arXiv subjects
Publications and source records attributed to Panki Kim.
In this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time fractional equations in metric measure spaces.
In this paper, we establish the precise asymptotic behaviors of the tail probability and the transition density of a large class of isotropic Lévy processes when the scaling order is between 0 and 2 including 2. We also obtain the precise asymptotic behaviors of the tail probability of subordinators when the scaling order is between 0 and 1 including 1. The asymptotic expressions are given in terms of the radial part of characteristic exponent $ψ$ and its derivative. In particular, when $ψ(λ)-\fracλ{2}ψ'(λ)$ varies regularly, as $\frac{tψ(r^{-1})^2}{ψ(r^{-1})-(2r)^{-1}ψ'(r^{-1})} \to 0$ the tail probability $\mathbb{P}(|X_t|\geq r)$ is asymptotically equal to a constant times $ t( ψ(r^{-1})-(2r)^{-1}ψ'(r^{-1})).$
In this paper we consider a large class of symmetric Markov processes $X=(X_t)_{t\ge0}$ on $\R^d$ generated by non-local Dirichlet forms, which include jump processes with small jumps of $α$-stable-like type and with large jumps of super-exponential decay. Let $D\subset \R^d$ be an open (not necessarily bounded and connected) set, and $X^D=(X_t^D)_{t\ge0}$ be the killed process of $X$ on exiting $D$. We obtain explicit criterion for the compactness and the intrinsic ultracontractivity of the Dirichlet Markov semigroup $(P^{D}_t)_{t\ge0}$ of $X^D$. When $D$ is a horn-shaped region, we further obtain two-sided estimates of ground state in terms of jumping kernel of $X$ and the reference function of the horn-shaped region $D$.
Let $J$ be the Lévy density of a symmetric Lévy process in $\mathbb{R}^d$ with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator $$ {\mathcal L}^κf(x):= \lim_{ε\downarrow 0} \int_{\{z \in \mathbb{R}^d: |z|>ε\}}(f(x+z)-f(x))κ(x,z)J(z)\, dz\, , $$ where $κ(x,z)$ is a Borel measurable function on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0<κ_0\le κ(x,z)\le κ_1$, $κ(x,z)=κ(x,-z)$ and $|κ(x,z)-κ(y,z)|\le κ_2|x-y|^β$ for some $β\in (0, 1)$. We construct the heat kernel $p^κ(t, x, y)$ of ${\mathcal L}^κ$, establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel $p^κ$.
In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels of a large class of symmetric (but not necessarily rotationally symmetric) Lévy processes on half spaces for all $t>0$. These Lévy processes may or may not have Gaussian component. When Lévy density is comparable to a decreasing function with damping exponent $β$,our estimate is explicit in terms of the distance to the boundary, the Lévy exponent and the damping exponent $β$ of Lévy density.
Based on two-sided heat kernel estimates for a class of symmetric jump processes on metric measure spaces, the laws of the iterated logarithm (LILs) for sample paths, local times and ranges are established. In particular, the LILs are obtained for $β$-stable-like processes on $α$-sets with $β>0$.
Minimal thinness is a notion that describes the smallness of a set at a boundary point. In this paper, we provide tests for minimal thinness for a large class of subordinate killed Brownian motions in bounded C1,1 domains, C1,1 domains with compact complements and domains above graphs of bounded C1,1 functions.
We prove non-explosion results for Schrödinger perturbations of symmetric transition densities and Hardy inequalities for their quadratic forms by using explicit supermedian functions of their semigroups.
Minimal thinness is a notion that describes the smallness of a set at a boundary point. In this paper, we provide tests for minimal thinness at finite and infinite minimal Martin boundary points for a large class of purely discontinuous symmetric Lévy processes.
In this paper, we discuss tangential limits for regular harmonic functions with respect to $ϕ(Δ):=-ϕ(-Δ)$ in the $C^{1,1}$ open set $D$ in $\mathbb{R}^d$, where $ϕ$ is the complete Bernstein function and $d \ge 2$. When the exterior function $f$ is local $L^p$-Hölder continuous of order $β$ on $D^c$ with $ p\in(1,\infty]$ and $β>1/p$, for a large class of Bernstein function $ϕ$, we show that the regular harmonic function $u_f$ with respect to $ϕ(Δ)$, whose value is $f$ on $D^c$, converges a.e. through a certain parabola that depends on $ϕ$ and $ϕ'$. Our result includes the case $ϕ(λ)=\log(1+λ^{α/2})$. Our proofs use both the probabilistic and analytic methods.
In this paper we study the Martin boundary of open sets with respect to a large class of purely discontinuous symmetric Lévy processes in ${\mathbb R}^d$. We show that, if $D\subset {\mathbb R}^d$ is an open set which is $κ$-fat at a boundary point $Q\in \partial D$, then there is exactly one Martin boundary point associated with $Q$ and this Martin boundary point is minimal.
In this paper, we introduce a class of stochastic partial differential equations (SPDEs) with fractional time-derivatives, and study the $L_2$-theory of the equations. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping.
In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in $C^{1,η}$ open sets. The processes are symmetric pure jump Markov processes with jumping intensity $κ(x,y) ψ_1 (|x-y|)^{-1} |x-y|^{-d-α}$, where $α\in (0,2)$. Here, $ψ_1$ is an increasing function on $[ 0, \infty )$, with $ψ_1(r)=1$ on $0 1$ for $β\in [0,\infty]$, and $ κ( x, y)$ is a symmetric function confined between two positive constants, with $|κ(x,y)-κ(x,x)|\leq c_5|x-y|^ρ$ for $|x-y|<1$ and $ρ>α/2$. We establish two-sided estimates for the transition densities of such processes in $C^{1,η}$ open sets when $η\in (α/2, 1]$. In particular, our result includes (relativistic) symmetric stable processes and finite-range stable processes in $C^{1,η}$ open sets when $η\in (α/2, 1]$.
In this paper, we consider a large class of subordinate Brownian motions $X$ via subordinators with Laplace exponents which are complete Bernstein functions satisfying some mild scaling conditions at zero and at infinity. We first discuss how such conditions govern the behavior of the subordinator and the corresponding subordinate Brownian motion for both large and small time and space. Then we establish a global uniform boundary Harnack principle in (unbounded) open sets for the subordinate Brownian motion. When the open set satisfies the interior and exterior ball conditions with radius $R >0$, we get a global uniform boundary Harnack principle with explicit decay rate. Our boundary Harnack principle is global in the sense that it holds for all $R>0$ and the comparison constants do not depend on $R$, and it is uniform in the sense that it holds for all balls with radii $r \le R$ and the comparison constants depend neither on $D$ nor on $r$. As an application, we give sharp two-sided estimates for the transition density and the Green function of such subordinate Brownian motions in the half-space.
In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels, in C^{1,1} open sets D in R^d, of a large class of subordinate Brownian motions with Gaussian components. When D is bounded, our sharp two-sided Dirichlet heat kernel estimates hold for all t>0. Integrating the heat kernel estimates with respect to the time variable t, we obtain sharp two-sided estimates for the Green functions, in bounded C^{1,1} open sets, of such subordinate Brownian motions with Gaussian components.
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 κ-fat open set, the sharp two-sided estimates are given in terms of surviving probabilities and the global transition density of the Levy process. When D is a C^{1, 1} open set and the Levy exponent of the process is given by Ψ(ξ)= ϕ(|ξ|^2) with ϕbeing a complete Bernstein function satisfying a mild growth condition at infinity, our two-sided estimates are explicit in terms of Ψ, the distance function to the boundary of D and the jumping kernel of X, which give an affirmative answer to the conjecture posted in [Potential Anal., 36 (2012) 235-261]. Our results are the first sharp two-sided Dirichlet heat kernel estimates for a large class of symmetric Levy processes with general Levy exponents. We also derive an explicit lower bound estimate for symmetric Levy processes on R^d in terms of their Levy exponents.
In this paper we prove a parabolic version of the Littlewood-Paley inequality for the operators of the type $ϕ(-Δ)$, where $ϕ$ is a Bernstein function. As an application, we construct an $L_p$-theory for the stochastic integro-differential equations of the type $du=(-ϕ(-Δ)u+f)dt +gdW_t$.
A subordinate Brownian motion $X$ is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $ϕ$ of the corresponding subordinator satisfies some mild conditions, we first prove the scale invariant boundary Harnack inequality for $X$ on arbitrary open sets. Then we give an explicit form of sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded $C^{1,1}$ open set. As a consequence, we prove the boundary Harnack inequality for $X$ on any $C^{1,1}$ open set with explicit decay rate. Unlike {KSV2, KSV4}, our results cover geometric stable processes and relativistic geometric stable process, i.e. the cases when the subordinator has the Laplace exponent $$ϕ(λ)=\log(1+λ^{α/2}) (0<α\leq 2, d > α)$$ and $$ϕ(λ)=\log(1+(λ+m^{α/2})^{2/α}-m) (0<α<2,\, m>0, d >2) .$$