Pathological Large Deviations of the KMP Process in Dimension $d\ge 2$

Technical Summary: Pathological Large Deviations of the KMP Process in Dimension $d \ge 2$

Problem Statement
This paper investigates the dynamic large deviations of the Kipnis–Marchioro–Presutti (KMP) process, a canonical Markov process modeling heat conduction on a discrete torus $\mathbb{T}^d_N$. The central object of study is the rate function governing the probability of rare trajectories deviating from the macroscopic heat equation $\partial_t u = \frac{1}{2}\Delta u$.

Previous formal analyses (e.g., Bertini, Gabrielli, Lebowitz) suggested a candidate rate function $I_\rho$ derived from a gradient flow structure involving a Fokker-Planck-like equation. However, the KMP process presents a unique challenge: while its diffusivity is linear, its mobility is quadratic. This contrasts with exclusion or zero-range processes where diffusivity and mobility are governed by the same function. The paper addresses whether the formal gradient flow structure fully characterizes the large deviations or if "pathological" trajectories—those lacking the regularity required to solve the associated skeleton equation—can occur with finite rate.

Methodology
The authors employ a combination of probabilistic large deviation techniques and the analysis of ill-posed partial differential equations (PDEs).

1. Restricted Lower Bound (Theorem 1):
   To establish a lower bound for the rate function on sufficiently regular paths ($u \in \mathcal{R}$), the authors utilize the tilting (change of measure) method. A significant technical hurdle is the lack of a "superexponential replacement estimate" for the quadratic mobility term, which is standard in other interacting particle systems.

   • Resolution: Instead of relying on a global superexponential estimate, the authors construct a specific change of measure $Q_{N,K}$ with a cutoff $K$ on the energy. By exploiting the regularity of the target path $u$, they prove an $L^2_{t,x}$ bound on the empirical measure under $Q_{N,K}$. This allows them to bypass the need for a superexponential estimate for unbounded observables, validating the lower bound for regular paths without requiring the full machinery usually needed for the upper bound.

2. Construction of Pathological Trajectories (Theorem 2):
   The core of the work involves constructing sequences of smooth paths $u_n$ that converge to singular limits $\Xi$ while maintaining a bounded (or vanishing) dynamic cost $J(u_n)$.

   • Skeleton Equation Analysis: The authors analyze the "skeleton" equation $\partial_t u = \frac{1}{2}\Delta u - \nabla \cdot (ug)$, where $g \in L^2_{t,x}$ is the control. They demonstrate that the balance between linear diffusion and quadratic mobility allows for the concentration of mass (spikes) that can move or appear/disappear with finite cost.
   • Dimensional Dependence: The construction relies on the strength of Sobolev embeddings and entropy dissipation estimates:
     • $d=1$: The cost of creating a singularity (a Dirac mass) is finite.
     • $d=2$: The Moser-Trudinger inequality allows for the construction of paths with countably many jumps (discontinuities in time) with finite cost.
     • $d \ge 3$: The entropy dissipation is no longer lower semicontinuous in the weak topology. This allows the construction of arbitrary relaxed measures (including those with singularities) that can be approximated by smooth paths with vanishing dynamic cost.

Key Results

• Theorem 1 (Restricted Lower Bound): For any dimension $d \ge 1$, the large deviation lower bound holds for all sufficiently regular paths $u$ (smooth, strictly positive). On this set, the candidate rate function $I_\rho$ is valid, and the dynamic cost $J$ coincides with the formal Fokker-Planck cost $J_{FP}$.
• Theorem 2 (Pathological Trajectories): The candidate rate function $I_\rho$ fails to describe all possible large deviations in dimensions $d \ge 2$.
  • $d=1$: There exist paths with a single instantaneous singularity (a Dirac mass appearing at time $t_0$) that occur with finite rate.
  • $d=2$: There exist paths with countably many instantaneous jumps (discontinuities in the weak topology of measures) that occur with finite rate.
  • $d \ge 3$: There exist paths converging to arbitrary relaxed measures (satisfying minimal energy bounds) with zero dynamic cost. This implies that in high dimensions, the large deviation principle is dominated by these singular trajectories, and the standard rate function is effectively zero for a vast class of non-smooth paths.

Significance and Claims
The paper claims that the obstacles preventing a full large deviation principle for the KMP process are not merely technical but fundamental.

• Failure of Formal Structure: The formal gradient flow structure, which successfully describes large deviations for other models (like the zero-range process), does not govern the KMP process in dimensions $d \ge 2$. The "pathological" trajectories (singularities and jumps) are not artifacts of the proof but are genuine large deviation events with finite (or zero) rate.
• Dimensional Sensitivity: The results highlight a sharp dependence on dimension. While the process is physically well-defined for all $d \ge 1$, the nature of the large deviations changes drastically: from singularities in $d=1$, to jumps in $d=2$, to a complete breakdown of the rate function's ability to penalize irregularity in $d \ge 3$.
• Methodological Contribution: The work provides a rigorous validation of the lower bound without superexponential estimates by tailoring the change of measure to the regularity of the target path. Furthermore, it connects the large deviation problem to the ill-posedness of parabolic-hyperbolic equations with rough drift, showing how the lack of compactness in higher dimensions permits the construction of trajectories that evade the standard regularity constraints.

The authors conclude that the large deviations of the KMP process in $d \ge 2$ include trajectories that are not continuous in time (in the weak topology) and cannot be interpreted as solutions to the skeleton equation, thereby demonstrating that the formal gradient flow structure is insufficient to describe the full landscape of rare events.