Comparison and well-posedness for fully nonlinear stochastic PDEs with general spatially dependent nonlinear noise (the Lions-Souganidis program)

The Lions-Souganidis theory of pathwise viscosity solutions gives degenerate elliptic PDE theory a stochastic counterpart: equations $du=F(D^2u,Du)\,dt+\sum_i H^i(Du)\circ dW_i$, with $F$ degenerate elliptic and $W$ Brownian, are solved without ever differentiating the driving signal, by testing against functions transported along the stochastic characteristics of the noise. The founding 1998 Note by Lions and Souganidis proved existence and stability for $H^i\in C^3$ but uniqueness only in two special cases, and only for noise Hamiltonians independent of the spatial variable, deferring the extensions to spatially dependent noise and more general equations to companion works. In the subsequent literature, the comparison principle for the spatially homogeneous equation is now treated as settled, nonsmooth and solution-dependent Hamiltonians are covered, and comparison is proved for special spatially dependent structures (Riemannian-quadratic Hamiltonians, separated potentials, linear-growth Hamiltonians, and semilinear rough-path coefficients). What remains open is the announced general case: a comparison principle, hence uniqueness and full well-posedness, for fully nonlinear second-order SPDEs with general nonlinear spatially dependent noise Hamiltonians $H^i(Du,u,x)$, which appear in the rough-path formulation of Buckdahn-Keller-Ma-Zhang as general nonlinear coefficients $g(t,x,\omega,u,\partial_xu)$. The problem asks for a proof covering this generality (with existence and stability, completing the theory) or an explicit counterexample to comparison within it.

Background

Classical viscosity solution theory handles fully nonlinear degenerate elliptic PDEs - equations whose nonlinearity $F(D^2u,Du)$ is merely continuous and degenerate elliptic, meaning $F(X,p)\le F(Y,p)$ whenever the symmetric matrices satisfy $X\le Y$, so that ellipticity may degenerate and solutions need not be classically differentiable - through a weak 'test function' notion: $u$ is a subsolution if whenever $u-\varphi$ has a local maximum at $(x,t)$ for a smooth $\varphi$, then $\partial_t\varphi+F(D^2\varphi,D\varphi)\le 0$ there. The well-posedness of that theory rests on the comparison principle (any subsolution lies below any supersolution with larger data), proved by the doubling-of-variables method: one maximizes $u(x,t)-v(y,s)$ minus quadratic penalties $\frac{1}{2\epsilon}|x-y|^2$ and $\frac{1}{2\delta}(t-s)^2$ and splits the second derivatives using the Crandall-Ishii lemma.

Lions and Souganidis extended this program to equations forced by Brownian noise,

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

where $W$ is an $m$-dimensional Brownian motion and $\circ dW$ is the Stratonovich differential (the form obtained as the limit of smoothly perturbed driving signals, which obeys the classical chain rule). Since Brownian paths are nowhere differentiable, $t\mapsto u(x,t)$ is typically nowhere differentiable and even the test-function definition must avoid differentiating the noise. The key idea is to test against functions that are themselves transported by the noise: for noise Hamiltonians $H^i=H^i(p)$, let $S(s,t)\varphi$ denote the solution at time $t$ of the noise-only stochastic Hamilton-Jacobi equation $dv=\sum_i H^i(Dv)\circ dW_i$ on $[s,t]$ with initial datum $\varphi$ (for linear $H^i(p)=\sigma_i^Tp$ this transport is explicit, $S(s,t)\varphi(x)=\varphi(x+\sigma(W_s-W_t))$; for general smooth $H^i$ it is computed by the method of characteristics, and $S(s,t)$ is called the stochastic characteristic flow). In one standard formulation, $u$ is a pathwise viscosity subsolution if whenever $(\bar x,\bar t)$ is a local maximum of $u(x,t)-S(t_0,t)\varphi(x)-g(t)$ over $t\in[t_0,\bar t]$, for a smooth $\varphi$ and a smooth nondecreasing penalty $g$ vanishing at $t_0$, one has $g'(\bar t)+F(D^2\varphi(\bar x),D\varphi(\bar x))\le 0$; supersolutions are defined with minima and the reverse inequality. Existence is obtained by replacing $W$ with smooth approximations $\zeta^\varepsilon$, solving the resulting deterministic degenerate elliptic equations in the viscosity sense, and passing to almost-sure uniform limits; stability of the notion makes this work. Uniqueness again reduces to a comparison principle, but the doubling-of-variables argument now requires sup- and inf-convolutions that commute with the transport $S(s,t)$ - and it is exactly here that smoothness and spatial homogeneity of the noise Hamiltonian enter critically: for $H^i=H^i(p)$ the flow $S(s,t)$ is a universal, solution-independent object; for $H^i=H^i(Du,x)$ a solution-independent, local-in-time characteristic flow $S(t,t_0)$ can still be constructed from the pathwise characteristic system (Seeger 2018, Section 3, Eqs. (3.1)-(3.8)), while for $u$-dependent noise Hamiltonians $H^i=H^i(Du,u,x)$ the noise characteristics couple to the very solution being tested and no such universal flow exists.

Two neighboring frameworks rephrase the same question. In the rough-path formulation of Buckdahn-Keller-Ma-Zhang one writes $du=f(t,x,\omega,u,\partial_xu,\partial_{xx}^2u)\,dt+g(t,x,\omega,u,\partial_xu)\,d\omega_t$, where $\omega$ is a geometric rough path (a continuous path enhanced with its iterated integrals satisfying Chen's relations - the minimal structure that gives meaning to nonlinear expressions in $d\omega$), and tests against functions $\varphi$ solving $\partial_\omega\varphi=g$ in the rough sense; their general coefficient $g(t,x,\omega,u,\partial_xu)$ is precisely the rough-path form of a general nonlinear, spatially dependent noise Hamiltonian. Parallel transform-based and path-dependent-PDE approaches (randomness removed by a change of variables, or the equation rewritten on the space of driving paths) cover the cases in which the noise coefficient is linear or otherwise transformable.

State of the art. For spatially homogeneous noise ($H^i=H^i(p)$, including Lipschitz difference-of-convex Hamiltonians) the theory is considered complete: existence, stability and comparison, with the comprehensive account in Souganidis's Lecture Notes in Mathematics chapter of 2019. For spatially dependent noise, comparison is proved only under special structures: quadratic Riemannian Hamiltonians $H(p,x)=\langle g^{-1}(x)p,p\rangle$ (Friz-Gassiat-Lions-Souganidis 2017); separated potentials $\frac{1}{2}|p|^2-f(x)$ and linear-growth Hamiltonians $(a(x)|p|^2+1)^{1/2}$ in the first-order, single-noise case (Seeger 2018); and semilinear $g$ (linear in $\partial_xu$) in the rough-path formulation (Buckdahn-Keller-Ma-Zhang 2020). Buckdahn-Keller-Ma-Zhang state that a comparison principle for general nonlinear SPDEs 'remain[s] open' and that a direct approach 'is currently lacking'; Gassiat-Seeger note that the existing spatially dependent results impose 'quite particular restrictions' on the Hamiltonians. The general case is the subject of this problem.

Problem Statement

Lions and Souganidis (1998) asked for a complete well-posedness theory - existence, uniqueness (comparison), and stability - in the pathwise viscosity framework for fully nonlinear second-order stochastic PDEs

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

with $F$ continuous degenerate elliptic and $H^i\in C^3$, proving everything except uniqueness (established only for $F\equiv 0$ and for linear $H^i(p)=\sigma_i^Tp$) and restricting to noise Hamiltonians independent of the spatial variable, the extensions to spatially dependent noise and more general equations being announced for companion works. The spatially homogeneous case, the nonsmooth and semilinear relaxations, and the applications have since been resolved; what remains open is the announced spatially dependent extension. The problem is therefore:

Establish, or refute, the comparison principle - and with it uniqueness and full well-posedness (existence, stability, continuous dependence) - for pathwise viscosity solutions of fully nonlinear second-order stochastic PDEs with general nonlinear, spatially dependent noise Hamiltonians,

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

where $F$ is continuous degenerate elliptic and the $H^i$ are nonlinear and depend on the spatial variable (and possibly on $u$), in the pathwise viscosity sense defined via test functions transported by the noise for spatially dependent noise $H^i(Du,x)$; stated in the parallel rough-path viscosity formulation of Buckdahn-Keller-Ma-Zhang for solution-dependent noise $H^i(Du,u,x)$, prove the comparison principle for $du=f(t,x,\omega,u,\partial_xu,\partial_{xx}^2u)\,dt+g(t,x,\omega,u,\partial_xu)\,d\omega_t$ for general nonlinear $g$ beyond the semilinear case. Prove comparison for general smooth spatially dependent Hamiltonians $H^i(Du,u,x)$ (matching the displayed equation, $u$-dependence included), each class in the solution sense assigned above (transported test functions for $H^i(Du,x)$, rough-path viscosity for solution-dependent coefficients), i.e., on a class that covers smooth spatially dependent $H^i$ in general and does not rest on the settled special structures (Riemannian-quadratic, separated-potential, linear-growth Hamiltonians; semilinear $g$) as an essential restriction, supplying the attendant existence and stability so the theory is well posed on that class; or exhibit, within the pathwise viscosity solution class for spatially dependent noise as defined in the cited literature - for $H^i(Du,x)$, test functions transported by the noise characteristic flow (Seeger 2018, Section 3, Definition 3.1; Souganidis, LNM 2253); for $u$-dependent noise coefficients, the rough-path viscosity definition of Buckdahn-Keller-Ma-Zhang via test functions $\varphi$ satisfying $\partial_\omega\varphi=g(t,x,\varphi,\partial_x\varphi)$ (equivalently their semi-jet definition) - an explicit equation of the announced generality together with a subsolution and a supersolution (or two distinct solutions) violating the comparison ordering, showing that no theory of the announced generality exists without further structural restrictions.

References

1. P.-L. Lions and P. E. Souganidis, 'Fully nonlinear stochastic partial differential equations', Comptes Rendus de l'Academie des Sciences - Series I - Mathematics 326 (1998) 1085-1092. DOI: 10.1016/S0764-4442(98)80067-0, https://doi.org/10.1016/S0764-4442(98)80067-0
2. M. G. Crandall, H. Ishii and P.-L. Lions, 'User's guide to viscosity solutions of second order partial differential equations', Bulletin of the American Mathematical Society 27 (1992) 1-67. DOI: 10.1090/S0273-0979-1992-00266-5, https://doi.org/10.1090/S0273-0979-1992-00266-5
3. P.-L. Lions and P. E. Souganidis, 'Fully nonlinear stochastic partial differential equations: non-smooth equations and applications', Comptes Rendus de l'Academie des Sciences - Series I - Mathematics 327 (1998) 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, 'Fully nonlinear stochastic pde with semilinear stochastic dependence', Comptes Rendus de l'Academie des Sciences - Series I - Mathematics 331 (2000) 617-624. DOI: 10.1016/S0764-4442(00)00583-8, https://doi.org/10.1016/S0764-4442(00)00583-8
5. P.-L. Lions and P. E. Souganidis, 'Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations', Comptes Rendus de l'Academie des Sciences - Series I - Mathematics 331 (2000) 783-790. DOI: 10.1016/S0764-4442(00)01597-4, https://doi.org/10.1016/S0764-4442(00)01597-4
6. R. Buckdahn and J. Ma, 'Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I', Stochastic Processes and their Applications 93 (2001) 181-204. DOI: 10.1016/S0304-4149(00)00093-4, https://doi.org/10.1016/S0304-4149(00)00093-4
7. R. Buckdahn and J. Ma, 'Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part II', Stochastic Processes and their Applications 93 (2001) 205-228. DOI: 10.1016/S0304-4149(00)00092-2, https://doi.org/10.1016/S0304-4149(00)00092-2
8. N. K. Yip and P. E. Souganidis, 'Uniqueness of motion by mean curvature perturbed by stochastic noise', Annales de l'Institut Henri Poincare C, Analyse non lineaire 21 (2004) 1-23. DOI: 10.1016/j.anihpc.2002.11.001, https://doi.org/10.1016/j.anihpc.2002.11.001
9. H. Oberhauser, M. Caruana and P. K. Friz, 'A (rough) pathwise approach to a class of non-linear stochastic partial differential equations', Annales de l'Institut Henri Poincare C, Analyse non lineaire 28 (2011), no. 1, 27-46. DOI: 10.1016/j.anihpc.2010.11.002, https://doi.org/10.1016/j.anihpc.2010.11.002; preprint arXiv:0902.3352, https://arxiv.org/abs/0902.3352
10. P. K. Friz, P. Gassiat, P.-L. Lions and P. E. Souganidis, 'Eikonal equations and pathwise solutions to fully non-linear SPDEs', Stochastics and Partial Differential Equations: Analysis and Computation 5 (2017) 256-277. DOI: 10.1007/s40072-016-0087-9, https://doi.org/10.1007/s40072-016-0087-9; preprint arXiv:1602.04746, https://arxiv.org/abs/1602.04746
11. B. Seeger, 'Perron's method for pathwise viscosity solutions', Communications in Partial Differential Equations 43 (2018) 998-1018. DOI: 10.1080/03605302.2018.1488262, https://doi.org/10.1080/03605302.2018.1488262; preprint arXiv:1605.01108, https://arxiv.org/abs/1605.01108
12. P. E. Souganidis, 'Pathwise solutions for fully nonlinear first- and second-order partial differential equations with multiplicative rough time dependence', Lecture Notes in Mathematics 2253, Springer, Cham (2019) 75-220. DOI: 10.1007/978-3-030-29545-5_3, https://doi.org/10.1007/978-3-030-29545-5_3
13. 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, https://doi.org/10.1186/s41546-020-00049-8; preprint arXiv:1501.06978, https://arxiv.org/abs/1501.06978
14. B. Seeger, 'Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations', The Annals of Applied Probability 30 (2020) 1784-1823. DOI: 10.1214/19-AAP1543, https://doi.org/10.1214/19-AAP1543; preprint arXiv:1802.04740, https://arxiv.org/abs/1802.04740
15. P. Gassiat and B. Seeger, 'The Neumann problem for fully nonlinear SPDE', The Annals of Applied Probability 34 (2024) 1730-1788. DOI: 10.1214/23-aap2001, https://doi.org/10.1214/23-aap2001; preprint arXiv:2110.10337, https://arxiv.org/abs/2110.10337