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Register a new userControlled viscosity solutions of fully nonlinear rough PDEs
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Massimiliano Gubinelli
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We propose a definition of viscosity solutions to fully nonlinear PDEs driven by a rough path via appropriate notions of test functions and rough jets. These objects will be defined as controlled processes with respect to the driving rough path. We show that this notion is compatible with the seminal results of Lions and Souganidis and with the recent results of Friz and coauthors on fully non-linear SPDEs with rough drivers.
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In this article, we adapt the definition of viscosity solutions to the obstacle problem for fully nonlinear path-dependent PDEs with data uniformly continuous in $(t,omega)$, and generator Lipschitz continuous in $(y,z,gamma)$. We prove that our definition of viscosity solutions is consistent with the classical solutions, and satisfy a stability result. We show that the value functional defined via the second order reflected backward stochastic differential equation is the unique viscosity solution of the variational inequalities.
This paper introduces a convenient solution space for the uniformly elliptic fully nonlinear path dependent PDEs. It provides a wellposedness result under standard Lipschitz-type assumptions on the nonlinearity and an additional assumption formulated on some partial differential equation defined locally by freezing the path.
In this paper we propose a new type of viscosity solutions for fully nonlinear path dependent PDEs. By restricting to certain pseudo Markovian structure, we remove the uniform non- degeneracy condition imposed in our earlier works [9, 10]. We establish the comparison principle under natural and mild conditions. Moreover, as applications we apply our results to two important classes of PPDEs: the stochastic HJB equations and the path dependent Isaacs equations, induced from the stochastic optimization with random coefficients and the path dependent zero sum game problem, respectively.
We study the focusing mass-critical rough nonlinear Schroedinger equations, where the stochastic integration is taken in the sense of controlled rough path. We obtain the global well-posedness if the mass of initial data is below that of the ground state. Moreover, the existence of minimal mass blow-up solutions is also obtained in both dimensions one and two. In particular, these yield that the mass of ground state is exactly the threshold of global well-posedness and blow-up of solutions in the stochastic focusing mass-critical case. Similar results are also obtained for a class of nonlinear Schroedinger equations with lower order perturbations.
We are concerned with the multi-bubble blow-up solutions to rough nonlinear Schrodinger equations in the focusing mass-critical case. In both dimensions one and two, we construct the finite time multi-bubble solutions, which concentrate at $K$ distinct points, $1leq K<infty$, and behave asymptotically like a sum of pseudo-conformal blow-up solutions in the pseudo-conformal space $Sigma$ near the blow-up time. The upper bound of the asymptotic behavior is closely related to the flatness of noise at blow-up points. Moreover, we prove the conditional uniqueness of multi-bubble solutions in the case where the asymptotic behavior in the energy space $H^1$ is of the order $(T-t)^{3+zeta}$, $zeta>0$. These results are also obtained for nonlinear Schrodinger equations with lower order perturbations, particularly, in the absence of the classical pseudo-conformal symmetry and the conversation law of energy. The existence results are applicable to the canonical deterministic nonlinear Schrodinger equation and complement the previous work [43]. The conditional uniqueness results are new in both the stochastic and deterministic case.
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