Let $u=\{u(t,x);t \in [0,T], x \in {\mathbb{R}}^{d}\}$ be the process solution of the stochastic heat equation $u_{t}=Δu+ \dot F, u(0,\cdot)=0$ driven by a Gaussian noise $\dot F$, which is white in time and has spatial covariance induced by the kernel $f$. In this paper we prove that the process $u$ is locally germ Markov, if $f$ is the Bessel kernel of order $α=2k,k \in \bN_{+}$, or $f$ is the Riesz kernel of order $α=4k,k \in \bN_{+}$.
We construct a nonstandard martingale from a discrete Markov chain. This is shown to be useful for solving the heat equation with a non smooth initial condition. We show that the nonstandard solution to the heat equation with a smooth initial condition specialises to the classical solution.
We prove sharp estimates on heat kernels and Green functions for subordinate Markov processes with both discrete an continuous time, under relatively weak assumptions about original processes as well as Laplace exponents of subordinators. We also show stability of heat kernel estimates in the discrete settings.
By making full use of heat kernel estimates, we establish the integral tests on the zero-one laws of upper and lower bounds for the sample path ranges of symmetric Markov processes. In particular, these results concerning on upper rate bounds are applicable for local and non-local Dirichlet forms, while lower rate bounds are investigated in both subcritical setting and critical setting.
We consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal Lévy processes with weak scaling conditions. First, we establish sharp two-sided heat kernel estimates for these processes in $C^{1,1}$ open sets. As corollaries of our main results, we obtain sharp two-sided Green function estimates and a scale invariant boundary Harnack inequality with explicit decay rates in $C^{1,1}$ open sets.
In this paper, we establish sharp two-sided heat kernel estimates for a large class of symmetric Markov processes in exterior $C^{1,η}$ open sets for all $t> 0$. The processes are symmetric pure jump Markov processes with jumping kernel intensity $$κ(x, y)ψ(|x-y|)^{-1}|x-y|^{-d-α}$$ where $α\in(0,2)$, $ψ$ is an increasing function on $[ 0, \infty)$ with $ψ(r)=1$ on $0<r\le 1$ and $c_1e^{c_2r^β}\le ψ(r)\le c_3e^{c_4r^β}$ on $r>1$ for $β\in[0, \infty]$. A symmetric function $κ(x, y)$ is bounded by two positive constants and $|κ(x, y)-κ(x,x)|\le c_5 |x-y|^ρ$ for $|x-y|<1$ and $ρ>α/2$. As a coroll…
It is now well known that curvature conditions à la Bakry-Emery are equivalent to contraction properties of the heat semigroup with respect to the classical quadratic Wasserstein distance. However, this curvature condition may include a dimensional correction which up to now had not induced any strenghtening of this contraction. We first consider the simplest example of the Euclidean heat semigroup, and prove that indeed it is so. To consider the case of a general Markov semigroup, we introduce a new distance between probability measures, based on the semigroup, and adapted to it. We prove tha…
Let $Z=(Z^{1}, \ldots, Z^{d})$ be the $d$-dimensional Lévy processes where $Z^{i}$'s are independent $1$-dimensional Lévy processes with jump kernel $J^{ϕ, 1}(u,w) =|u-w|^{-1}ϕ(|u-w|)^{-1}$ for $u, w\in \mathbb R$. Here $ϕ$ is an increasing function with weak scaling condition of order $\underline α, \overline α\in (0, 2)$. Let $J(x,y) \asymp J^ϕ(x,y)$ be the symmetric measurable function where \begin{align*} J^ϕ(x,y):=\begin{cases} J^{ϕ, 1}(x^i, y^i)\qquad&\text{ if $x^i \ne y^i$ for some $i$ and $x^j = y^j$ for all $j \ne i$}\\ 0\qquad&\text{ if $x^i \ne y^i$ for more than one index $i$.} \e…
In this paper, we establish the coincidence of two classes of $L^p$-Kato class measures in the framework of symmetric Markov processes admitting upper and lower estimates of heat kernel under mild conditions. One class of $L^p$-Kato class measures is defined by the $p$-th power of positive order resolvent kernel, another is defined in terms of the $p$-th power of Green kernel depending on some exponents related to the heat kernel estimates. We also prove that $q$-th integrable functions on balls with radius $1$ having uniformity of its norm with respect to centers are of $L^p$-Kato class if $q…
In this paper, we establish sharp two-sided estimates for transition densities of a large class of subordinate Markov processes. As applications, we show that the parabolic Harnack inequality and Hölder regularity hold for parabolic functions of such processes, and derive sharp two-sided Green function estimates.
It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of \cite{CKS} for Nash-type inequalities with power order considerably and also extend that of \cite{Gri} for second order differential operators on a complete non-compact manifold.
In this paper, we study the transition densities of pure-jump symmetric Markov processes in $ {\mathbb R}^d$, whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of transition densities (heat kernel estimates) for such processes. This is the first study on global heat kernel estimates of jump processes (including non-Lévy processes) whose weak scaling index is not necessarily strictly less than 2. As an application, we proved that the finite second moment…
A stable-like process is a Feller process $(X_t)_{t\geq 0}$ taking values in $\mathbb{R}^d$ and whose generator behaves, locally, like an $α$-stable Lévy process, but the index $α$ and all other characteristics may depend on the state space. More precisely, the jump measure need not to be symmetric and it strongly depends on the current state of the process; moreover, we do not require the gradient term to be dominated by the pure jump part. Our approach is to understand the above phenomena as suitable microstructural perturbations. We show that the corresponding martingale problem is well-pos…
We show two-sided bounds of heat kernel for anisotropic non-singular symmetric pure jump Markov process whose jump kernel $J(x,y)$ is comparable to $\frac{{\bf 1}_{\mathcal{V}}(x-y)}{|x-y|^{d+α}}$, where $\mathcal{V}$ is a union of symmetric cones, $0<α<2$ and $x,y\in\mathbb{R}^d$.
In this paper, we investigate a class of hybrid stochastic heat equations. By explicit formulae of solutions, we not only reveal the sample Lyapunov exponents but also discuss the $p$th moment Lyapnov exponents. Moreover, several examples are established to demonstrate that unstable (deterministic or stochastic) dynamical systems can be stabilized by Markovian switching.
We apply convergence results for discrete Markov chains, to prove the existence of an equilibrium limit in the nonstandard heat equation. We construct a nonstandard backward martingale from a nonstandard solution, and show, using the Feynman-Kac method, how to derive an explicit formula for such solutions, when the initial condition is S-continuous. Finally, we prove that that the nonstandard solution to the heat equation, with a smooth initial condition, specialises to the classical solution.
We consider a stochastic heat equation driven by a space-time white noise and with a singular drift, where a local-time in space appears. The process we study has an explicit invariant measure of Gibbs type, with a non-convex potential. We obtain existence of a Markov solution, which is associated with an explicit Dirichlet form. Moreover we study approximations of the stationary solution by means of a regularization of the singular drift or by a finite-dimensional projection.
In this paper we study the transition densities for a large class of non-symmetric Markov processes whose jumping kernels decay exponentially or subexponentially. We obtain their upper bounds which also decay at the same rate as their jumping kernels. When the lower bounds of jumping kernels satisfy the weak upper scaling condition at zero, we also establish lower bounds for the transition densities, which are sharp.
In this paper we show that Dirichlet heat kernel estimates for a class of (not necessarily symmetric) Markov processes are stable under non-local Feynman-Kac perturbations. This class of processes includes, among others, (reflected) symmetric stable-like processes on closed $d$-sets in $\bR^d$, killed symmetric stable processes, censored stable processes in $C^{1, 1}$ open sets as well as stable processes with drifts in bounded $C^{1, 1}$ open sets.
We show that solutions of the stochastic heat equation driven by space-time white noise, although not smooth, meaningfully solve the two-dimensional Toda equations. Then by extending our arguments we show the time evolution of the multilayer process introduced by O'Connell and Warren is conjugate to a flow induced by the stochastic heat equation. In particular this establishes a Markov property conjectured by O'Connell and Warren. It also defines, for the first time, the multilayer process started from a general initial condition.
Heat kernel estimates for Markov processes with blowing-up jump kernels
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Seoul National University（首尔国立大学）；Dr. Franjo Tuđman Defense and Security University（弗兰约·图季曼国防与安全大学）；University of Zagreb（萨格勒布大学）
We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovasz and Kannan, can be refined to apply to the maximum relative deviation $|p^n(x,y)/π(y)-1|$ of the distribution at time $n$ from the stationary distribution $π$. We then extend our results to Markov chains on infinite state spaces and to continu…
In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on $\mathbb{R}^+$ or $\mathbb{R}$ with values in a general Riemannian maifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper \cite{RWZZ17} on finite volume case. Moveover, we also obtain some functional inequalities associated to these Markov processes. This implies that on infinite volume case, the exponential ergodicity of the solution if the Ricci curvature is strictly positive and the non-ergodicit…
In this article we develop a general method by which one can explicitly evaluate certain sums of $n$-th powers of products of $d\geq 1$ elementary trigonometric functions evaluated at $\mathbf{m}=(m_1,\ldots,m_d)$-th roots of unity. Our approach is to first identify the individual terms in the expression under consideration as eigenvalues of a discrete Laplace operator associated to a graph whose vertices form a $d$-dimensional discrete torus $G_{\mathbf{m}}$ which depends on $\mathbf{m}$. The sums in question are then related to the $n$-th step of a Markov chain on $G_{\mathbf{m}}$. The Marko…
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 study the long-time behavior of the Cesaro means of fundamental solutions for fractional evolution equations corresponding to random time changes in the Brownian motion and other Markov processes. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order derivatives.
