arXiv subjects
Publications and source records attributed to Longjie Xie.
In this note, we consider the blowup phenomenon of Grushin's operator. By using the knowledge of probability, we first get expression of heat kernel of Grushin's operator. Then by using the properties of heat kernel and suitable auxiliary function, we get that the solutions will blow up in finite time.
In this paper, we study damped Langevin stochastic differential equations with singular velocity fields. We prove the strong well-posedness of such equations. Moreover, by combining the technique of Lyapunov functions with Krylov's estimate, we also establish the exponential ergodicity for the unique strong solution.
Consider the following time-dependent stable-like operator with drift $$ \mathscr{L}_tφ(x)=\int_{\mathbb{R}^d}\big[φ(x+z)-φ(x)-z^{(α)}\cdot\nablaφ(x)\big]σ(t,x,z)ν_α(d z)+b(t,x)\cdot\nabla φ(x), $$ where $d\geq 1$, $ν_α$ is an $α$-stable type Lévy measure with $α\in(0,1]$ and $z^{(α)}=1_{α=1}1_{|z|\leq1}z$, $σ$ is a real-valued Borel function on $\mathbb{R}_+\times\mathbb{R}^d\times\mathbb{R}^d$ and $b$ is an $\mathbb{R}^d$-valued Borel function on $\mathbb{R}_+\times\mathbb{R}^d$. By using the Littlewood-Paley theory, we establish the well-posedness for the martingale problem associated with $\mathscr{L}_t$ under the sharp balance condition $α+β\geq1$, where $β$ is the Hölder index of $b$ with respect to $x$. Moreover, we also study a class of stochastic differential equations driven by Markov processes with generators of the form $\mathscr{L}_t$. We prove the pathwise uniqueness of strong solutions for such equations when the coefficients are in certain Besov spaces.
By using Duhamel's formula, we prove sharp two-sided estimates for the heat kernel of spectral fractional Laplacian with time-dependent gradient perturbation in bounded $C^{1,1}$ domains. Moreover, we also obtain gradient estimate as well as Hölder continuity of the gradient of the heat kernel.
We give a direct proof of the sharp two-sided estimates, recently established in [4,9], for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in $C^{1, 1}$ open sets by using Duhamel formula. We also obtain a gradient estimate for the Dirichlet heat kernel. Our assumption on the open set is slightly weaker in that we only require $D$ to be $C^{1,θ}$ for some $θ\in (α/2, 1]$.
We show the existence and uniqueness of strong solutions for stochastic differential equation driven by partial $α$-stable noise and partial Brownian noise with singular coefficients. The proof is based on the regularity of degenerate mixed type Kolmogorov equation.
Using heat kernel estimates, we prove the pathwise uniqueness for strong solutions of irregular stochastic differential equation driven by a family of Markov process, whose generator is a non-local and non-symmetric Lévy type operator. Due to the extra term $1_{[0,σ(X_{s-},z)]}(r)$ in multiplicative noise, we need to derive some new regularity results for the generator and use a trick of mixing $L_1$ and $L_2$-estimates by Kurtz and Protter \cite{Ku-Po}.
We show the strong well-posedness of SDEs driven by general multiplicative Lévy noises with Sobolev diffusion and jump coefficients and integrable drift. Moreover, we also study the strong Feller property, irreducibility as well as the exponential ergodicity of the corresponding semigroup when the coefficients are time-independent and singular dissipative. In particular, the large jump is allowed in the equation. To achieve our main results, we present a general approach for treating the SDEs with jumps and singular coefficients so that one just needs to focus on Krylov's {\it apriori} estimates for SDEs.
We show the pathwise uniqueness for stochastic partial differential equation driven by a cylindrical $α$-stable process with Hölder continuous drift, thus obtaining an infinite dimensional generalization of the result of Priola [Osaka J. Math., 2012] in the case $H=\mathbb{R}^d$. The proof is based on an infinite dimensional Kolmogorov equation with non-local operator.
For $d\geq 2$, we establish the existence and uniqueness of heat kernels for a large class of time-dependent second order diffusion operator with jumps, which is the sum of time-dependent of a second order elliptic differential operators non-divergence form and a non-local $α$-stable-type operator with bounded time-dependent coefficient. Moreover, we obtain sharp two-sided estimates, gradient estimate and fractional derivative estimate for the heat kernels under some mild conditions. Our approach is mainly analytic but also uses some probabilistic techniques.
In this work, by using Levi's parametrix method we first construct the fundamental solution of the critical non-local operator perturbed by gradient. Then, we use the obtained estimates to prove the existence and uniqueness of strong solutions for stochastic differential equation driven by Markov process with irregular coefficients, whose generator is a non-local and non-symmetric Lévy type operator.
By establishing a characterization for Sobolev differentiability of random fields, we prove the weak differentiability of solutions to stochastic differential equations with local Sobolev and super-linear growth coefficients with respect to the starting point. Moreover, we also study the strong Feller property and the irreducibility of the associated diffusion semigroup.
In this work we construct the heat kernel of the 1/2-order Laplacian perturbed by the first-order gradient term in Hölder space and the zero-order potential term in generalized Kato's class, and obtain sharp two-sided estimates as well as the gradient estimate of the heat kernel.