# Stochastic differential equations with critical drifts

@article{Nam2020StochasticDE, title={Stochastic differential equations with critical drifts}, author={Kyeongsik Nam}, journal={Stochastic Processes and their Applications}, year={2020} }

We establish the well-posedness of SDE with the additive noise when a singular drift belongs to the critical spaces. We prove that if the drift belongs to the Orlicz-critical space $L^{q,1}([0,T],L^p_x)$ for $p,q\in (1,\infty)$ satisfying $\frac{2}{q}+\frac{d}{p} =1$, then the corresponding SDE admits a unique strong solution. We also derive the Sobolev regularity of a solution under the Orlicz-critical condition.

#### 9 Citations

Weak well-posedness of SDEs with drifts in critical spaces.

- Mathematics
- 2020

We prove the unique weak solvability of time-inhomogeneous stochastic differential equations with additive noises and drifts in critical Lebsgue space $L^q([0,T]; L^{p}(\mathbb{R}^d))$ with… Expand

SDEs with critical time dependent drifts: strong solutions

- Mathematics
- 2021

This paper is a continuation of [RZ20]. Based on a compactness criterion for random fields in Wiener-Sobolev spaces, in this paper, we prove the unique strong solvability of timeinhomogeneous… Expand

Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness

- Mathematics
- Electronic Journal of Probability
- 2019

In this paper linear stochastic transport and continuity equations with drift in critical $L^{p}$ spaces are considered. In this situation noise prevents shocks for the transport equation and… Expand

$L^q(L^p)$-theory of stochastic differential equations

- Mathematics
- 2019

In this paper we show the weak differentiability of the unique strong solution with respect to the starting point $x$ as well as Bismut-Elworthy-Li's derivative formula for the following stochastic… Expand

Stochastic processes and Markov chains in shape and material optimization problems of composite structures

- Mathematics
- 2017 7th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO)
- 2017

Commonly shape and material optimization for complex structures having complicated boundary and loading conditions requires FE analysis in order to determine accurately enough (especially for… Expand

Strong solutions of stochastic differential equations with square integrable drift

- Bulletin des Sciences Mathématiques
- 2021

On time inhomogeneous stochastic Itô equations with drift in

- Mathematics, Physics
- 2020

We prove the solvability of Ito stochastic equations with uniformly nondegenerate, bounded, measurable diffusion and drift in $L_{d+1}(\mathbb{R}^{d+1})$. Actually, the powers of summability of the… Expand

SDEs with critical time dependent drifts: weak solutions.

- Mathematics
- 2020

We prove the unique weak solvability of time-inhomogeneous stochastic differential equations with additive noises and drifts in critical Lebsgue space $L^q([0,T]; L^{p}(\mathbb{R}^d))$ with… Expand

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