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-posed, and its solution is a strong Feller process which admits a transition density. For the transition density we obtain a representation as a sum of an explicitly given principal term -- this is essentially the density of an $α$-stable random variable whose parameters depend on the current state $x$ -- and a residual term; the $L^\infty\otimes L^1$-norm of the residual term is negligible and so is, under an additional structural assumption, the $L^\infty\otimes L^\infty$-norm. Concrete examples illustrate the relation between the assumptions and possible transition density estimates.
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.
In this paper we show that we can use Markov kernels as a model for optimal transport. This new framework can be easily translated into the standard coupling formulation of optimal transport, and we show that we can use a coupling as a Markov kernel for many optimal transport problems. Using kernels allows us to extend optimal transport to signed measures and treats the support of the measure as the salient feature. This approach reveals additional structure for one-dimensional signed optimal transport.
A known property of conditional expectation is extended to the framework of Markov kernels. Its meaning in terms of densities is provided. Some examples located in the field of clinical diagnosis are presented to delimit the main result of the paper.
All one-condition generalized inverses of the Markovian kernel I - P, where P is the transition matrix of a finite irreducible Markov chain, can be uniquely specified in terms of the stationary probabilities and the mean first passage times of the underlying Markov chain. Special sub-families include the group inverse of I - P, Kemeny and Snell's fundamental matrix of the Markov chain and the Moore-Penrose g-inverse. The elements of some sub-families of the generalized inverses can also be re-expressed involving the second moments of the recurrence time variables. Some applications to Kemeny's…
In this paper we investigate the kernel estimator of the density for a stationary reversible Markov chain. The proofs are based on a new central limit theorem for a triangular array of reversible Markov chains obtained under conditions imposed to covariances, which has interest in itself.
The Markov commutator associated to a finite Markov kernel P is the convex semigroup consisting of all Markov kernels commuting with P. Its interest comes from its relation with the hypergroup property and with the notion of Markovian duality by intertwining. In particular, it is shown that the discrete analogue of the Achour-Trim{è}che's theorem, asserting the preservation of non-negativity by the wave equations associated to certain Metropolis birth and death transition kernels, cannot be extended to all convex potentials. But it remains true for symmetric and monotone convex potentials. Key…
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.
Let $\{X_n\}_{n\in\N}$ be a Markov chain on a measurable space $\X$ with transition kernel $P$ and let $V:\X\r[1,+\infty)$. The Markov kernel $P$ is here considered as a linear bounded operator on the weighted-supremum space $\cB_V$ associated with $V$. Then the combination of quasi-compactness arguments with precise analysis of eigen-elements of $P$ allows us to estimate the geometric rate of convergence $ρ_V(P)$ of $\{X_n\}_{n\in\N}$ to its invariant probability measure in operator norm on $\cB_V$. A general procedure to compute $ρ_V(P)$ for discrete Markov random walks with identically dist…
Resolvent metrics are generalization of the resistance metric and provide unified treatment of heat kernel estimates of sub-Gaussian type under minimal conditions.
This article investigates the heat kernel of the two-dimensional uniform spanning tree. We improve previous work by demonstrating the occurrence of log-logarithmic fluctuations around the leading order polynomial behaviour for the on-diagonal part of the quenched heat kernel. In addition we give two-sided estimates for the averaged heat kernel, and we show that the exponents that appear in the off-diagonal parts of the quenched and averaged versions of the heat kernel differ. Finally, we derive various scaling limits for the heat kernel, the implications of which include enabling us to sharpen…
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.
The solution to a multivariate linear Stochastic Differential Equation (SDE) with constant initial state is well known to be a Gaussian Markov process, but its covariance kernel involves the solution to an integral equation in the general case. We show that the covariance kernel has a simpler semi-parametric form for families of such solutions representing increments of a common process. We also show that a covariance kernel of a particular parametric form is necessary and sufficient for a solution to possess stationary increments and for a Gaussian process, in considerable generality, to have…
We work out some relations between duality and intertwining in the context of discrete Markov chains, fixing up the background of previous relations first established for birth and death chains and their Siegmund duals. In view of the results, the monotone properties resulting from the Siegmund dual of birth and death chains are revisited in some detail, with emphasis on the non neutral Moran model. We also introduce an ultrametric type dual extending the Siegmund kernel. Finally we discuss the sharp dual, following closely the Diaconis-Fill study.
We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.
These notes focus on the applications of the stochastic Taylor expansion of solutions of stochastic differential equations to the study of heat kernels in small times. As an illustration of these methods we provide a new heat kernel proof of the Chern-Gauss-Bonnet theorem.
We survey the recent progress in the study of heat kernels for a class of non-symmetric non-local operators. We focus on the existence and sharp two-sided estimates of the heat kernels and their connection to jump diffusions.
Let $P$ be a Markov kernel on a measurable space $\X$ and let $V:\X\r[1,+\infty)$. We provide various assumptions, based on drift conditions, under which $P$ is quasi-compact on the weighted-supremum Banach space $(\cB_V,\|\cdot\|_V)$ of all the measurable functions $f : \X\r\C$ such that $\|f\|_V := \sup_{x\in \X} |f(x)|/V(x) < \infty$. Furthermore we give bounds for the essential spectral radius of $P$. Under additional assumptions, these results allow us to derive the convergence rate of $P$ on $\cB_V$, that is the geometric rate of convergence of the iterates $P^n$ to the stationary distri…
In this paper we show that two-sided heat kernel estimates for a class of (not necessarily symmetric) diffusions with jumps are stable under non-local Feynman-Kac perturbations.
We obtain matching two sided estimates of the heat kernel on a connected sum of parabolic manifolds, each of them satisfying the Li-Yau estimate. The key result is the on-diagonal upper bound of the heat kernel at a central point. Contrary to the nonparabolic case (which was settled in [15]), the on-diagonal behavior of the heat kernel in our case is determined by the end with the maximal volume growth function. As examples, we give explicit heat kernel bounds on the connected sums $R^2#R^2$ and $R^1#R^2$ where $R^1 = R_+\timesS^1$.
We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the Hölder continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes.
An asymptotic model for extreme behavior of certain Markov chains is the "tail chain". Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and non-extreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular va…
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.
We relate Gruet formula for the heat kernel on real hyperbolic spaces to the commonly used one derived from Millson induction. The bridge between both formulas is settled by Yor result on the joint distribution of a Brownian motion and of its exponential functional at fixed time. This result allows further to relate Gruet formula with real parameter to the heat kernel of the hyperbolic Jacobi operator and to derive a new integral representation for the heat kernel of the Maass Laplacian. When applied to harmonic AN groups, Yor result yields also new a integral representation of their correspon…
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 $x,y\in\mathbb{R}^d$.
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.
We study discrete time random walks in an environment of i.i.d. non-negative bounded conductances in $\mathbb{Z}^d$. We are interested in the anomaly of the heat-kernel decay. We improve recent results and techniques.
Gradient and stability type estimates of heat kernel associated with fractional power of a uniformly elliptic operator are obtained. $L^p$-operator norm of semigroups associated with fractional power of two uniformly elliptic operators are also obtained.
We demonstrate a relationship between the heat kernel on a finite weighted Abelian Cayley graph and Gaussian functions on lattices. This can be used to prove a new inequality for the heat kernel on such a graph: when $t \leq t'$, $$\frac{H_t(u, v)}{H_t(u,u)} \leq \frac{H_{t'}(u, v)}{H_{t'}(u,u)}$$ This was an open problem posed by Regev and Shinkar.
We study measures associated to Brownian motions on infinite-dimensional Heisenberg-like groups. In particular, we prove that the associated path space measure and heat kernel measure satisfy a strong definition of smoothness.
We prove existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular domains, in the context of general metric measure spaces. As a corollary, we prove uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R^d with positive continuous density of the Lévy measure; stable-like processes in R^d and in domains; …
In this paper, we address the problem of fitting multivariate Hawkes processes to potentially large-scale data in a setting where series of events are not only mutually-exciting but can also exhibit inhibitive patterns. We focus on nonparametric learning and propose a novel algorithm called MEMIP (Markovian Estimation of Mutually Interacting Processes) that makes use of polynomial approximation theory and self-concordant analysis in order to learn both triggering kernels and base intensities of events. Moreover, considering that N historical observations are available, the algorithm performs l…
We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schrödinger operator with negative Hardy potential $$Δ^{α/2} -λ|x|^{-α}$$ on $\RR^d$, where $α\in(0,d\wedge 2)$ and $λ>0$. The proof is purely analytical but elementary. In particular, for upper bounds of heat kernel we use the Chapman-Kolmogorov equation and adopt self-improving argument.
It is known that the couple formed by the two dimensional Brownian motion and its Lévy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequal…
Quenched and annealed heat kernel estimates are established for Fontes-Isopi-Newman (FIN) processes on spaces equipped with a resistance form. These results are new even in the case of the one-dimensional FIN diffusion, and also apply to fractals such as the Sierpinski gasket and carpet.
We show how the essential spectral radius of a bounded positive kernel, acting on bounded functions, is linked to its lower approximation by certain absolutely continuous kernels. The standart Doeblin's condition can be interpreted in this context, and, when suitably reformulated, it leads to a formula for the essential spectral radius. This results may be used to characterize the Markov kernels having a quasi-compact action on a space of measurable functions bounded with respect to some test function, when no irreducibilty and aperiodicity are assumed.
The purpose of this article is to describe all possible beliefs of market participants on objective measures under Markovian environments when a risk-neutral measure is given. To achieve this, we employ the Martin integral representation of Markovian pricing kernels. Then, we offer economic and financial implications of this representation. This representation is useful to analyze the long-term behavior of the state variable in the market. The Ross recovery theorem and the long-term behavior of cash flows are discussed as applications.
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.
We show how the evolving set methodology of Morris and Peres can be used to show Cheeger inequalities for bounding the spectral gap of a finite Markov kernel. This leads to sharp versions of several previous Cheeger inequalities, including ones involving edge-expansion, vertex-expansion, and mixtures of both. A bound on the smallest eigenvalue also follows.
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 estimate the heat kernel of the smooth open set for the isotropic unimodal pure-jump Lévy process with infinite Lévy measure and weakly scaling Lévy-Kchintchine exponent.
We study the subelliptic heat kernels of the CR three dimensional solvable Lie groups. We first classify all left-invariant sub-Riemannian structures on three dimensional solvable Lie groups and obtain representations of these groups. We give expressions for the heat kernels on these groups and obtain heat semigroup gradient bounds using a new type of curvature-dimension inequality.
Under some mild assumptions on the Levy measure and the symbol we obtain gradient estimates of Dirichlet heat kernels for pure-jump isotropic unimodal Levy processes in $R^d$.
This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the $L^p$ norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds in this setting.
We study strong ratio limit properties and the exact long time asymptotics of the heat kernel of a general second-order parabolic operator which is defined on a noncompact Riemannian manifold.
In this paper we study interior potential-theoretic properties of purely discontinuous Markov processes in proper open subsets $D\subset \mathbb{R}^d$. The jump kernels of the processes may be degenerate at the boundary in the sense that they may vanish or blow up at the boundary. Under certain natural conditions on the jump kernel, we show that the scale invariant Harnack inequality holds for any proper open subset $D\subset \mathbb{R}^d$ and prove some interior regularity of harmonic functions. We also prove a Dynkin-type formula and several other interior results.
We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical-order singularity, by transferring it to appropriate weighted space with singular weight.
We provide sharp two-sided estimates of the Fourier-Bessel heat kernel and we give sharp two-sided estimates of the transition probability density for the Bessel process in (0,1) killed at 1 and killed or reflected at 0.
We study the kernel estimator of the transition density of bifurcating Markov chains. Under some ergodic and regularity properties, we prove that this estimator is consistent and asymptotically normal. Next, in the numerical studies, we propose two data-driven methods to choose the bandwidth parameters. These methods are based on the so-called two bandwidths approach.
We propose an adaptive estimator for the stationary distribution of a bifurcating Markov Chain on $\mathbb R^d$. Bifurcating Markov chains (BMC for short) are a class of stochastic processes indexed by regular binary trees. A kernel estimator is proposed whose bandwidth is selected by a method inspired by the works of Goldenshluger and Lepski [18]. Drawing inspiration from dimension jump methods for model selection, we also provide an algorithm to select the best constant in the penalty.
