A quantitative primal-dual certificate for stochastic Mather measures

Definitions and mathematical statement

Diogo Gomes's source question asks for extensions of Aubry--Mather and Monge--Kantorovich techniques to infinite-dimensional linear programming, citing stochastic Mather measures for second-order Hamilton--Jacobi equations as an example. It specifies no single class of linear programs. The aid supplies the following restricted smooth, uniformly diffusive, quadratic model on $\mathbb T^d$.

Fix $\nu>0$, smooth $b:\mathbb T^d\to{\mathbb{R}}^d$ and $U:\mathbb T^d\to{\mathbb{R}}$, and set $L(x,v)=\tfrac12|v-b(x)|^2+U(x)$. A stochastic-holonomic occupation measure is a Borel probability $\mu$ on $\mathbb T^d\times{\mathbb{R}}^d$ with finite second velocity moment satisfying \[ \int(\nu\Delta\phi(x)+v\cdot D\phi(x))\,{\,\mathrm{d}}\mu(x,v)=0 \quad(\phi\in C^\infty(\mathbb T^d)). \] Let $J(\mu)=\int L\,{\,\mathrm{d}}\mu$ and $\beta=\inf_{\mu\text{ feasible}}J(\mu)$. With these explicit sign and diffusion conventions, the aid's attained Hamilton--Jacobi dual is \[ \nu\Delta u+b\cdot Du-\tfrac12|Du|^2+U=\beta, \qquad \beta=\sup_{w\in C^2(\mathbb T^d)}\min_x \left(\nu\Delta w+b\cdot Dw-\tfrac12|Dw|^2+U\right). \] It reports a smooth corrector $u$, obtained from a principal positive eigenfunction, and no duality gap. The corrector is understood modulo constants.

Set $a(x)=b(x)-Du(x)$, and let $m>0$ be the normalized invariant density \[ \nu\Delta m-\operatorname{div}(a m)=0,\qquad\int_{\mathbb T^d}m\,{\,\mathrm{d}} x=1. \] The reported unique primal optimizer is $\mu_*=m(x)\,{\,\mathrm{d}} x\,\delta_{a(x)}({\,\mathrm{d}} v)$. For every feasible finite-cost $\mu$ the exact certificate is \[ J(\mu)-\beta=\tfrac12\int|v-a(x)|^2\,{\,\mathrm{d}}\mu(x,v). \] This measures velocity deviation from the feedback graph, not an unspecified distance between probability measures. The aid also reports, for a fixed admissible Hölder exponent $0<\alpha<1$ and the corresponding elliptic Poisson estimate, \[ \|\pi_{x\#}\mu-m\,{\,\mathrm{d}} x\|_{(C^\alpha)^*} \le C\,[J(\mu)-\beta]^{1/2}, \] where $\pi_x$ is spatial projection and the dual norm is the supremum of absolute integration against test functions of $C^\alpha$ norm at most one. The constant depends on the fixed model and exponent, not on $\mu$; the supplied summary does not state its quantitative dependence or the exact regularity range. This model-specific certificate does not resolve the source's general infinite-dimensional-programming agenda.

Short English statement

The broad question seeks variational and transport tools for infinite-dimensional linear programs. The supplied result handles a positive-diffusion quadratic stochastic-Mather model: its primal and dual attain the same value, and objective excess exactly measures departure from the optimal feedback graph and controls the spatial marginal.

Sources

• Local UnsolvedMath registry, permanent ID 20002490, source code catalogue entry; created and updated 14 August 2026. Canonical aim-pdes-notes.json, zero-based notes[101], tag problem; background/extracted item 4 attributed to Diogo Gomes.
• AIM "New connections between dynamical systems and PDE's," version dated 15 August 2003: https://aimath.org/WWN/dynpde/dynpde.pdf. The supplied verification identifies the official item as Problem 34, rather than the truncated extracted 4. Wording is marked exact; original remarks and literature field are empty.
• Research attempt 1: partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low. No complete authoritative paper citation or full theorem statement for the quantitative model is supplied.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.