arXiv subjects

Rongrong Tian

Publications and source records attributed to Rongrong Tian.

5 recordsLinked to original sources

arXiv2025

SDEs with subcritical Lebesgue--Hölder drift and driven by $α$-stable processes

We obtain the unique weak and strong solvability for time inhomogeneous stochastic differential equations with the drift in subcritical Lebesgue--Hölder spaces $L^p([0,T];{\mathcal C}_b^β({\mathbb R}^d;{\mathbb R}^d))$ and driven by $α$-stable processes for $α\in (0,2)$. The weak well-posedness is derived for $β\in (0,1)$, $α+β>1$ and $p>α/(α+β-1)$ through Prohorov's theorem, Skorohod's representation and the regularity estimates of solutions for a class of fractional parabolic partial differential equations. The pathwise uniqueness and Davie's type uniqueness are proved for $β>1-α/2$ by using Itô--Tanaka's trick. Moreover, we give a counterexample to the pathwise uniqueness for the supercritical Lebesgue--Hölder drifts to explain the present result is sharp.

math.PR↗

arXiv2025

Uniqueness of strong solutions for SDEs with Hölder diffusions

This paper is concerned with the Itô stochastic differential equations with $\mR^{d\times k}$ diffusions in class of Hölder spaces and continuous $\mR^d$ drifts. We derive a uniqueness result of strong solutions for $\cC^α\ (α\geq \frac{1}{2})$ coefficients and this result is new. Our proof is supported by Itô's formula and a finer analysis on cut-off and smoothing techniques.

math.AP↗

arXiv2025

Malliavin Calculus and Stochastic Differential Equations

This paper is devoted to a study on SDEs with a bounded Borel drift b. We first remark that the original integration by parts formula due to P. Malliavin can be used to deal with derivatives with respect to space variables, then we obtain a link between the product of heat kernels and iterated divergences in Malliavin calculus. An explicit estimate for the derivative of solutions to SDE is obtained in terms of the L-infinity norm of b; as a result, we prove that the SDE defines a continuous flow of maps in Sobolev spaces.

math.PR↗

arXiv2021

Strong solutions of stochastic differential equations with square integrable drift

We prove the existence and uniqueness of strong solutions for stochastic differential equations in which the drift coefficient is square integrable in time variable and Hölder continuous in space variable. Moreover, we prove that the unique strong solution has a continuous modification, which is $β$-Hölder continuous in space variable for every $β\in (0,1)$, and as an $L^2(Ω\times (0,T))$ valued function, it is differentiable as well.

math.AP↗

arXiv2018

Well-posedness of nonlinear transport equation by stochastic perturbation

We are concerned with multidimensional nonlinear stochastic transport equation driven by Brownian motions. For irregular fluxes, by using stochastic BGK approximations and commutator estimates, we gain the existence and uniqueness of stochastic entropy solutions. Besides, for $BV$ initial data, the $BV$ and Hölder regularities are also derived for the unique stochastic entropy solution. Particularly, for the transport equation, we gain a regularization result, i.e. while the existence fails for the transport equation, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be another explicit example (the first example is given in [22]) of a PDE of fluid dynamics that becomes well-posed under the influence of a multiplicative Brownian type noise.

math.PR↗