $$dY(t)+[\mu\mathcal{A}Y(t)+\mathfrak{B}(Y(t))+\alpha Y(t)+\beta\mathfrak{C}(Y(t))]\,dt=\sqrt{Q}\,dW(t)+\int_Z G(z)\,\pi(dt,dz),$$2.

Two structural restrictions deserve emphasis. First, the result requires $$dY(t)+[\mu\mathcal{A}Y(t)+\mathfrak{B}(Y(t))+\alpha Y(t)+\beta\mathfrak{C}(Y(t))]\,dt=\sqrt{Q}\,dW(t)+\int_Z G(z)\,\pi(dt,dz),$$3 in two dimensions and $$dY(t)+[\mu\mathcal{A}Y(t)+\mathfrak{B}(Y(t))+\alpha Y(t)+\beta\mathfrak{C}(Y(t))]\,dt=\sqrt{Q}\,dW(t)+\int_Z G(z)\,\pi(dt,dz),$$4 (plus the critical case $$dY(t)+[\mu\mathcal{A}Y(t)+\mathfrak{B}(Y(t))+\alpha Y(t)+\beta\mathfrak{C}(Y(t))]\,dt=\sqrt{Q}\,dW(t)+\int_Z G(z)\,\pi(dt,dz),$$5 with $$dY(t)+[\mu\mathcal{A}Y(t)+\mathfrak{B}(Y(t))+\alpha Y(t)+\beta\mathfrak{C}(Y(t))]\,dt=\sqrt{Q}\,dW(t)+\int_Z G(z)\,\pi(dt,dz),$$6) in three dimensions; the upper bound $$dY(t)+[\mu\mathcal{A}Y(t)+\mathfrak{B}(Y(t))+\alpha Y(t)+\beta\mathfrak{C}(Y(t))]\,dt=\sqrt{Q}\,dW(t)+\int_Z G(z)\,\pi(dt,dz),$$7 in 3D arises from the interpolation argument controlling $$dY(t)+[\mu\mathcal{A}Y(t)+\mathfrak{B}(Y(t))+\alpha Y(t)+\beta\mathfrak{C}(Y(t))]\,dt=\sqrt{Q}\,dW(t)+\int_Z G(z)\,\pi(dt,dz),$$8 against the available moments. Second, the authors explicitly note they cannot prove closability of $$dY(t)+[\mu\mathcal{A}Y(t)+\mathfrak{B}(Y(t))+\alpha Y(t)+\beta\mathfrak{C}(Y(t))]\,dt=\sqrt{Q}\,dW(t)+\int_Z G(z)\,\pi(dt,dz),$$9 as an operator on $\mathcal{A}$0, so the gradient term is controlled only along the extended domain $\mathcal{A}$1.

Consequences: Carré du Champ identity and resolvent theory

From the main theorem the paper derives the identité du carré du champ for the integro-differential Kolmogorov operator:

$\mathcal{A}$2

valid for all $\mathcal{A}$3, with the accompanying bounds $\mathcal{A}$4 and the analogous jump-difference bound. The jump contribution to this identity appears not to have been previously recorded in the literature on integro-differential Kolmogorov operators. Resolvent estimates follow directly: for $\mathcal{A}$5,

$\mathcal{A}$6

A perturbation result shows that for bounded Borel $\mathcal{A}$7, the operator $\mathcal{A}$8 has resolvent containing $\mathcal{A}$9, obtained by a contraction argument on $\mathfrak{B}(y)=(y\cdot\nabla)y$0, whose norm is bounded by $\mathfrak{B}(y)=(y\cdot\nabla)y$1. Semigroup identities of the form $\mathfrak{B}(y)=(y\cdot\nabla)y$2 extend by density to all of $\mathfrak{B}(y)=(y\cdot\nabla)y$3.

Application to infinite-horizon optimal control

The framework is applied to the controlled SCBF system with control entering through $\mathfrak{B}(y)=(y\cdot\nabla)y$4, constrained to the ball of radius $\mathfrak{B}(y)=(y\cdot\nabla)y$5, minimizing the discounted cost $\mathfrak{B}(y)=(y\cdot\nabla)y$6. Dynamic programming leads to the stationary HJB equation

$\mathfrak{B}(y)=(y\cdot\nabla)y$7

where $\mathfrak{B}(y)=(y\cdot\nabla)y$8 is the Legendre transform of the running control cost $\mathfrak{B}(y)=(y\cdot\nabla)y$9. Because $\mathfrak{C}(y)=\mathscr{P}(|y|^{r-1}y)$0 is not closed, the equation is interpreted through its closure, i.e., with $\mathfrak{C}(y)=\mathscr{P}(|y|^{r-1}y)$1 in place of $\mathfrak{C}(y)=\mathscr{P}(|y|^{r-1}y)$2, and solved in the mild sense via the resolvent fixed-point formulation $\mathfrak{C}(y)=\mathscr{P}(|y|^{r-1}y)$3.

The main solvability theorem asserts that if $\mathfrak{C}(y)=\mathscr{P}(|y|^{r-1}y)$4 is Lipschitz continuous and $\mathfrak{C}(y)=\mathscr{P}(|y|^{r-1}y)$5, then for discount factors satisfying $\mathfrak{C}(y)=\mathscr{P}(|y|^{r-1}y)$6, the HJB equation admits a unique mild solution, constructed by Banach's fixed point theorem on the product space $\mathfrak{C}(y)=\mathscr{P}(|y|^{r-1}y)$7 using the resolvent estimates above. In the appendix, for the quadratic running cost $\mathfrak{C}(y)=\mathscr{P}(|y|^{r-1}y)$8 inside the ball and $\mathfrak{C}(y)=\mathscr{P}(|y|^{r-1}y)$9 outside, an explicit verification argument yields existence of an optimal pair with feedback law $m$00 when $m$01, saturated at norm $m$02 otherwise, and optimal cost equal to the value function $m$03.

Limitations and open questions

Several qualifications attach to these results. The essential $m$04-dissipativity theorem covers only the absorption regimes $m$05 (2D), $m$06 and critical $m$07 with $m$08 (3D); whether the argument extends to subcritical exponents $m$09 without exponential moments remains open. The HJB solvability condition couples the Lipschitz norm of the Hamiltonian to the discount factor, so large-gain or small-discount regimes fall outside the contraction argument. The closability of $m$10 in $m$11 is left unresolved, restricting the regularity theory available for mild solutions. Finally, the treatment is confined to additive jump noise with spatially constant coefficient $m$12; multiplicative Lévy perturbations would require derivative estimates for the noise channel that are not developed here.

Conclusion

The paper establishes the essential $m$13-dissipativity of the Kolmogorov operator for the SCBF system driven by additive Lévy noise, apparently for the first time in the jump-noise setting for equations of Navier–Stokes type. Its methodological contribution is to replace exponential moment estimates — unavailable or prohibitively difficult for Lévy-driven fluid models — with coercivity supplied directly by the Forchheimer absorption term, thereby simplifying the proof that $m$14 closes $m$15. The resulting carré du champ identity, resolvent bounds, and perturbation theory support a complete mild-solution theory for the infinite-dimensional stationary HJB equation and an explicit optimal feedback law, providing a systematic template applicable to other dissipative SPDEs with strong absorption nonlinearities.