Degenerate parabolic stochastic partial differential equations: Quasilinear case
By Arnaud Debussche, Martina Hofmanov'a, Julien Vovelle. 2013
129 citations recorded by the source corpus.
We study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an $L^1$-contraction property. In comparison to the first-order case (Debussche and Vovelle, 2010) and to the semilinear degenerate parabolic case (Hofmanova, 2013), the present result contains two new ingredients: a generalized Ito formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.
Subfields: Applied Mathematics, Finance
View Paper: https://www.semanticscholar.org/paper/b17844563e31694a7e1357cb84e895772df548bc
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
- Samy Tindel
- A. Deya
- Massimiliano Gubinelli
- Martina Hofmanov'a
- Benjamin Gess
- Panagiotis E. Souganidis
- S. Armstrong
- Peter K. Friz
In Finance
- Samy Tindel
- A. Deya
- Benjamin Gess
- Massimiliano Gubinelli
- Panagiotis E. Souganidis
- Martina Hofmanov'a
- Martin Hairer
- Peter K. Friz
Citations in the corpus
- Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws (2014, background): “The higher dimensional case has first studied by Chen and Frid in [11] and was subsequently generalized by Chen and Perthame [12] and Debussche and Vovelle [18].”
- A priori estimates for rough PDEs with application to rough conservation laws (2016, background): “This direction also appears in several works on stochastic conservation laws and degenerate parabolic SPDEs, see [12], [15], [13], [14], [42], [41] and in the (rough) pathwise works [30], [29], [43], [50], [52].”
- Stochastic non-isotropic degenerate parabolic–hyperbolic equations (2016, background): “In particular, Debussche, Hofmanova and Vovelle [11] and Bauzet, Vallet andWittbold [5] studied SPDE of the type (1.10) additionally including a second order quasilinear term.”