Pathwise well-posedness of fully nonlinear stochastic Hamilton–Jacobi equations with spatially inhomogeneous noise and nonsmooth Hamiltonians

The Lions–Souganidis pathwise viscosity theory (initiated in 1998) gives existence, uniqueness, and stability for fully nonlinear stochastic PDEs of the form $du = F(D^2u,Du)\,dt + \sum_i H^i(Du)\circ dW_i$ when $F$ is continuous, degenerate elliptic, and satisfies a mild structural condition, and each Hamiltonian $H^i$ is Lipschitz and a difference of two convex functions, with the noise spatially homogeneous. The unfinished part of that program — identified in the authors' own 2018/2019 overview as "the next step in the development of the theory" and still open — is the extension to spatially inhomogeneous equations $du = F(D^2u,Du,u,x)\,dt + \sum_i H^i(Du,x)\circ dW_i$ with spatially regular (in $x$) driving noise at the same generality of nonsmoothness. Well-posedness is currently known in the spatially dependent setting only for smooth Hamiltonians under growth restrictions and for convex Hamiltonians, while the difference-of-convex theory in the spatially dependent case and Hamiltonians depending jointly on $u$ and $Du$ remain open, with counterexamples already delimiting parts of the boundary.

Background

Hamilton–Jacobi equations are first-order nonlinear evolution equations such as $u_t + H(Du) = 0$, where $Du$ denotes the spatial gradient and $H$ the Hamiltonian; their second-order fully nonlinear analogues $u_t = F(D^2u,Du,u,x,t)$, with $D^2u$ the Hessian matrix, include the level-set equations for interface motion — e.g. mean curvature flow, the motion of an interface with normal velocity equal to its mean curvature, which is tracked by solving an equation for a function whose zero level set is the interface. Since classical solutions of such equations generally break down, the standard notion of solution is the viscosity solution: a locally bounded function satisfying appropriate differential inequalities when tested against smooth functions at local maxima and minima. The central tool of the theory is the comparison principle — the statement that a subsolution lies below a supersolution with ordered data, from which uniqueness follows; for second-order equations it requires degenerate ellipticity, i.e. $F(X,p) \le F(Y,p)$ whenever $X \le Y$ as symmetric matrices. When the equation is forced by a Brownian motion $W=(W^1,\dots,W^m)$, the natural objects are stochastic PDEs (SPDEs) written in Stratonovich form, $\circ\,dW$, where the Stratonovich (pathwise) interpretation is the limit of equations driven by smooth approximations of the noise path.

In 1998 Lions and Souganidis founded a pathwise theory of stochastic viscosity solutions for fully nonlinear SPDEs

$$du = F(D^2u,Du)\,dt + \sum_{i=1}^m H^i(Du)\circ dW_i, \qquad u(\cdot,0)=u_0,$$

announced in two notes in the Comptes Rendus: the first (C. R. Acad. Sci. Paris 326 (1998) 1085–1092) introducing the theory, the second being the source note for the present problem. The source note, 'Fully nonlinear stochastic partial differential equations: non-smooth equations and applications' (C. R. Acad. Sci. Paris 327 (1998) 735–741), extends the theory to nonsmooth Hamiltonians: its main theorem states that if the initial datum $u_0$ is bounded and uniformly continuous, $F$ is continuous and degenerate elliptic and satisfies a mild structural condition on doubled variables (the doubling-of-variables technique, standard in comparison proofs, introduces a function of two copies of the space variable; the condition is a degenerate-elliptic function $G$ controlling $F(X,p)-F(Y,p)$ in the doubled test), and each $H^i$ is Lipschitz and can be written as a difference of two convex functions (a DC Hamiltonian), then the solutions of the equations driven by smooth path approximations are Cauchy and converge, uniformly in $(x,t)$ and almost surely in the noise realization, to a unique solution depending continuously on the driving path. The DC structure is what lets the argument bypass smoothness of $H$ via the convexification mechanism of level-set front-propagation theory. The note's abstract announces applications to pathwise stochastic control and to the propagation of fronts in random environments, and the note defers the full treatment of the spatially homogeneous ($x$-independent) theory to companion papers listed in its reference list as 'in preparation'; the note's abstract and reference list do not document a deferral of the spatially inhomogeneous extension, which is instead announced in the authors' own 2018/2019 overview described below.

The subsequent development of the program, consolidated in Souganidis's 2018/2019 overview 'Pathwise Solutions for Fully Nonlinear First- and Second-Order Partial Differential Equations with Multiplicative Rough Time Dependence', is as follows. For spatially homogeneous noise the theory is essentially complete: for smooth $H$ and for DC nonsmooth $H$; existence by Perron's method (the construction of the solution as the envelope of subsolutions, avoiding explicit approximation schemes; Seeger 2018); convergent approximation schemes with error estimates (Seeger 2020); Neumann boundary problems (Gassiat–Seeger 2024); and eikonal equations — the front-propagation equations $u_t = c(x)|Du|$ — with quadratic Hamiltonians tied to Riemannian geometry (Friz–Gassiat–Lions–Souganidis 2017). Moreover, Lions–Seeger–Souganidis (Indiana Univ. Math. J. 2022) proved that DC Hamiltonians form the largest class for which the pathwise Hamilton–Jacobi problem is well-posed for all continuous driving paths, so within the spatially homogeneous, all-paths setting the 'beyond DC' question is closed negatively. For noise that is linear in $Du$ (semilinear stochastic dependence), spatial dependence of the noise is also well understood, via Doss–Sussmann-type transformations (changes of variables that remove noise entering linearly through the gradient) in the work of Buckdahn and Ma, recast with rough paths — the deterministic theory of equations driven by signals encoded through their iterated integrals — by Caruana–Friz–Oberhauser (2011), and unified with path-dependent PDEs (PDEs whose state includes the whole past of the driving path) by Buckdahn–Keller–Ma–Zhang (2020), where however a full comparison principle is available only in the semilinear case.

The genuinely nonlinear spatially inhomogeneous case is where the theory stops. For $du = F(D^2u,Du,u,x)\,dt + \sum_i H^i(Du,x)\circ dW_i$ (equivalently, multiplicative noise whose coefficient is regular in $x$), well-posedness is established only for smooth Hamiltonians under growth restrictions — essentially quadratic growth in $p$, with special structure needed for faster growth, and with a Lions–Souganidis argument for multiple Brownian paths — and for convex spatially dependent Hamiltonians, where Gassiat–Gess–Lions–Souganidis (2019, 2024) proved finite speed of propagation (solutions at a point depend, uniformly in the path, only on data in a bounded cone) and long-time convergence. On the obstruction side, Gassiat (2017) constructed a stochastic Hamilton–Jacobi equation with a nonconvex Hamiltonian and infinite speed of propagation, already in the spatially homogeneous setting, showing that the domain-of-dependence and cancellation structure fails without convexity; the same authors' 2024 paper explicitly lists non-convex Hamiltonians, multiple spatially inhomogeneous noises, and mean-curvature flow with spatially inhomogeneous noise among the open questions. The nonsmooth (difference-of-convex) spatially dependent theory remains unsettled: the 2018 overview describes the spatially dependent extension as 'the next step in the development of the theory' and defers its nonsmooth details to a Lions–Souganidis manuscript 'in preparation' that has not appeared in the literature to date. Hamiltonians depending jointly on $u$ and $Du$ are separately stated in the overview — in its spatially homogeneous, smooth setting — to be 'an open problem' apart from special cases.

Problem Statement

Establish or refute well-posedness of pathwise viscosity solutions — existence, comparison (hence uniqueness), and continuous dependence on the driving path — for the spatially inhomogeneous, fully nonlinear initial value problem

$$du = F(D^2u, Du, u, x)\,dt + \sum_{i=1}^m H^i(Du, x)\circ dW_i \quad\text{in } \mathbb{R}^d \times (0,\infty), \qquad u(\cdot,0)=u_0,$$

formulated on $\mathbb{R}^d$, as in the 1998 theory and in the spatially dependent results of the 2018/2019 overview, where $W=(W^1,\dots,W^m)$ is an $m$-dimensional Brownian motion (or, more generally, a continuous driving path), $F$ is continuous and degenerate elliptic (with the argument list $(D^2u,Du,u,x)$ as displayed, so $F$ may depend on the state $u$ and the spatial variable $x$), and the Hamiltonians $H^i(p,x)$ are nonsmooth in $p$ at the generality of the 1998 spatially homogeneous result — the natural target class being $H^i$ Lipschitz in $p$ with $H^i(\cdot,x)$ a difference of two convex functions locally uniformly in $x$ (the coefficients $H^i(p,x)$ being regular in the $x$-variable, which is the sense in which the driving noise is spatially regular; the exact $x$-regularity class is part of what a solution must make precise — the published smooth spatially dependent theory uses explicit bounded-derivative assumptions, while the nonsmooth spatially dependent theory was deferred to an unpublished manuscript — at the level required by the test-function and doubled-equation constructions). This is the extension of the Lions–Souganidis pathwise theory beyond its spatially homogeneous difference-of-convex hypotheses that the 2018/2019 overview describes as 'the next step in the development of the theory', deferred there to the unpublished Lions–Souganidis manuscript 'Pathwise solutions for nonlinear partial differential equations with rough signals' (in preparation); it has not been executed in the published literature. Where full well-posedness fails, a refutation by an explicit counterexample within the corresponding spatially dependent nonsmooth setting — two distinct pathwise viscosity solutions for the same data and driving path, or two sequences of smooth path approximations converging uniformly to the same driving path whose solutions converge to different limits — settles the same target. The completion boundary is: a well-posedness theorem covering the stated class, or such an explicit counterexample refuting well-posedness within the same setting.

References

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