相关论文: Heat kernel of supercritical SDEs with unbounded d…

We prove sharp two-sided bounds of the fundamental solution for an integro-differential operator of order $\alpha \in (0,2)$ that generates a $d$-dimensional Markov process. The corresponding Dirichlet form is comparable to that of $d$…

Let $d\geq 2$. In this paper, we investigate the following stochastic differential equation (SDE) in ${\mathbb R}^d$ driven by Brownian motion $$ {\rm d} X_t=b(t,X_t){\rm d} t+\sqrt{2}{\rm d} W_t, $$ where $b$ belongs to the space ${\mathbb…

For $d \ge 2$, $\alpha \in (0,2)$ and $M > 0$, we consider the gradient perturbation of a family of nonlocal operators $\{\Delta+a^\alpha\Delta^{\alpha/2}, a\in (0,M]\}$. We establish the existence and uniqueness of the fundamental solution…

In this work, by using the Malliavin calculus, under H\"ormander's condition, we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated Brownian motions.…

For $d\ge 2$ and $0<\beta<\alpha<2$, consider a family of non-local operators $\mathcal{L}^{b}=\Delta^{\alpha/2}+\mathcal{S}^{b}$ on $\mathbb{R}^d$, where $$ \mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-\beta)\int_{ \{z\in…

We investigate the stochastic heat equation driven by space-time white noise defined on an abstract Hilbert space, assuming that the drift and diffusion coefficients are both merely H\"older continuous. Random field SPDEs are covered as…

In this paper, we consider a large class of purely discontinuous rotationally symmetric Levy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a…

We prove that the heat kernel associated to the Schr\"odinger type operator $A:=(1+|x|^\alpha)\Delta-|x|^\beta$ satisfies the estimate $$k(t,x,y)\leq…

Let L=d^2/dx^2+u(x) be the one-dimensional Schrodinger operator and H(x,y,t) be the corresponding heat kernel. We prove that the nth Hadamard's coefficient H_n(x,y) is equal to 0 if and only if there exists a differential operator M of…

We study the heat kernel of the supercritical fractional diffusion equation with the drift in the critical H\"{o}lder space. We show that such a drift can have point irregularities strong enough to make the heat kernel vanish at a point for…

We establish global two-sided heat kernel estimates (for full time and space) of the Schr\"odinger operator $-\frac{1}{2}\Delta+V$ on $\R^d$, where the potential $V(x)$ is locally bounded and behaves like $c|x|^{-\alpha}$ near infinity with…

We consider the heat equation $u_t=Lu$ where $L$ is a second-order difference operator in a discrete variable $n$. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients…

We study an inhomogeneous semilinear heat inequality on the unit sphere \(\mathbb S^N\), \(N\ge3\), in an exterior geodesic domain associated with a fixed pole. The equation involves the singular Hardy-type potential \(\lambda/\sin^2 r\),…

In this paper, we study two types of purely discontinuous symmetric Markov processes $X$ in bounded smooth subsets of $\mathbb R^d$: conservative processes and processes killed either upon approaching the boundary of the set or by a killing…

We give the upper and the lower estimates of heat kernels for Schr\"odinger operators $H=-\Delta+V$, with nonnegative and locally bounded potentials $V$ in $\mathbb{R}^d$, $d \geq 1$. We observe a factorization: the contribution of the…

Consider the Schr\"odinger operator $ \mathcal L^V=-\Delta+V $ on $\R^d$, where $V:\R^d\to [0,\infty)$ is a nonnegative and locally bounded potential on $\R^d$ so that for all $x\in \R^d$ with $|x|\ge 1$, $c_1g(|x|)\le V(x)\le c_2g(|x|)$…

Let $\displaystyle L = -\frac{1}{w} \, \mathrm{div}(A \, \nabla u) + \mu$ be the generalized degenerate Schr\"odinger operator in $L^2_w(\mathbb{R}^d)$ with $d\ge 3$ with suitable weight $w$ and measure $\mu$. The main aim of this paper is…

We prove existence and smoothness of the density of the solution to a nonlinear stochastic heat equation on $L^2(\mathcal{O})$ (evaluated at fixed points in time and space), where $\mathcal{O}$ is an open bounded domain in $\mathbb{R}^d$.…

We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric $\alpha$-stable L\'evy processes with values in $R^d$ under some mild H{\"o}lder regularity…

We consider the Schr\"odinger type operator ${\mathcal A}=(1+|x|^{\alpha})\Delta-|x|^{\beta}$, for $\alpha\in [0,2]$ and $\beta\ge 0$. We prove that, for any $p\in (1,\infty)$, the minimal realization of operator ${\mathcal A}$ in…