Related papers: Stochastic equations with time-dependent singular …

We prove existence, uniqueness and Sobolev regularity of weak solution of the Cauchy problem of the stochastic transport equation with drift in a large class of singular vector fields containing, in particular, the $L^d$ class, the weak…

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 $d/p+2/q=1$. The weak uniqueness is obtained by…

In this paper, we prove the strong Feller property for stochastic delay (or functional) differential equations with singular drift. We extend an approach of Maslowski and Seidler to derive the strong Feller property of those equations. The…

Using the method of Krylov's estimates, we prove the existence of weak solutions of stochastic differential equations driven by purely discontinuous Levy processes satisfying an additional assumption. The diffusion coefficient is assumed to…

In this paper, we study the weak differentiability of global strong solution of stochastic differential equations, the strong Feller property of the associated diffusion semigroups and the global stochastic flow property in which the…

We extend Krylov and R\"{o}ckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b: [0,T]\times{\mathbb…

We consider a stochastic delay differential equation driven by a general Levy process. Both, the drift and the noise term may depend on the past, but only the drift term is assumed to be linear. We show that the segment process is…

We prove the unique weak solvability of stochastic differential equations with time-inhomogeneous drift in essentially the largest (scaling-invariant) Morrey class, i.e.\,with integrability parameter $q>1$ close to $1$. The constructed weak…

Consider stochastic differential equations (SDEs) in $\Rd$: $dX_t=dW_t+b(t,X_t)\d t$, where $W$ is a Brownian motion, $b(\cdot, \cdot)$ is a measurable vector field. It is known that if $|b|^2(\cdot, \cdot)=|b|^2(\cdot)$ belongs to the Kato…

We study functional stochastic differential equations with a locally unbounded, functional drift focusing on well-posedness, stability and the strong Feller property. Following the non-functional case, we only consider integrability…

We consider the problem of constructing weak solutions to the It\^{o} and to the Stratonovich stochastic differential equations having critical-order singularities in the drift and critical-order discontinuities in the dispersion matrix.

The attracting inverse-square drift provides a prototypical counterexample to solvability of singular SDEs: if the coefficient of the drift is larger than a certain critical value, then no weak solution exists. We prove a positive result on…

In this paper, we consider a class of Mckean-Vlasov stochastic differential equation with oblique reflection over an non-smooth time dependent domain. We establish the existence and uniqueness results of this class, address the propagation…

This paper investigates a time-dependent multidimensional stochastic differential equation with drift being a distribution in a suitable class of Sobolev spaces with negative derivation order. This is done through a careful analysis of the…

In this article we introduce a new method for the construction of unique strong solutions of a larger class of stochastic delay equations driven by a discontinuous drift vector field and a Wiener process. The results obtained in this paper…

Recently Krylov established weak existence of solutions to SDEs for integrable drifts in mixed Lebesgue spaces, whose exponents satisfy the condition $1/q+d/p\leq 1$, thus going below the celebrated Ladyzhenskaya-Prodi-Serrin condition. We…

Based on a compactness criterion for random fields in Wiener-Sobolev spaces, in this paper, we prove the unique strong solvability of time-inhomogeneous stochastic differential equations with drift coefficients in critical Lebesgue spaces,…

Consider the following McKean-Vlasov SDE: $$ d X_t=\sqrt{2}d W_t+\int_{{\mathbb R}^d}K(t,X_t-y)\mu_{X_t}(dy)d t,\ \ X_0=x, $$ where $\mu_{X_t}$ stands for the distribution of $X_t$ and $K(t,x): {\mathbb R}_+\times{\mathbb R}^d\to{\mathbb…

We study a multidimensional stochastic differential equation with additive noise: \[ d X_t=b(t, X_t) dt +d \xi_t, \] where the drift $b$ is integrable in space and time, and $\xi$ is either a fractional Brownian motion or a L\'evy process.…

We study in this article the existence and uniqueness of solutions to a class of stochastic transport equations with irregular coefficients. Asking only boundedness of the divergence of the coefficients (a classical condition in both the…