Comparison and uniqueness of pathwise viscosity solutions for fully nonlinear stochastic PDEs with non-smooth (difference-of-convex) Hamiltonians

The Lions-Souganidis theory of stochastic viscosity solutions establishes well-posedness of $du=F(D^{2}u,Du)\,dt+\sum_{i=1}^{m}H_{i}(Du)\circ dW^{i}$ on $\mathbb{R}^{d}$ with bounded uniformly continuous initial data $u_{0}$ and $F$ continuous, degenerate elliptic, but the published comparison theorem requires smooth Hamiltonians ($H\in C^{3}$ in the 2000 source note; $H\in C^{2}$ in later expositions), with detailed proofs given only for $m=1$ and $F$ independent of $(x,t)$. The extension to non-smooth Hamiltonians, namely globally Lipschitz differences of two convex functions, the class containing the geometric Hamiltonian $H(p)=|p|$ and the weakest class for which the solution operator extends to arbitrary continuous driving paths, was announced by the authors in 2002 and is asserted in later peer-reviewed work, but the companion publication containing complete proofs never appeared; complete published proofs exist only under additional restrictions (smooth $H$; or convex domains with structural conditions and Lipschitz-type assumptions on $F$). The open problems are: (i) a complete comparison principle for the whole-space second-order problem with difference-of-convex (Lipschitz) Hamiltonians and merely continuous degenerate elliptic $F$, covering $m>1$ and $(x,t)$-dependent $F$ under the standard structural condition; and (ii) the minimal Hamiltonian regularity sufficient at Brownian path regularity, that is, whether $H\in C^{1}$ or even $H\in C^{0,1}$ without the difference-of-convex structure still yields uniqueness, a question explicitly posed as an open goal in the 2019 survey literature and in the 2022 interpolation paper.

Background

Consider the fully nonlinear stochastic PDE $du=F(D^{2}u,Du)\,dt+\sum_{i=1}^{m}H_{i}(Du)\circ dW^{i}$, where $W=(W^{1},\dots,W^{m})$ is an $m$-dimensional Brownian motion and $\circ\,dW^{i}$ denotes the Stratonovich stochastic differential, that is, the chain-rule-preserving form of the stochastic integral; here $F:\mathbb{S}^{d}\times\mathbb{R}^{d}\to\mathbb{R}$ is continuous and degenerate elliptic, where $\mathbb{S}^{d}$ is the space of real symmetric $d\times d$ matrices ordered by $X\leq Y$ when $Y-X$ is nonnegative definite and degenerate ellipticity means $X\leq Y$ implies $F(X,p)\leq F(Y,p)$, and the initial datum is $u_{0}\in BUC(\mathbb{R}^{d})$, the space of bounded uniformly continuous functions. Such equations arise in stochastic control and differential games and in front-propagation problems, including stochastically perturbed mean-curvature flow, where the Hamiltonian is $H(p)=|p|$. Classical SPDE theory does not apply: the equation is fully nonlinear in $D^{2}u$, so no Ito calculus in infinite-dimensional or variational form is available, and the multiplicative gradient noise cannot be removed by a change of variables precisely because of this nonlinearity. Lions and Souganidis (1998-2000) therefore introduced a pathwise, deterministic-for-fixed-noise notion of viscosity solution. (A viscosity solution is the standard weak notion for fully nonlinear equations, in which sub- and supersolutions are tested not by membership in a classical function space but by smooth test functions touching them from above or below.) In their pathwise version, for a fixed continuous path, sub- and supersolutions are tested not with smooth functions but with functions evolved by the solution operator $S(s,t)\varphi$ of the auxiliary stochastic Hamilton-Jacobi equation $dv=\sum_{i=1}^{m}H_{i}(Dv)\circ dW^{i}$, which is solved explicitly by stochastic characteristics, that is, by transporting the initial datum along the characteristic curves of the Hamiltonian flow carried out pathwise for the fixed noise realization, when the $H_{i}$ are smooth. The central well-posedness result is a comparison principle, from which existence and uniqueness, stability, and convergence of numerical schemes all follow; its proof requires a doubling-of-variables argument (the standard comparison device of testing the subsolution and the supersolution at a jointly varying pair of points $(x,y)$ coupled by a quadratic penalty that is then sent to zero) in which the test functions are transported by the stochastic characteristics and the usual sup/inf-convolution approximations (regularizations of a function as the pointwise supremum or infimum of quadratically penalized translates) are replaced by modified convolutions that commute with the characteristic flow, combined with Krylov-type estimates (measure bounds on the set where the Hessian of a semiconvex function is large) and an Aleksandrov-type maximum principle for parabolic expansions (a parabolic analogue of the Aleksandrov-Bakelman-Pucci bound on the maximum of a function in terms of the determinant measure of its Hessian). A key structural discovery is the role of Hamiltonians that are differences of two convex functions: for the first-order equation ($F\equiv 0$) it is proved that the solution operator extends to arbitrary continuous driving paths if and only if the Hamiltonians are of this difference-of-convex form, so this class is the largest well-posed one for rough paths; it is also the minimal setting needed for geometric equations with $H(p)=|p|$. A further regularity interplay is known: driving paths of Holder exponent $\alpha$ require $H\in C^{2(1-\alpha)+\varepsilon}$ for the smooth extension, while for Brownian paths ($\alpha=1/2$) the sufficiency of $H\in C^{1,\delta}$ (continuously differentiable with $\delta$-Hölder continuous derivative) is stated in the survey literature without proof, and whether $H\in C^{1}$ or even $H\in C^{0,1}$ (globally Lipschitz) suffices is an explicitly open goal. In the second-order setting, the published comparison theorems assume $H\in C^{2}$ (or $H\in C^{3}$ in the original note), and the difference-of-convex case is covered in the published record only on convex domains with structural restrictions (Gassiat-Seeger, Neumann problem), while the whole-space statement is asserted with attribution to unpublished Lions-Souganidis manuscripts. For context on deterministic viscosity-solution comparison, see the User's Guide of Crandall, Ishii, and Lions, whose structural condition (properness together with growth restrictions on the dependence of $F$ on $(u,x,t)$) is the model for the condition on $F$ needed when $F$ depends on $(x,t)$.

Problem Statement

Prove a comparison principle, and thereby uniqueness, of stochastic (pathwise) viscosity solutions of the Cauchy problem $du=F(D^{2}u,Du)\,dt+\sum_{i=1}^{m}H_{i}(Du)\circ dW^{i}$ on $\mathbb{R}^{d}$ with $u_{0}\in BUC(\mathbb{R}^{d})$, where $F$ is continuous and degenerate elliptic and is allowed to depend on $(x,t)$ under the standard structural condition (the analogue of the comparison condition in the User's Guide of Crandall, Ishii, and Lions), in the regime of non-smooth Hamiltonians: each $H_{i}$ is globally Lipschitz and a difference of two convex functions, the weakest class for which the solution operator extends to arbitrary continuous driving paths and the class containing the geometric Hamiltonian $H(p)=|p|$; the proof should cover $m\geq 1$. The complementary part of the problem, flagged by the authors themselves, is the minimal Hamiltonian regularity at Brownian path regularity: determine whether $H\in C^{1}$, or even $H\in C^{0,1}$ without the difference-of-convex structure, still yields uniqueness, noting that currently only $H\in C^{1,\delta}$, meaning continuously differentiable with $\delta$-Hölder continuous derivative (stated without proof), and the difference-of-convex case (announced, with complete proofs available only in restricted settings) are available. The 2000 source note established the theory for $H\in C^{3}$ with detailed proofs only for $m=1$ and $F$ independent of $(x,t)$ and deferred precisely these extensions to a companion publication that has not appeared, so both parts remain to be closed in the published record.

References

1. P.-L. Lions and P. E. Souganidis, Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Ser. I Math. 331 (2000), no. 10, 783-790. DOI: 10.1016/S0764-4442(00)01597-4, https://doi.org/10.1016/S0764-4442(00)01597-4
2. P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Ser. I Math. 326 (1998), no. 9, 1085-1092. DOI: 10.1016/S0764-4442(98)80067-0, https://doi.org/10.1016/S0764-4442(98)80067-0
3. P.-L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations: non-smooth equations and applications, C. R. Acad. Sci. Paris Ser. I Math. 327 (1998), no. 8, 735-741. DOI: 10.1016/S0764-4442(98)80161-4, https://doi.org/10.1016/S0764-4442(98)80161-4
4. P.-L. Lions and P. E. Souganidis, Viscosity solutions of fully nonlinear stochastic partial differential equations, RIMS Kokyuroku 1287 (2002) 58-65, https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1287-6.pdf
5. P. E. Souganidis, Pathwise solutions for fully nonlinear first- and second-order partial differential equations with multiplicative rough time dependence, in Singular Random Dynamics, Lecture Notes in Mathematics 2253, Springer (2019), pp. 75-220. DOI: 10.1007/978-3-030-29545-5_3, arXiv:1809.01748, https://arxiv.org/abs/1809.01748
6. B. Seeger, Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations, Ann. Appl. Probab. 30 (2020), no. 4. DOI: 10.1214/19-AAP1543, arXiv:1802.04740, https://arxiv.org/abs/1802.04740
7. P. Gassiat and B. Seeger, The Neumann problem for fully nonlinear SPDE, Ann. Appl. Probab. 34 (2024), no. 2. DOI: 10.1214/23-AAP2001, arXiv:2110.10337, https://arxiv.org/abs/2110.10337
8. P.-L. Lions, B. Seeger, and P. E. Souganidis, Interpolation results for pathwise Hamilton-Jacobi equations, Indiana Univ. Math. J. 71 (2022), no. 5, 2126-2193. DOI: 10.1512/iumj.2022.71.9174, arXiv:2007.02440, https://arxiv.org/abs/2007.02440
9. P.-L. Lions and P. 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, arXiv:1909.05672, https://arxiv.org/abs/1909.05672
10. P. K. Friz, P. Gassiat, P.-L. Lions, and P. E. Souganidis, Eikonal equations and pathwise solutions to fully non-linear SPDEs, Stoch. Partial Differ. Equ. Anal. Comput. 5 (2017), no. 2, 256-277. DOI: 10.1007/s40072-016-0087-9, arXiv:1602.04746, https://arxiv.org/abs/1602.04746
11. R. Buckdahn, C. Keller, J. Ma, and J. Zhang, Fully nonlinear stochastic and rough PDEs: classical and viscosity solutions, Probability, Uncertainty and Quantitative Risk 5 (2020), article 7. DOI: 10.1186/s41546-020-00049-8, arXiv:1501.06978, https://arxiv.org/abs/1501.06978
12. B. Seeger, Perron's method for pathwise viscosity solutions, Comm. Partial Differential Equations 43 (2018), no. 6, 998-1018. DOI: 10.1080/03605302.2018.1488262, arXiv:1605.01108, https://arxiv.org/abs/1605.01108
13. A. Agresti, A quasilinear approach to fully nonlinear parabolic (S)PDEs on $\mathbb{R}^{d}$, arXiv:1802.06395, https://arxiv.org/abs/1802.06395
14. M. G. Crandall, H. Ishii, and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), no. 1, 1-67. DOI: 10.1090/S0273-0979-1992-00266-5, https://doi.org/10.1090/S0273-0979-1992-00266-5