# Multidimensional SDE with distributional drift and Lévy noise

@article{Kremp2020MultidimensionalSW,
title={Multidimensional SDE with distributional drift and L{\'e}vy noise},
author={Helena Kremp and Nicolas Perkowski},
journal={arXiv: Probability},
year={2020}
}
• Published 12 August 2020
• Mathematics
• arXiv: Probability
We solve multidimensional SDEs with distributional drift driven by symmetric, $\alpha$-stable Levy processes for $\alpha\in (1,2]$ by studying the associated (singular) martingale problem and by solving the Kolmogorov backward equation. We allow for drifts of regularity $(2-2\alpha)/3$, and in particular we go beyond the by now well understood "Young regime", where the drift must have better regularity than $(1-\alpha)/2$. The analysis of the Kolmogorov backward equation in the low regularity…
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## References

SHOWING 1-10 OF 72 REFERENCES
Rough paths and 1d SDE with a time dependent distributional drift: application to polymers
• Mathematics
• 2014
Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar–Parisi–Zhang (KPZ) equation, we reconsider
Strong existence and uniqueness for stable stochastic differential equations with distributional drift
• Mathematics
• 2018
We consider the stochastic differential equation $$dX_t = b(X_t) dt + dL_t,$$ where the drift $b$ is a generalized function and $L$ is a symmetric one dimensional $\alpha$-stable L\'evy processes,
On Multidimensional stable-driven Stochastic Differential Equations with Besov drift
• Mathematics
• 2019
We establish well-posedness results for multidimensional non degenerate $\alpha$-stable driven SDEs with time inhomogeneous singular drifts in $\mathbb{L}^r-{\mathbb B}_{p,q}^{-1+\gamma}$ with
Recurrence and transience properties of multi-dimensional diffusion processes in selfsimilar and semi-selfsimilar random environments
• Mathematics
• 2017
This note is a short review of the papers [8] and [9]. It is well-known that a multi-dimensional standard Brownian motion, which consists of $d$ independent one-dimensional standard Brownian motions,
Zero White Noise Limit through Dirichlet forms, with application to diffusions in a random medium
SummaryWe study the Zero White Noise Limit for diffusions in a continuous multidimensional medium: given a continuous function on ℝn,W, we consider diffusions whose drift term is the gradient ofW and
SOME SDES WITH DISTRIBUTIONAL DRIFT PART I: GENERAL CALCULUS
• Mathematics
• 2003
Diffusions in a generalized sense were studied by several authors. First, we mention a classical book by N.I. Portenko ([21]) which, however, remains in the framework of semimartingales. The point of
Strong solutions of stochastic equations with singular time dependent drift
• Mathematics
• 2005
Abstract.We prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time. We only assume local
Stochastic differential equations for Dirichlet processes
• Mathematics
• 2001
Abstract. We consider the stochastic differential equation dXt = a(Xt)dWt + b(Xt)dt, where W is a one-dimensional Brownian motion. We formulate the notion of solution and prove strong existence and
Some SDEs with distributional drift.
• Mathematics
• 2004
In dimension 1 we study a martingale problem related to a parabolic PDE operator L with continuous (non-degenerate) diffusion term and with drift being the derivative of a continuous function. We
Stochastic analysis, rough path analysis and fractional Brownian motions
• Mathematics
• 2002
Abstract. In this paper we show, by using dyadic approximations, the existence of a geometric rough path associated with a fractional Brownian motion with Hurst parameter greater than 1/4. Using the