We consider the Cauchy problem for a pseudo-differential operator which has a translation-invariant and analytic symbol. For a certain set of initial conditions, a formal solution is obtained by a perturbative expansion. The series so obtained can be re-expressed in terms of generalized Feynman graphs and Feynman rules. The logarithm of the solution can then be represented by a series containing only the connected Feynman graphs. Under some conditions, it is shown that the formal solution uniquely determines the real solution by means of Borel transforms. The formalism is then applied to probabilistic Lévy distributions. Here, the Gaussian part of such a distribution is re-interpreted as a initial condition and a large diffusion expansion for Lévy densities is obtained. It is outlined how this expansion can be used in statistical problems that involve Lévy distributions.
Let $X$ be a symmetric strong Markov process on a Luzin space. In this paper, we present criteria of the $L^p$-independence of spectral bounds for generalized non-local Feynman-Kac semigroups of $X$ that involve continuous additive functionals of $X$ having zero quadratic variations and discontinuous additive functionals of $X$.
This work develops further a probabilist approach to the asymptotic behavior of growth-fragmentation semigroups via the Feynman-Kac formula, which was introduced in a joint article with A.R. Watson [4]. Here, it is first shown that the sufficient condition for a Malthusian behavior which was established in [4], is also necessary. We then provide a simple criterion to ensure exponential speed of convergence, which enables us to treat cases than were not covered previously in the literature.
In the first part of this paper, we give a useful criterion for uniform integrability of exponential martingales in the context of Markov processes. The condition of this criterion is easy to verify and is, in general, much weaker than the commonly used Novikov's condition. In the second part of this paper, we present a new approach to the study of spectral bounds of Feynman-Kac semigroups for a large class of symmetric Markov processes. We first establish criteria for the $L^p$-independence of spectral bounds for Feynman-Kac semigroups generated by continuous additive functionals, using gauge…
We prove Bismut-type formulae for the first and second derivatives of a Feynman-Kac semigroup on a complete Riemannian manifold. We derive local estimates and give bounds on the logarithmic derivatives of the integral kernel. Stationary solutions are also considered. The arguments are based on local martingales, although the assumptions are purely geometric.
For a (non-symmetric) strong Markov process $X$, consider the Feynman--Kac semigroup \[T_t^Af(x):=\mathbb {E}^x\bigl[e^{A_t}f(X_t)\bigr],\quad x\in {\mathbb {R}^n}, t>0,\] where $A$ is a continuous additive functional of $X$ associated with some signed measure. Under the assumption that $X$ admits a transition probability density that possesses upper and lower bounds of certain type, we show that the kernel corresponding to $T_t^A$ possesses the density $p_t^A(x,y)$ with respect to the Lebesgue measure and construct upper and lower bounds for $p_t^A(x,y)$. Some examples are provided.
Suppose that $α\in (0,2)$ and that $X$ is an $α$-stable-like process on $\R^d$. Let $F$ be a function on $\R^d$ belonging to the class $\bf{J_{d,α}}$ (see Introduction) and $A_{t}^{F}$ be $\sum_{s \le t}F(X_{s-},X_{s}), t> 0$, a discontinuous additive functional of $X$. With neither $F$ nor $X$ being symmetric, under certain conditions, we show that the Feynman-Kac semigroup $\{S_{t}^{F}:t \ge 0\}$ defined by $$ S_{t}^{F}f(x)=\mathbb{E}_{x}(e^{-A_{t}^{F}}f(X_{t}))$$ has a density $q$ and that there exist positive constants $C_1,C_2,C_3$ and $C_4$ such that $$C_{1}e^{-C_{2}t}t^{-\frac{d}α}(1 \w…
We introduce a stochastic process and functional that should describe the semigroup generated by the stochastic Bessel operator. Recently Gorin and Shkolnikov showed that the largest eigenvalues for certain random matrix ensembles with soft edge behavior can be understood by analyzing large powers of tridiagonal matrices, which converge to operators in the stochastic Airy semigroup. In this article we make some progress towards realizing Gorin and Shkolnikov's program at the random matrix hard edge. We analyze large powers of a suitable tridiagonal matrix model (a slight modification of the $β…
Feynman-Kac semigroups appear in various areas of mathematics: non-linear filtering, large deviations theory, spectral analysis of Schrodinger operators among others. Their long time behavior provides important information, for example in terms of ground state energy of Schrodinger operators, or scaled cumulant generating function in large deviations theory. In this paper, we propose a simple and natural extension of the stability of Markov chains for these non-linear evolutions. As other classical ergodicity results, it relies on two assumptions: a Lyapunov condition that induces some compact…
Long time behavior of killed Feynman-Kac semigroups with singular Schr{ö}dinger potentials
Université Clermont Auvergne（克莱蒙费朗大学）；CNRS（法国国家科学研究中心）；School of Mathematics and Statistics, Yunnan University（云南大学数学与统计学院）；School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）；Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）
Quantum tunnelling phenomenon allows a particle in Schrödinger mechanics tunnels through a barrier that it classically could not overcome. Even the infinite potentials do not always form impenetrable barriers. We discuss an answer to the following question: What is a critical magnitude of potential, which creates impenetrable barrier and for which the corresponding Schrödinger evolution system separates? In addition we describe some quantitative estimates for the separating effect in terms of cut-off potentials.
A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of $n$-fold iterated integrals of some elementary functions as $n\to\infty$. In this note we obtain some Feynman formulae for a class of semigroups associated with Feller processes. Finite dimensional integrals in the Feynman formulae give approximations for functional integrals in some Feynman--Kac formulae corresponding to the underlying processes. Hence, these Feynman formulae gi…
Free regular convolution semigroups describe the distribution of free subortinators, while Bondesson class convolution semigroups correspond to classical subordinators with completely monotone Levy density. We show that these two classes of convolution semigroups are in bijection with the class of complete Bernstein functions and we establish an integral identity linking the two semigroups. We provide several explicit examples that illustrate this result.
This expository paper is intended for a short self-contained introduction to the theory of infinite convolutions of probability measures on Polish semigroups. We give the proofs of the Rees decomposition theorem of completely simple semigroups, the Ellis--Żelazko theorem, the convolution factorization theorem of convolution idempotents, and the convolution factorization theorem of cluster points of infinite convolutions.
Let $(X_t)_{t\ge 0}$ be a symmetric strong Markov process generated by non-local regular Dirichlet form $(D,\D(D))$ as follows \begin{equation*} \begin{split} & D(f,g)=\int_{\R^d}\int_{\R^d}\big(f(x)-f(y)\big)\big(g(x)-g(y)\big) J(x,y)\,dx\,dy, \quad f,g\in \D(D) \end{split} \end{equation*} where $J(x,y)$ is a strictly positive and symmetric measurable function on $\R^d\times \R^d$. We study the intrinsic hypercontractivity, intrinsic supercontractivity and intrinsic ultracontractivity for the Feynman-Kac semigroup $$ T^V_t(f)(x)=\Ee^x\left(\exp\Big(-\int_0^tV(X_s)\,ds\Big)f(X_t)\right),\,\, x…
Extension of two known facts concerning subordination is made. The first fact is that, in subordination of 1-dimensional Brownian motion with drift, selfdecomposability is inherited from subordinator to subordinated. This is extended to subordination of cone-parameter convolution semigroups. The second fact is that, in subordination of strictly stable cone-parameter convolution semigroups on $\mathbb{R}^d$, selfdecomposability is inherited from subordinator to subordinated. This is extended to semi-selfdecomposability.
It is proved that a general non-differentiable skew convolution semigroup associated with a strongly continuous semigroup of linear operators on a real separable Hilbert space can be extended to a differentiable one on the entrance space of the linear semigroup. A cadlag strong Markov process on an enlargement of the entrance space is constructed from which we obtain a realization of the corresponding Ornstein-Uhlenbeck process. Some explicit characterizations of the entrance spaces for special linear semigroups are given.
Our goal is to find classes of convolution semigroups on Lie groups $G$ that give rise to interesting processes in symmetric spaces $G/K$. The $K$-bi-invariant convolution semigroups are a well-studied example. An appealing direction for the next step is to generalise to right $K$-invariant convolution semigroups, but recent work of Liao has shown that these are in one-to-one correspondence with $K$-bi-invariant convolution semigroups. We investigate a weaker notion of right $K$-invariance, but show that this is, in fact, the same as the usual notion. Another possible approach is to use genera…
A special type of immigration associated with measure-valued branching processes is formulated by using skew convolution semigroups. We give characterization for a general inhomogeneous skew convolution semigroup in terms of probability entrance laws. The related immigration process is constructed by summing up measure-valued paths in the Kuznetsov process determined by an entrance rule. The behavior of the Kuznetsov process is then studied, which provides insights into trajectory structures of the immigration process. Some well-known results on excessive measures are formulated in terms of st…
This note is devoted to representation of some evolution semigroups. The semigroups are generated by pseudo-differential operators, which are obtained by different (parametrized by a number $τ$) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains functions which are second order polynomials with respect to the momentum variable and also some other functions. The considered semigroups are represented as limits of $n$-fold iterated integrals when $n$ tends to infinity (such representations are called Feynman formulae). Some of…
We study (weakly) continuous convolution semigroups of probability measures on a Lie group G or a homogeneous space G/K, where K is a compact subgroup. We show that such a convolution semigroup is the convolution product of its initial measure and a continuous convolution semigroup with initial measure at the identity of G or the origin of G/K. We will also obtain an extension of Dani-McCrudden's result on embedding an infinitely divisible probability measure in a continuous convolution semigroup on a Lie group to a homogeneous space.
We introduce a class of Lévy processes subject to specific regularity conditions, and consider their Feynman-Kac semigroups given under a Kato-class potential. Using new techniques, first we analyze the rate of decay of eigenfunctions at infinity. We prove bounds on $λ$-subaveraging functions, from which we derive two-sided sharp pointwise estimates on the ground state, and obtain upper bounds on all other eigenfunctions. Next, by using these results, we analyze intrinsic ultracontractivity and related properties of the semigroup refining them by the concept of ground state domination and asym…
Skew convolution semigroups play an important role in the study of generalized Mehler semigroups and Ornstein-Uhlenbeck processes. We give a characterization for a general skew convolution semigroup on real separable Hilbert space whose characteristic functional is not necessarily differentiable at the initial time. A connection between this subject and catalytic branching superprocesses is established through fluctuation limits, providing a rich class of non-differentiable skew convolution semigroups. Path regularity of the corresponding generalized Ornstein-Uhlenbeck processes in different t…
In this paper we will take under consideration subordinators and their inverse processes (hitting-times). We will present in general the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore we will discuss the concept of time-changed $C_0$-semigroup in case the time-change is performed by means of the hitting-time of a subordinator. We will show that such time-change give rise to bounded linear operators not preserving the semigroup property and we will present their governing equations by using aga…
We consider three magnetic relativistic Schrödinger operators which correspond to the same classical symbol $\sqrt{(ξ-A(x))^2+m^2}+V(x)$ and whose heat semigroups admit the Feynman-Kac-Itô type path integral representation $E[e^{-S^m(x,t, X)}g(x+X(t))]$. Using these representations, we prove the convergence of these heat semigroups when the mass--parameter $m$ goes to zero. Its proof reduces to the convergence of $e^{-S^m(x,t;X)}$, which yields a limit theorem for exponentials of semimartingales as functionals of Lévy processes $X$.
We design a theoretic tree-based functional representation of a class of Feynman-Kac particle distributions, including an extension of the Wick product formula to interacting particle systems. These weak expansions rely on an original combinatorial, and permutation group analysis of a special class of forests. They provide refined non asymptotic propagation of chaos type properties, as well as sharp $\LL\_p$-mean error bounds, and laws of large numbers for $U$-statistics. Applications to particle interpretations of the top eigenvalues, and the ground states of Schrödinger semigroups are also d…
Consider the symmetric non-local Dirichlet form $(D,\D(D))$ given by $$ D(f,f)=\int_{\R^d}\int_{\R^d}\big(f(x)-f(y)\big)^2 J(x,y)\,dx\,dy $$with $\D(D)$ the closure of the set of $C^1$ functions on $\R^d$ with compact support under the norm $\sqrt{D_1(f,f)}$, where $D_1(f,f):=D(f,f)+\int f^2(x)\,dx$ and $J(x,y)$ is a nonnegative symmetric measurable function on $\R^d\times \R^d$. Suppose that there is a Hunt process $(X_t)_{t\ge 0}$ on $\R^d$ corresponding to $(D,\D(D))$, and that $(L,\D(L))$ is its infinitesimal generator. We study the intrinsic ultracontractivity for the Feynman-Kac semigrou…
We establish convolution inequalities for Besov spaces $B_{p,q}^s$ and Triebel--Lizorkin spaces $F_{p,q}^s$. As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces $A_{p,q}^s$, $A \in \{B,F\}$. Our results apply to a wide class of convolution semigroups including the Gauß--Weierstraß semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian $(-Δ)^m$, and we can derive various caloric smoothing estimates.
In this paper we introduce the concept of a convolution type operation of functionals on Wiener space. It contains several kinds of the concepts of convolution products on Wiener space, which have been studied by many authors. We then investigate fundamental relationships between generalized analytic Fourier--Feynman transforms and convolution type operations. Both of the generalized analytic Fourier--Feynman transform of the convolution type operation and the convolution type operation of the generalized analytic Fourier--Feynman transforms are represented as a product of the generalized anal…
We access the edge of Gaussian beta ensembles with one spike by analyzing high powers of the associated tridiagonal matrix models. In the classical cases beta=1, 2, 4, this corresponds to studying the fluctuations of the largest eigenvalues of additive rank one perturbations of the GOE/GUE/GSE random matrices. In the infinite-dimensional limit, we arrive at a one-parameter family of random Feynman-Kac type semigroups, which features the stochastic Airy semigroup of Gorin and Shkolnikov [13] as an extreme case. Our analysis also provides Feynman-Kac formulas for the spiked stochastic Airy opera…
In this note the Chernoff Theorem is used to approximate evolution semigroups constructed by the procedure of subordination. The considered semigroups are subordinate to some original, unknown explicitly but already approximated by the same method, counterparts with respect to subordinators either with known transitional probabilities, or with known and bounded Lévy measure. These results are applied to obtain approximations of semigroups corresponding to subordination of Feller processes, and (Feller type) diffusions in Euclidean spaces, star graphs and Riemannian manifolds. The obtained appr…
We investigate relations between additive convolution semigroup and max-convolution semigroup through the law of large numbers for the free multiplicative convolution. Based on the relation, we give a formula related with Belinschi-Nica semigroup and max-Belinschi-Nica semigroup. Finally, we give several limit theorems for classical, free and Boolean extreme values.
Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a Lévy process in $\mathbb{R}^d$ and $Ω$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. In this article we consider the quantity $H (t) = \int_Ω\mathbb{P}_{x} (X_t\in Ω^c) dx$ which is called the heat content. We study its asymptotic behaviour as $t$ goes to zero for isotropic Lévy processes under some mild assumptions on the characteristic exponent. We also treat the class of Lévy processes with finite variation in full generality.
We prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on $\mathbb{R} ^d$ under the assumption that its Lévy--Khintchine exponent varies slowly. We also derive some new estimates of the transition densities and Green functions.
We give sharp bounds for the isotropic unimodal probability convolution semigroups when their Lévy-Khintchine exponent has Matuszewska indices strictly between 0 and 2.
The approximation of the Feynman-Kac semigroups by systems of interacting particles is a very active research field, with applications in many different areas. In this paper, we study the parallelization of such approximations. The total population of particles is divided into sub-populations, referred to as \emph{islands}. The particles within each island follow the usual selection / mutation dynamics. We show that the evolution of each island is also driven by a Feynman-Kac semigroup, whose transition and potential can be explicitly related to ones of the original problem. Therefore, the sam…
With appropriate notions of Hermitian vector bundles and connections over weighted graphs which we allow to be locally infinite, we prove Feynman-Kac-type representations for the corresponding semigroups and derive several applications thereof.
We prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on $\mathbb{R}^d$ under the assumption that its Lévy--Khintchine exponent is regularly varying of index between $0$ and $2$.
It was recently shown by the authors that deformations of hypergroup convolutions w.r.t. positive semicharacters can be used to explain probabilistic connections between the Gelfand pairs (SL(d,C), SU(d)) and Hermitian matrices. We here study connections between general convolution semigroups on commutative hypergroups and their deformations. We are able to develop a satisfying theory, if the underlying positive semicharacter has some growth property. We present several examples which indicate that this growth condition holds in many interesting cases.
A general affine Markov semigroup is formulated as the convolution of a homogeneous one with a skew convolution semigroup. We provide some sufficient conditions for the regularities of the homogeneous affine semigroup and the skew convolution semigroup. The corresponding affine Markov process is constructed as the strong solution of a system of stochastic equations with non-Lipschitz coefficients and Poisson-type integrals over some random sets. Based on this characterization, it is proved that the affine process arises naturally in a limit theorem for the difference of a pair of reactant proc…
Let $μ$ be a probability measure (or corresponding random variable) such that all moments $μ_n$ exist. Knowledge of the moments is not sufficient to determine infinite divisibility of the measure; we show also that infinitely divisible, and in particular lognormal, distributions lose infinitely divisibilty when censored in certain ways even if all moments are arbitrarily close to those of the uncensored distribution. The moments of a composition of k copies of $μ$ are expressed as combinatorial compositions of the $μ_n$. We express the moments of the compositions in the context of occupancy pr…
Let $μ$ denote a Borel probability measure and let $\{ μ_{t} \}_{t\geq 1}$ denote the free additive convolution semigroup of Nica and Speicher. We show that the support of these measures varies continuously in the Hausdorff metric for $t >1$. We utilize complex analytic methods and, in particular, a characterization of the absolutely continuous portion of these supports due to Huang.
The free convolution is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent unitarily invariant square random matrices or of a sum of free operators in a non commutative probability space. In the same way, the rectangular free convolution allows to deduce, from the individual singular distributions, the singular distribution of a sum of independent unitarily invariant rectangular random matrices. In this paper, we consider the regularization properties of t…
In a 1999 paper, Bercovici and Pata showed that a natural bijection between the classically, free and Boolean infinitely divisible measures held at the level of limit theorems of triangular arrays. This result was extended to include monotone convolution by the authors. In recent years, operator-valued versions of free, Boolean and monotone probability have also been developed. Belinschi, Popa and Vinnikov showed that the Bercovici-Pata bijection holds for the operator-valued versions of free and Boolean probability. In this article, we extend the bijection to include monotone probability theo…
The subject of this paper is the study of convolution semigroups of states on a locally compact quantum group, generalising classical families of distributions of a Lévy process on a locally compact group. In particular a definitive one-to-one correspondence between symmetric convolution semigroups of states and noncommutative Dirichlet forms satisfying the natural translation invariance property is established, extending earlier partial results and providing a powerful tool to analyse such semigroups. This is then applied to provide new characterisations of the Haagerup Property and Property …
We study the Feynman-Kac semigroup generated by the Schr{ö}dinger operator based on the fractional Laplacian $-(-Δ)^{α/2} - q$ in $\Rd$, for $q \ge 0$, $α\in (0,2)$. We obtain sharp estimates of the first eigenfunction $ϕ_1$ of the Schr{ö}dinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials $q$ such that $\lim_{|x| \to \infty} q(x) = \infty$ and comparable on unit balls we obtain that $ϕ_1(x)$ is comparable to $(|x| + 1)^{-d - α} (q(x) + 1)^{-1}$ and intrinsic ultracontractivity holds iff $\lim_{|x| \to \infty} q(x)/\log|x| = \i…
We consider a pair of probability measures $μ,ν$ on the unit circle such that $Σ_λ(η_ν(z))=z/η_μ(z)$. We prove that the same type of equation holds for any $t\geq 0$ when we replace $ν$ by $ν\boxtimesλ_t$ and $μ$ by $\mathbb{M}_t(μ)$, where $λ_t$ is the free multiplicative analogue of the normal distribution on the unit circle of $\mathbb{C}$ and $\mathbb{M}_t$ is the map defined by Arizmendi and Hasebe. These equations are a multiplicative analogue of equations studied by Belinschi and Nica. In order to achieve this result, we study infinite divisibility of the measures associated with subord…
The wrapping transformation $W$ is a homomorphism from the semigroup of probability measures on the real line, with the convolution operation, to the semigroup of probability measures on the circle, with the multiplicative convolution operation. We show that on a large class $\mathcal{L}$ of measures, $W$ also transforms the three non-commutative convolutions---free, Boolean, and monotone---to their multiplicative counterparts. Moreover, the restriction of $W$ to $\mathcal{L}$ preserves various qualitative properties of measures and triangular arrays. We use these facts to give short proofs of…
We prove a Feynman path integral formula for the unitary group $ \exp(-itL_{v,θ})$, $t\geq 0$, associated with a discrete magnetic Schrödinger operator $L_{v,θ}$ on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate $$ |\exp(-itL_{v,θ})(x,y)|\leq \exp(-tL_{-\mathrm{deg},0})(x,y), $$ which controls the unitary group uniformly in the potentials in terms of a Schrödinger semigroup, where the potential $\mathrm{deg}$ is the weighted degree function of the graph.
Belinschi and Nica introduced a composition semigroup on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know whether a probability measure is freely infinitely divisible or not. In this paper we further investigate this indicator, introduce a multiplicative version of it and are able to show many properties. Specifically, on the first half of the paper, we calculate how the indicator changes with respect to free and Boolean powers; we prove that free and Boolean 1/2-stable laws have free divisibility indicators equal to …
We study the multiplicative convolution for c-monotone independence. This convolution unifies the monotone, Boolean and orthogonal multiplicative convolutions. We characterize convolution semigroups for the c-monotone multiplicative convolution on the unit circle. We also prove that an infinitely divisible distribution can always be embedded in a convolution semigroup. We furthermore discuss the (non)-uniqueness of such embeddings including the monotone case. Finally connections to the multiplicative Boolean convolution are discussed.
We realize the Belinschi-Nica semigroup of homomorphisms as a free multiplicative subordination. This realization allows to define more general semigroups of homomorphisms with respect to free multiplicative convolution. For these semigroups we show that a differential equation holds, generalizing the complex Burgers equation. We give examples of free multiplicative subordination and find a relation to the Markov-Krein transform, Boolean stable laws and monotone stable laws. A similar idea works for additive subordination, and in particular we study the free additive subordination associated t…
We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and of Schur functions. We consider the set of probability distributions as a semigroup $\bold M$ equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of $\bold M$ contains either indecomposable ("prime") factors or it belongs to a class, say $I_0$, of distributions without indecomposable factors. In contrast to the classical convolution semigroup in the free ad…
