Open core of the stochastic regularization program for pathwise Hamilton–Jacobi equations: the law of the asymptotic random constant $u∞$, $C^{1,1}$ regularization in $d>1$, and exceptional time sets beyond Brownian driving paths

Lions and Souganidis (2020) proved sharp, Hamiltonian-independent regularity estimates for viscosity solutions of uniformly convex Hamilton–Jacobi equations and used them to obtain intermittent stochastic regularization and long-time convergence of pathwise (stochastic) viscosity solutions of $du=H(Du)·dζ$ driven by a continuous rough path $ζ$. The sharp theory is confined to three regimes: one space dimension, pure-quadratic Hamiltonians, and Brownian driving paths. Three linked questions remain open. (1) The distribution of the asymptotic random constant $u∞$, to which space-periodic solutions converge almost surely, is unknown for general uniformly convex $H$ — even in one space dimension with a single Brownian noise — and is characterized only for $H(p)=|p|$, where it is the pushforward of normalized Lebesgue measure by the initial datum. (2) No mechanism is known that produces sharp two-sided weighted $C^{1,1}$ bounds in dimension $d>1$ for non-quadratic uniformly convex $H$; unconditional sharp propagation provably holds if and only if $H$ is a pure quadratic. (3) The exceptional time sets at which the intermittent bounds degenerate are quantified only for Brownian paths, where the set has Hausdorff dimension $1/2$; nothing is known for other classes of rough paths. The 2024 follow-up by Gassiat, Gess, Lions and Souganidis resolves one special case of (1) and explicitly lists the characterization of the random limiting constant among the most intriguing open questions.

Background

The deterministic estimates. A Hamilton–Jacobi (HJ) equation is a first-order fully nonlinear PDE $u_t=±H(Du)$ on $ℝ^d×(0,∞)$, where $u_t$ and $Du$ denote the time derivative and spatial gradient of $u$ and $H:ℝ^d→ℝ$ is the Hamiltonian. Its weak solutions are understood in the viscosity sense of Crandall and Lions: sub- and supersolutions are characterized by inequalities imposed on smooth test functions touching $u$ from above or below, a definition under which comparison and uniqueness hold. Throughout, $H$ is uniformly convex, i.e. $H∈C²$ and $θI≤D²H(p)≤ΘI$ for all $p$ with constants $0<θ≤Θ$ ($I$ the identity matrix, $D²H$ the Hessian); the solved special case $H(p)=|p|$ used below lies outside this class, being degenerate (not $C²$ at the origin, with $D²H$ not bounded below by $θI$). The classical regularizing effect, obtained from the Lax–Oleinik variational representation of solutions, says that solutions instantly become semiconcave — roughly, $−D²u(·,t)≤I/(θt)$ — but these bounds depend on the convexity constants $θ,Θ$ and blow up under iteration: alternating forward and backward evolutions over time increments $a,b$ produces factors like $1/(θa−Θb)_+$, which cannot be controlled for arbitrary increments. Lions and Souganidis (2020) corrected this defect with sharp estimates in the intrinsic weight $F(p)=√(D²H(p))$ (the matrix square root of the Hessian): $−F(Du)D²uF(Du)≤I/t$ with constants independent of $θ$ and $Θ$, together with a two-sided propagation principle. Crucially, unconditional propagation — valid from arbitrary merely Lipschitz initial data — holds in exactly two situations: one space dimension, or pure-quadratic Hamiltonians $H(p)=(Ap,p)$; in $d>1$ for non-quadratic $H$ it provably fails (an if-and-only-if theorem), while from $C^{1,1}$ initial data (bounded second derivatives) propagation still holds. Pathwise (stochastic) viscosity solutions and intermittent regularization. The estimates are tailored to equations with rough multiplicative time dependence $du=H(Du)·dζ$ in $ℝ^d×(0,∞)$, where $ζ$ is merely a continuous path starting at $0$ — for instance a Brownian sample path; for semimartingale $ζ$ the dot is the Stratonovich differential. The pathwise (stochastic) viscosity solutions of Lions and Souganidis are defined as uniform limits of solutions driven by smooth approximations of $ζ$, which makes the equation well posed for every continuous path. Since such a $ζ$ is monotone on no usable partition of time, the solution is governed by alternating compositions of the forward and backward deterministic solution operators, so regularity must survive arbitrarily fast sign changes of the forcing — precisely what the Hamiltonian-independent estimates deliver. Writing $M(t)=max_{s≤t}ζ(s)$ and $m(t)=min_{s≤t}ζ(s)$ for the running maximum and minimum, the intermittent regularization theorem states that, for $d=1$ or quadratic $H$, $−I/(M(t)−ζ(t))≤−F(Du(·,t))D²u(·,t)F(Du(·,t))≤I/(ζ(t)−m(t))$ in the viscosity sense: at every time $t$ with $m(t)<ζ(t)<M(t)$ the solution is $C^{1,1}$ with bounds depending only on the oscillation of the path. For general uniformly convex $H$ in $d>1$ only the weaker Lipschitz bound $‖Du(·,t)‖∞≤√(2‖u(·,t)‖∞/(θ(M(t)−m(t))))$ survives. For Brownian $ζ$ the degenerate times are exactly those at which the path sits at its running maximum or minimum; classical fluctuation results — for which Peres's theorem on points of increase of random walks is the standard reference — show that this exceptional set is uncountable, has no isolated points, and has Hausdorff dimension $1/2$ (Hausdorff dimension being the standard fractal size notion), so regularization holds off a set of times of dimension $1/2$. Long-time behavior. If the path oscillates without bound in the sense $M(t_n)−m(t_n)→∞$, the Lipschitz bound forces space-periodic solutions to converge uniformly along $t_n$ to a constant. For Brownian forcing this holds almost surely, and the limit $u∞$ is genuinely random: its law is nondegenerate, i.e. not a point mass, as shown by an explicit one-dimensional example with $H(p)=|p|$ and piecewise-linear periodic initial data. The quadratic-Hamiltonian predecessor of this program is the regularization-by-noise result of Gassiat and Gess (2019). The natural ergodic-theoretic question — what is the law of $u∞$, and how does it depend on the initial datum and the Hamiltonian? — is answered in a single special case: Gassiat, Gess, Lions and Souganidis (2024) proved for $H(p)=|p|$ in one space dimension that $Law(u∞)=(u₀)_*dx$, the pushforward of normalized Lebesgue measure on the torus under the initial datum $u₀$ (equivalently, the distribution of $u₀(X)$ for $X$ uniformly distributed on the torus), exploiting that monotone transformations of solutions are again solutions — a structural property general Hamiltonians lack. The same work establishes qualitative almost-sure convergence to constants in any dimension for multiple spatially homogeneous noises (by a mechanism not based on $C^{1,1}$ regularization), convergence for spatially inhomogeneous convex Hamiltonians toward statistically stationary solutions, quantitative decay rates in one dimension, and accelerated decay of oscillations for the stochastically forced mean-curvature flow (the level-set equation for interfaces moving by mean curvature with stochastic forcing). Its closing section explicitly lists the characterization of the random limiting constant among the most intriguing open questions of the subject. What remains is the open core of the program: extending the sharp theory beyond the three regimes — one space dimension, quadratic Hamiltonians, Brownian driving paths — in which every currently known sharp statement lives.

Problem Statement

The common scientific objective is to complete the sharp theory of intermittent stochastic regularization and its long-time consequences for pathwise (stochastic) viscosity solutions of $du=H(Du)·dζ$ with convex Hamiltonians, beyond the three regimes in which every sharp statement currently lives: one space dimension, pure-quadratic Hamiltonians, and Brownian driving paths. Throughout, solutions are space-periodic and $ζ$ is a continuous driving path. Three linked targets: (1) Law of the asymptotic random constant. Determine the distribution of $u∞$ — and its finer statistical properties, such as its dependence on the initial datum $u₀$ and on the Hamiltonian — for general uniformly convex Hamiltonians, beyond the special case $H(p)=|p|$ where $Law(u∞)=(u₀)_*dx$. This is open even in one space dimension with a single Brownian noise and a general uniformly convex $H$: the known facts are almost-sure existence of $u∞$, nondegeneracy of its law in general, the $|p|$ identification $Law(u∞)=(u₀)_*dx$, and — for one-dimensional $H$ a difference of two convex functions (a class that contains every convex $H$) driven by an almost-surely unbounded continuous martingale — the first-moment identity $𝔼[u∞]=∫_𝕋u₀dx$ (Theorem 4.1 of the 2024 paper) and the delimitation, in its Remark 4.2, that there is, in general, no additional information on $u∞$. Martingale and non-semimartingale driving paths, several spatially homogeneous noises, and $d>1$ are all of interest. (2) $C^{1,1}$ regularization in several dimensions. For $d>1$ and non-quadratic uniformly convex $H$, determine whether any mechanism — stochastic, iterative, or of another kind — produces sharp Hamiltonian-independent two-sided bounds on $−F(Du)D²uF(Du)$ from initial data that are merely Lipschitz. Since unconditional sharp propagation holds if and only if $H$ is a pure quadratic, an affirmative answer would require a mechanism circumventing this obstruction. (3) Exceptional time sets for general driving paths. Describe quantitatively — by Hausdorff dimension or measure, and by prevalence in time — the exceptional set of times at which the intermittent bounds degenerate, i.e. the set of times at which the driving path sits at its running maximum or minimum, $\{t:ζ(t)\in\{M(t),m(t)\}\}$, and correspondingly the typical-time regularity wherever intermittent bounds are available, for driving paths beyond Brownian motion (whose exceptional set has Hausdorff dimension $1/2$). Natural classes include other continuous semimartingales, fractional Brownian paths, and non-semimartingale rough paths. Each of the three targets is independently open and independently resolvable, and within a target any single one of the listed classes of driving paths is of interest.

References

1. Pierre-Louis Lions and Panagiotis E. Souganidis, New regularity results for Hamilton–Jacobi equations and long time behavior of pathwise (stochastic) viscosity solutions, Research in the Mathematical Sciences 7 (2020), no. 3, article 17, DOI 10.1007/s40687-020-00214-7, https://doi.org/10.1007/s40687-020-00214-7 (preprint: arXiv:1909.05672, https://arxiv.org/abs/1909.05672).
2. Paul Gassiat, Benjamin Gess, Pierre-Louis Lions, and Panagiotis E. Souganidis, Long-time behavior of stochastic Hamilton-Jacobi equations, Journal of Functional Analysis 286 (2024), no. 4, article 110269, DOI 10.1016/j.jfa.2023.110269, https://doi.org/10.1016/j.jfa.2023.110269 (preprint: arXiv:2211.12099, https://arxiv.org/abs/2211.12099).
3. Paul Gassiat and Benjamin Gess, Regularization by noise for stochastic Hamilton–Jacobi equations, Probability Theory and Related Fields 173 (2019), no. 3–4, 1063–1098, DOI 10.1007/s00440-018-0848-7, https://doi.org/10.1007/s00440-018-0848-7.
4. Paul Gassiat, Benjamin Gess, Pierre-Louis Lions, and Panagiotis E. Souganidis, Speed of propagation for Hamilton–Jacobi equations with multiplicative rough time dependence and convex Hamiltonians, Probability Theory and Related Fields 176 (2020), no. 1–2, 421–448, DOI 10.1007/s00440-019-00921-5, https://doi.org/10.1007/s00440-019-00921-5.
5. Yuval Peres, Points of increase for random walks, Israel Journal of Mathematics 95 (1996), no. 1, 341–347, DOI 10.1007/BF02761045, https://doi.org/10.1007/BF02761045.
6. Pierre-Louis Lions, Benjamin Seeger, and Panagiotis E. Souganidis, Interpolation results for pathwise Hamilton-Jacobi equations, Indiana University Mathematics Journal 71 (2022), no. 5, 2126–2193, DOI 10.1512/iumj.2022.71.9174, https://doi.org/10.1512/iumj.2022.71.9174 (preprint: arXiv:2007.02440, https://arxiv.org/abs/2007.02440).
7. Pierre Cardaliaguet and Benjamin Seeger, Hölder regularity of Hamilton-Jacobi equations with stochastic forcing, Transactions of the American Mathematical Society 374 (2021), no. 10, 7197–7233, DOI 10.1090/tran/8435, https://doi.org/10.1090/tran/8435.