Stochastic Homogenization of Non-local Hamilton-Jacobi-Bellman equations

Technical Summary: Stochastic Homogenization of Non-Local Hamilton-Jacobi-Bellman Equations

Problem Statement
This paper addresses the stochastic homogenization of non-local Hamilton-Jacobi-Bellman (HJB) equations in a stationary ergodic random medium. The specific equation under study is:
$$\begin{cases} \partial_t u^\epsilon(t, x) = \epsilon^{\alpha-1} \mathcal{L} u^\epsilon(t, x) + H\left(\nabla u^\epsilon(t, x), \frac{x}{\epsilon}, \omega\right), & (t, x) \in [0, \infty) \times \mathbb{R}^n, \\ u^\epsilon(0, x) = u_0(x), & x \in \mathbb{R}^n, \end{cases}$$
where $\alpha \in (1, 2)$, $\omega$ represents an element of a probability space $(\Omega, \mathcal{F}, P)$, and $H$ is a convex, super-linearly growing Hamiltonian with rapid random oscillations. The operator $\mathcal{L}$ is a non-local integro-differential operator defined by a Lévy measure of the form $K(\hat{y})|y|^{-n-\alpha}dy$, where $K$ is a smooth, non-negative, even, and uniformly non-degenerate function on the unit sphere. This operator serves as the infinitesimal generator of an $\alpha$-stable type jump-diffusion process.

The objective is to prove that as $\epsilon \to 0$, the solution $u^\epsilon$ converges locally uniformly (almost surely) to the solution $u$ of a deterministic homogenized equation:
$$\begin{cases} \partial_t u(t, x) = \bar{H}(\nabla u(t, x)), \\ u(0, x) = u_0(x), \end{cases}$$
where $\bar{H}$ is a deterministic, convex effective Hamiltonian independent of the spatial variable $x$ and the random parameter $\omega$.

Methodology
The authors adapt the method developed by Kosygina, Rezakhanlou, and Varadhan (CPAM 2006) for local HJB equations with vanishing Laplacian terms to the non-local setting involving jump-diffusions. The methodology relies on three main pillars:

1. Stochastic Optimal Control Representation: The solution $u^\epsilon$ is represented via a stochastic control problem driven by a jump-diffusion process. The authors establish that the non-local operator $\mathcal{L}$ can be rewritten in a "divergence form" $\mathcal{L}u = \nabla \cdot (\mathcal{R}\nabla u)$, where $\mathcal{R}$ is a regular integral operator. This structural characterization is crucial for handling the non-local terms in the homogenization process.

2. Environmental Process and Effective Hamiltonian: Following the approach of [24], the authors lift the jump-diffusion processes from the physical space $\mathbb{R}^n$ to the probability space $\Omega$ to define an "environmental process" $\theta_t$. Using the ergodicity of this process, they define the effective Lagrangian $\bar{L}$ and subsequently the effective Hamiltonian $\bar{H}$ via a variational formula involving invariant measures of the environmental process.

3. Approximate Super-Correctors: To establish the upper bound of the convergence (the difficult direction), the authors construct approximate super-solutions (super-correctors). This involves finding a random vector field $v$ such that a specific inequality involving the non-local divergence operator $J$ and the Hamiltonian holds:
   $$Jv(\omega) + H(p + v(\omega), \omega) \leq \bar{H}(p).$$
   The construction of these correctors is the primary technical novelty, requiring the handling of the non-local operator's singularity and the lack of standard elliptic regularity.

Key Contributions and Results

• Main Theorem (Theorem 1.1): Under standard stationary ergodic assumptions and convexity/super-linearity conditions (Assumption A), the paper proves the almost sure locally uniform convergence of $u^\epsilon$ to $u$. The result holds under either of two additional technical assumptions:

  • Assumption (B): A specific convexity-continuity condition on the Hamiltonian $H$ (related to separation of variables).
  • Assumption (C): A condition requiring the growth exponent $\beta$ of the Hamiltonian to satisfy $\beta > n$, which allows for the application of Sobolev embedding theorems.

• Divergence Form Characterization: The paper explicitly characterizes the non-local operator $\mathcal{L}$ as the divergence of a regular integral operator acting on the gradient (Section 2.2). This representation, $\mathcal{L} = J \circ \nabla = \nabla \cdot \mathcal{R} \circ \nabla$, is essential for defining the correctors in the random environment.

• Regularity Estimates: The authors derive explicit regularity estimates for the solution $u^\epsilon$ with respect to space and time, showing dependence on $\epsilon$ (Theorem 4.1). These estimates are used to upgrade pointwise convergence to locally uniform convergence.

• Construction of Super-Correctors: The paper details the construction of approximate super-correctors for the non-local case. Unlike the local case where correctors are often constructed via PDE methods, here the authors utilize convex analysis on the space of random variables and mollification techniques to handle the non-local operator $J$.

Significance and Claims
The paper claims to be the first work addressing the stochastic homogenization of non-local HJB equations with vanishing non-local terms in a stationary random environment.

• Extension of Existing Theory: The work extends the seminal results of Kosygina, Rezakhanlou, and Varadhan [24] from local (diffusion) settings to non-local (jump-diffusion) settings.
• Methodological Adaptation: It demonstrates that the control representation and environmental process techniques, originally designed for vanishing Laplacians, can be successfully adapted to jump processes. The authors note that while viscosity solution theory for non-local HJB equations has been studied recently (e.g., [6, 17, 13]), their approach relies primarily on stochastic control representations and abstract environmental processes rather than deep viscosity solution theory.
• Scope and Limitations: The authors maintain a modest scope, focusing on qualitative homogenization. They explicitly state that they do not address the rate of convergence (which is known to be difficult even in the local case) nor the case of dynamic random environments (time-dependent coefficients), though they suggest these as natural directions for future work. The result is restricted to the case where the non-local term vanishes ($\epsilon^{\alpha-1} \mathcal{L}$) as $\epsilon \to 0$.

In summary, the paper establishes a rigorous foundation for the homogenization of non-local HJB equations driven by jump-diffusions, bridging the gap between stochastic control theory for jump processes and homogenization theory in random media.