Strong solutions of stochastic equations with singular time dependent drift

@article{Krylov2005StrongSO,
  title={Strong solutions of stochastic equations with singular time dependent drift},
  author={Nicolai V. Krylov and Michael R{\"o}ckner},
  journal={Probability Theory and Related Fields},
  year={2005},
  volume={131},
  pages={154-196}
}
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 Lq_Lp-integrability of b in ℝ×G with d/p+2/q<1. We also prove strong Feller properties in this case. If b is the gradient in x of a nonnegative function ψ blowing up as G∋x→∂G, we prove that the conditions 2Dtψ≤Kψ,2Dtψ+Δψ≤Keɛψ,ɛ ∈ [0,2), imply that the explosion time is infinite and the… 
Strong solution for stochastic transport equations with irregular drift: existence and non-existence
We prove some existence, uniqueness and non-existence results of stochastic strong solutions for a class of stochastic transport equations with a $q$-integrable (in time), bounded and
SDEs with critical time dependent drifts: weak solutions.
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
Ergodicity of stochastic differential equations with jumps and singular coefficients
We show the strong well-posedness of SDEs driven by general multiplicative Levy noises with Sobolev diffusion and jump coefficients and integrable drift. Moreover, we also study the strong Feller
Stochastic differential equations with Sobolev diffusion and singular drift
In this paper we study properties of solutions to stochastic differential equations with Sobolev diffusion coefficients and singular drifts. The properties we study include stability with respect to
Strong Uniqueness of Singular Stochastic Delay Equations
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
Stochastic transport equation with singular drift
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
Strong Uniqueness for Stochastic Evolution Equations with Unbounded Measurable Drift Term
We consider stochastic evolution equations in Hilbert spaces with merely measurable and locally bounded drift term $$B$$B and cylindrical Wiener noise. We prove pathwise (hence strong) uniqueness in
SDEs with critical time dependent drifts: strong solutions
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
Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift
We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing
Weak well-posedness of SDEs with drifts in critical spaces.
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
...
...

References

SHOWING 1-10 OF 31 REFERENCES
Some Properties of Traces for Stochastic and Deterministic Parabolic Weighted Sobolev Spaces
Abstract The results presented provide information on the behavior near x1=0 for fixed t>0 in the sense of Lp-spaces with weights for functions u(t)=u(t, x1, x′) (x1⩾0, x′∈ R d−1, t⩾0) satisfying
Uniqueness of solutions of stochastic differential equations
It follows from a theorem of Veretennikov [4] that (1) has a unique strong solution, i.e. there is a unique process x(t), adapted to the filtration of the Brownian motion, satisfying (1).
Existence of strong solutions for Itô's stochastic equations via approximations
SummaryGiven strong uniqueness for an Itô's stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on “any” probability space by using, for example,
ON STRONG SOLUTIONS AND EXPLICIT FORMULAS FOR SOLUTIONS OF STOCHASTIC INTEGRAL EQUATIONS
Conditions are obtained under which the stochastic equation has a strong solution. In particular, in the multidimensional case where the diffusion matrix is the identity matrix and the drift vector
On the Rate of Convergence of Splitting-up Approximations for SPDEs
We consider the stochastic PDE $$du(t,x) = (Lu(t,x) + f(t,x))dt + ({{M}_{k}}u(t) + {{g}_{k}}(t,x)) \circ d{{W}^{k}},$$ where L and M k are second and first order partial differential operators,
The Heat Equation in Lq((0, T), Lp)-Spaces with Weights
Existence and uniqueness theorems are presented for the heat equation in Lp spaces with or without weights allowing derivatives of solutions to blow up near the boundary. It is allowed for the powers
Energy forms, Hamiltonians, and distorted Brownian paths
We study the Hamiltonians for nonrelativistic quantum mechanics—and for the heat equation—in terms of energy forms ∫∇f∇fd’gm, where dμ is a positive, not necessarily finite measure on Rn. We cover
Statistics of random processes
1. Essentials of Probability Theory and Mathematical Statistics.- 2. Martingales and Related Processes: Discrete Time.- 3. Martingales and Related Processes: Continuous Time.- 4. The Wiener Process,
...
...