Stochastic generalized porous media and fast diffusion equations

By Jiagang Ren, Michael Röckner, Feng-Yu Wang. 2006

171 citations recorded by the source corpus.

Abstract We present a generalization of Krylov–Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for σ -finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called N -functions in the theory of Orlicz spaces.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/c7a7676fd61271b91bc90706f0b1ff9712fe83d4

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

1. Wei Liu
2. Michael Röckner
3. Tusheng Zhang
4. Feng-Yu Wang
5. Benjamin Gess
6. Wei David Liu
7. Shijie Shang
8. Giuseppe Da Prato

In Computational Theory and Mathematics

1. Wei Liu
2. Michael Röckner
3. Tusheng Zhang
4. Wei David Liu
5. Benjamin Gess
6. Feng-Yu Wang
7. Giuseppe Da Prato
8. Shijie Shang

In Finance

1. Wei Liu
2. Michael Röckner
3. Tusheng Zhang
4. Shijie Shang
5. Benjamin Gess
6. Feng-Yu Wang
7. Giuseppe Da Prato
8. Wei David Liu

Citations in the corpus

• Existence of strong solutions for stochastic porous media equation under general monotonicity conditions (2007, methodology): “As a matter of fact, in the line of the classical work of Krylov and Rozovskii [10] the approach used in [15] is a variational one, that is, one considers the stochastic equation (1.”
• Strong solutions for SPDE with locally monotone coefficients driven by L\'{e}vy noise (2011, background): “Ren et al [49], Röckner and Wang [51] and Zhang [59].”
• Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise (2008, background): “(ii) This theorem can not be applied to stochastic fast-diffusion equations in [23, 27] since (A2) fails to satisfy.”
• Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces (2012, methodology): “There are also methods that apply to fully nonlinear problems, see for instance [4, 54] for porous medium equations.”
• SPDE in Hilbert space with locally monotone coefficients (2010, background): “For general results on the existence and uniqueness of solutions to SPDE we refer to [25, 16, 12, 27, 35].”
• A mild Itô formula for SPDEs (2010, background): “We refer to [38, 39, 68, 83, 85, 90, 91] for other Itô type formulas with the particular test function φ(v) = ‖v‖2H , v ∈ H.”
• Well-posedness of stochastic partial differential equations with Lyapunov condition (2013, background): “[29], Röckner and Wang [30], Zhang [41] (and the references therein).”
• Non-monotone stochastic generalized porous media equations☆ (2008, methodology): “Since F ∗ e and L(2)(m) are generally incomparable, the solutions constructed in [15] do not automatically provide solutions starting from points in L(2)(m) \ F ∗ e .”
• Martingale solutions and Markov selections for stochastic partial differential equations (2009, unknown): “in [12], [6], [19], [3], [4], [22], [20](see also [18]).”
• Random attractors for a class of stochastic partial differential equations driven by general additive noise (2010, background): “For general results on the existence and uniqueness of variational solutions to SPDE we refer to [23, 28, 38, 40, 43, 51].”
• Harnack inequality and strong Feller property for stochastic fast-diffusion equations☆ (2007, methodology): “For more general stochastic evolution equations in Hilbert space we refer to [8, 12, 13, 14] and the references within.”
• Harnack inequality and applications for stochastic evolution equations with monotone drifts (2008, methodology): “This type of SPDE has been studied intensively in recent years, we refer to [8, 13, 20, 28, 18, 27, 39] (and references therein) for various generalizations and applications.”
• Well-posedness of stochastic partial differential equations with fully local monotone coefficients (2022, methodology): “We refer to [73, 76, 77, 93] and references therein for the early applications of variational approaches to SPDEs.”
• Local and global well-posedness of SPDE with generalized coercivity conditions☆ (2012, unknown): “to [23, 42, 51] for some further generalizations.”