Stochastic Homeomorphism Flows of SDEs with Singular Drifts and Sobolev Diffusion Coefficients

By Xicheng Zhang. 2010

169 citations recorded by the source corpus.

In this paper we prove the stochastic homeomorphism flow property and the strong Feller property for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients. Moreover, the local well posedness under local assumptions are also obtained. In particular, we extend Krylov and Rockner's results in [10] to the case of non-constant diffusion coefficients.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/80fdec638f1ebb72d9e19311ea99d787321e0d3b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Longjie Xie
2. Xicheng Zhang
3. Feng-Yu Wang
4. Franco Flandoli
5. Renming Song
6. E. Fedrizzi
7. Xing Huang
8. Zhen-Qing Chen

In Control and Systems Engineering

1. Feng-Yu Wang
2. Longjie Xie
3. Xicheng Zhang
4. Renming Song
5. Zhen-Qing Chen
6. Xing Huang
7. Franco Flandoli
8. E. Fedrizzi

In Finance

1. Xicheng Zhang
2. Feng-Yu Wang
3. Longjie Xie
4. Xing Huang
5. Renming Song
6. Franco Flandoli
7. E. Fedrizzi
8. Zhen-Qing Chen

Citations in the corpus

• Noise Prevents Singularities in Linear Transport Equations (2012, background): “The existence of an Hölder continuous stochastic flow has been proved in [12], [13], [21].”
• Stochastic differential equations with Sobolev diffusion and singular drift and applications (2016, methodology): “(4) It follows by generalized Itô’s formula (see [11] or [30], Lemma 4.”
• Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance (2020, background): “According to [25], assumption (Aσ) together with (Ab) or (A ′ b) implies the strong wellposedness, where under (Ab) the non-explosion is implied by (1.”
• Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients (2014, methodology): “Hence, letting θ > 1 so that q θ > d2 + 1, by [37], Theorem 2.”
• Ergodicity of stochastic differential equations with jumps and singular coefficients (2017, background): “The interested readers are referred to [17, 18, 31, 34, 45, 52, 54] and references therein.”
• Distribution-Dependent SDEs for Landau Type Equations (2016, background): “the SDE is non-degenerate), this condition has been largely weakened as |b|+ |∇σ| ∈ Lploc(dx) for some p > d, see [29] and references within.”
• Stochastic flows for Lévy processes with Hölder drifts (2015, background): “Since then, many people have made contributions to the pathwise uniqueness problem for SDEs driven by Brownian motion (see [12, 8, 7, 25] and references therein).”