- Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations. [arXiv, abstract]
- Perron's method for pathwise viscosity solutions. [arXiv, abstract]
- Homogenization of pathwise Hamilton-Jacobi equations. Journal de Mathématiques Pures et Appliquées, 2018 [link, arXiv, abstract]
The aim of this paper is to develop a general method for constructing approximation schemes for viscosity solutions of fully nonlinear pathwise stochastic partial differential equations, and for proving their convergence. Our results apply to approximations such as explicit finite difference schemes and Trotter-Kato type mixing formulas. The irregular time dependence disrupts the usual methods from the classical viscosity theory for creating schemes that are both monotone and convergent, an obstacle that cannot be overcome by incorporating higher order correction terms, as is done for numerical approximations of stochastic or rough ordinary differential equations. The novelty here is to regularize those driving paths with non-trivial quadratic variation in order to guarantee both monotonicity and convergence. We present qualitative and quantitative results, the former covering a wide variety of schemes for second-order equations. An error estimate is established in the Hamilton-Jacobi case, its merit being that it depends on the path only through the modulus of continuity, and not on the derivatives or total variation. As a result, it is possible to choose a regularization of the path so as to obtain efficient rates of convergence. This is demonstrated in the specific setting of equations with multiplicative white noise in time, in which case the convergence holds with probability one. We also present an example using scaled random walks that exhibits convergence in distribution.
We use Perron's method to construct viscosity solutions of fully nonlinear degenerate parabolic pathwise (rough) partial differential equations. This provides an intrinsic method for proving the existence of solutions that relies only on a comparison principle, rather than considering equations driven by smooth approximating paths. The result covers the case of multidimensional geometric rough path noise, where the noise coefficients depend nontrivially on space and on the gradient of the solution. Also included in this note is a discussion of the comparison principle and a summary of the pathwise equations for which one has been proved.
We present qualitative and quantitative homogenization results for pathwise Hamilton-Jacobi equations with “rough” multiplicative driving signals. When there is only one such signal and the Hamiltonian is convex, we show that the equation, as well as equations with smooth approximating paths, homogenize. In the multi-signal setting, we demonstrate that blow-up or homogenization may take place. The paper also includes a new well-posedness result, which gives explicit estimates for the continuity of the solution map and the equicontinuity of solutions in the spatial variable.