- 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 provide a general framework for the construction of approximation schemes for viscosity solutions of fully nonlinear pathwise (stochastic) partial differential equations. The main examples of interest are explicit finite difference schemes, although convergence results are general enough to include other examples such as Trotter-Kato type mixing formulas. The arguments are adapted from the classical viscosity solution setting, with a more careful analysis needed to treat the irregular time dependence. In particular, paths with non-trivial quadratic variation need to be regularized in order to preserve the monotonicity of the scheme. The main result is qualitative, and uses an adaptation of the method of half-relaxed limits from the classical viscosity setting to prove the convergence of schemes satisfying general assumptions. Rates of convergence are presented in the Hamilton-Jacobi case through an error estimate, whose merit is that it depends on the path only through its modulus of continuity, and not on its derivatives or total variation. Specific examples are given in the case of equations driven by Brownian motion, including a scheme that uses scaled random works to establish convergence in law and a scheme with a random partition of the time interval.
In this paper, Perron's method is used to construct viscosity solutions of fully nonlinear degenerate parabolic pathwise (stochastic) partial differential equations. This provides an alternative method for proving existence for such equations that relies solely on a comparison principle, rather than considering limits of solutions of equations driven by smooth approximating paths. The result covers the most general case of multiplicative noise, that is, equations driven by a multidimensional geometric rough path and Hamiltonians with nontrivial spatial dependence. Also included in this note is a list of examples of equations satisfying the comparison principle, so that, in particular, the results proved here imply that the Cauchy problem for these equations is well-posed.
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.