The classical Ray-Knight theorems for Brownian motion determine the law of its local time process either at the first hitting time of a given value a by the local time at the origin, or at the first hitting time of a given position b by Brownian motion. We extend these results by describing the local time process jointly for all a and all b, by means of stochastic integral with respect to an appropriate white noise. Our result applies to $μ$-processes, and has an immediate application: a $μ$-process is the height process of a Feller continuous-state branching process (CSBP) with immigration (Lambert [10]), whereas a Feller CSBP with immigration satisfies a stochastic differential equation driven by a white noise (Dawson and Li [7]); our result gives an explicit relation between these two descriptions and shows that the stochastic differential equation in question is a reformulation of Tanaka's formula.
We study the bijection between binary Galton--Watson trees in continuous time and their exploration process, both in the sub- and in the supercritical cases. We then take the limit over renormalized quantities, as the size of the population tends to infinity. We thus deduce Delmas' generalization of the second Ray--Knight theorem.
We provide a short proof of the Ray-Knight second generalized Theorem, using a martingale which can be seen (on the positive quadrant) as the Radon-Nikodym derivative of the reversed vertex-reinforced jump process measure with respect to the Markov jump process with the same conductances. Next we show that a variant of this process provides an inversion of that Ray-Knight identity. We give a similar result for the Ray-Knight first generalized Theorem.
We provide a simple forest model to encode the genealogical structure of a multitype Galton-Watson process with immigration. We provide two encodings of these forests by stochastic processes. We show, under appropriate conditions, the depth-first encodings of each particular type converge to a solution to a system of stochastic integral equations involving height processes perturbed by functionals of their local times. The forest picture allows us to extend the Ray-Knight theorem and show that local time of the solution to the system of equations form a multitype continuous state branching pro…
We consider a discrete model of population with interaction where the birth and death rates are non linear functions of the population size. After proceeding to renormalization of the model parameters, we obtain in the limit of large population that the population size evolves as a diffusion solution of the SDE Z^x_t =x+\int_0^t f(Z^x_s)ds+2\int_0^t\int_0^{Z^x_s}W(ds,du), where W(ds,du) is a time space white noise on ([0,\infty))^2. We give a Ray-Knight representation of this diffusion in terms of the local times of a reflected Brownian motion H with a drift that depends upon the local time ac…
We study composition-valued continuous-time Markov chains that appear naturally in the framework of Chinese Restaurant Processes (CRPs). As time evolves, new customers arrive (up-step) and existing customers leave (down-step) at suitable rates derived from the ordered CRP of Pitman and Winkel (2009). We relate such up-down CRPs to the splitting trees of Lambert (2010) inducing spectrally positive Lévy processes. Conversely, we develop theorems of Ray-Knight type to recover more general up-down CRPs from the heights of Lévy processes with jumps marked by integer-valued paths. We further establi…
In 1981 F. Knight published an article with a partial solution to a problem proposed by Ito-McKean see [Ito,[p.217]]. In this paper Knight, see [Knight], characterized the Levy measures of gap diffusions also known as quasi-diffusions. The proof is very elegant but it uses quite a lot functional analysis, more specifically spectral Krein Theory. We present a new proof of Knight's Theorem, defined at the beginning of the Introduction as well as the new proof of the same theorem referred as Theorem 2.6 in Section 2.
We use generalized Ray-Knight theorems introduced by Bálint Tóth in 1996 together with techniques developed for excited random walks as main tools for establishing positive and negative results concerning convergence of some classes of diffusively scaled self-interacting random walks (SIRWs) to Brownian motions perturbed at extrema (BMPE). Tóth's work studied two classes of SIRWs: asymptotically free and polynomially self-repelling walks. For both classes Toth has shown, in particular, that the distribution function of a scaled SIRW observed at independent geometric times converges to that of …
A stochastic flow of homeomorphisms of the real line previously studied by Bass and Burdzy is shown to arise in describing a Brownian motion conditional on knowing its local times on hitting a fixed level. This makes it possible to connect Ray-Knight type results for the flow with the classical Ray-Knight theorems for Brownian motion.
The purpose of this note is to prove a central limit theorem for the $L^3$-modulus of continuity of the Brownian local time using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^3$-modulus of the Brownian local time.
We investigate the local times of a continuous-time Markov chain on an arbitrary discrete state space. For fixed finite range of the Markov chain, we derive an explicit formula for the joint density of all local times on the range, at any fixed time. We use standard tools from the theory of stochastic processes and finite-dimensional complex calculus. We apply this formula in the following directions: (1) we derive large deviation upper estimates for the normalized local times beyond the exponential scale, (2) we derive the upper bound in Varadhan's \chwk{l}emma for any measurable functional o…
The purpose of this note is to prove a central limit theorem for the $L^2$-modulus of continuity of the Brownian local time obtained in \cite{CLMR}, using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^2$-modulus of the Brownian local time.
We consider continuous-time random interlacements on a transient weighted graph. We prove an identity in law relating the field of occupation times of random interlacements at level u to the Gaussian free field on the weighted graph. This identity is closely linked to the generalized second Ray-Knight theorem, and uniquely determines the law of occupation times of random interlacements at level u.
We derive a Ray-Knight type theorem for the local time process (in the space variable) of a skew Brownian motion up to an independent exponential time. It is known that the local time seen as a density of the occupation measure and taken with respect to the Lebesgue measure has a discontinuity at the skew point (in our case at zero), but the local time taken with respect to the speed measure is continuous. In this paper we discuss this discrepancy by characterizing the dynamics of the local time process in both these cases. The Ray-Knight type theorem is applied to study integral functionals o…
We study an ordinary differential equation controlled by a stochastic process. We present results on existence and uniqueness of solutions, on associated local times (Trotter and Ray-Knight theorems), and on time and direction of bifurcation. A relationship with Lipschitz approximations to Brownian paths is also discussed.
We consider the continuous time version of the `true' or `myopic' self-avoiding random walk with site repulsion in 1d. The Ray-Knight-type method which was applied to the discrete time and edge repulsion case, is applicable to this model with some modifications. We present a limit theorem for the local time of the walk and a local limit theorem for the displacement.
We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a new method of investigations based on a study of the Radon-Nikodym density of the ERW distribution wit…
In this note we first consider local times of random walks killed at leaving positive half-axis. We prove that the distribution of the properly rescaled local time at point $N$ conditioned on being positive converges towards an exponential distribution. The proof is based on known results for conditioned random walks, which allow to determine the asymptotic behaviour of moments of local times. Using this information we also show that the field of local times of a reflected random walk converges in the sense of finite dimensional distributions. This is in the spirit of the seminal result by Kni…
We proof a limit theorem for moments in space of the increments of Brownian local time. As special cases for the second and third moments, previous results by Chen et al. (Ann. Prob. 38, 2010, no. 1) and Rosen (Stoch. Dyn. 11, 2011, no. 1), which were later reproven by Hu and Nualart (Electron. Commun. Probab. 14, 2009; Electron. Commun. Probab. 15, 2010) and Rosen (Séminaire de Probabilités XLIII, Springer, 2011) are included. Furthermore, a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a fundamentally different approach by exc…
We prove an exponential concentration bound for cover times of general graphs in terms of the Gaussian free field, extending the work of Ding-Lee-Peres and Ding. The estimate is asymptotically sharp as the ratio of hitting time to cover time goes to zero. The bounds are obtained by showing a stochastic domination in the generalized second Ray-Knight theorem, which was shown to imply exponential concentration of cover times by Ding. This stochastic domination result appeared earlier in a preprint of Lupu, but the connection to cover times was not mentioned.
In this expository paper we describe the pathwise behaviour of the integral functional $\int_0^t f(Y_u)\,\dd u$ for any $t\in[0,ζ]$, where $ζ$ is (a possibly infinite) exit time of a one-dimensional diffusion process $Y$ from its state space, $f$ is a nonnegative Borel measurable function and the coefficients of the SDE solved by $Y$ are only required to satisfy weak local integrability conditions. Two proofs of the deterministic characterisation of the convergence of such functionals are given: the problem is reduced in two different ways to certain path properties of Brownian motion where ei…
By the methods of probability and duality technique, we give some comparison theorems for the solutions of infinite horizon forward-backwad stochastic differential equations.
We investigate a specific infinite urn scheme first considered by Karlin (1967). We prove functional central limit theorems for the total number of urns with at least k balls for different k.
Under quasi-monotone assumptions for coefficients, we show one kind of comparison theorem for multi-dimensional\textbf{\}backward doubly stochastic differential equations on infinite horizon. An example is given as well.
This is a note on a local ergodic theorem for a symmetric exclusion process defined on an infinite tower of coverings, which is associated with a finitely generated residually finite amenable group.
We study the infinite urn scheme when the balls are sequentially distributed over an infinite number of urns labelled 1,2,... so that the urn $j$ at every draw gets a ball with probability $p_j$, $\sum_j p_j=1$. We prove functional central limit theorems for discrete time and the poissonised version for the urn occupancies process, for the odd-occupancy and for the missing mass processes extending the known non-functional central limit theorems.
We present a new approach to the bootstrap for chains of infinite order taking values on a finite alphabet. It is based on a sequential Bootstrap Central Limit Theorem for the sequence of canonical Markov approximations of the chain of infinite order. Combined with previous results on the rate of approximation this leads to a Central Limit Theorem for the bootstrapped estimator of the sample mean which is the main result of this paper.
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.
We consider a variant of self-repelling random walk on the integer lattice Z where the self-repellence is defined in terms of the local time on oriented edges. The long-time asymptotic scaling of this walk is surprisingly different from the asymptotics of the similar process with self-repellence defined in terms of local time on unoriented edges. We prove limit theorems for the local time process and for the position of the random walker. The main ingredient is a Ray-Knight-type of approach. At the end of the paper, we also present some computer simulations which show the strange scaling behav…
A necessary and sufficient condition is established for an asymptotically stable renewal process to satisfy the strong renewal theorem. This result is valid for all alpha in (0, 1), thus completing a result for alpha in (1/2, 1) which was proved in the 1963 paper of Garsia and Lamperti [6]. This paper is superseded by arXiv:1612.07635.
In this paper, we first establish the reflected backward stochastic difference equations with finite state (FS-RBSDEs for short). Then we explore the Existence and Uniqueness Theorem as well as the Comparison Theorem by "one step" method. The connections between FS-RBSDEs and optimal stopping time problems are investigated and we also show that the optimal stopping problems with multiple priors under Knightian uncertainty is a special case of our FS-RBSDEs. As a byproduct we develop the general theory of g-martingales in discrete time with finite state including Doob-Mayer Decomposition Theore…
Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation.
We study rigidity phenomenons for infinite dimension diffusion operators of positive curvature using semigroup interpolations. In particular, for such diffusions, an analogous statement of Obata's theorem is established. Moreover, the same rigidity holds for the Bakry--Ledoux isoperimetric comparison Theorem - a result due to Franck Morgan. Recently, Mossel and Neeman have exploited the semigroup proof of Bakry and Ledoux to derive dimension free bounds for the Gaussian isoperimetry. We extend theirs arguments to obtain in particular new quantitative bounds on the spherical isoperimetric inequ…
A Riemann-Roch theorem on graph was initiated by M. Baker and S. Norine. In their article [2], a Riemann-Roch theorem on a finite graph with uniform vertex-weight and uniform edge-weight was established and it was suggested a Riemann-Roch theorem on an infinite graph was feasible. In this article, we take an edge-weighted infinite graph and focus on the importance of the spectral gaps of the Laplace operators defined on its finite subgraphs naturally given by Q-valued positive weights on the edges. We build a potential theoretic scheme for proof of a Riemann-Roch theorem on the edge-weighted i…
We investigate the local times of a continuous-time Markov chain on an arbitrary discrete state space. For fixed finite range of the Markov chain, we derive an explicit formula for the joint density of all local times on the range, at any fixed time. We use standard tools from the theory of stochastic processes and finite-dimensional complex calculus. We apply this formula in the following directions: (1) we derive large deviation upper estimates for the normalized local times beyond the exponential scale, (2) we derive the upper bound in Varadhan's lemma for any measurable functional of the l…
The Radon transform is one of the most useful and applicable tools in functional analysis. First constructed by John Radon in 1917 it has now been adapted to several settings. One of the principle theorems involving the Radon transform is the Support Theorem. In this paper, we discuss how the Radon transform can be constructed in the white noise setting. We also develop a Support Theorem in this setting.
In this paper we consider stochastic integration with respect to cylindrical Brownian motion in infinite dimensional spaces. We study weak characterizations of stochastic integrability and present a natural continuation of results of van Neerven, Weis and the second named author. The limitation of weak characterizations will be demonstrated with a nontrivial counterexample. The second subject treated in the paper addresses representation theory for random variables in terms of stochastic integrals. In particular, we provide an infinite dimensional version of Dudley's representation theorem for…
Local actions of $\mathbb{P}_\mathbb{N}$, the group of finite permutations on $\mathbb{N}$, on quasi-local algebras are defined and proved to be $\mathbb{P}_\mathbb{N}$-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of $C^*$-algebras, which include the CAR algebra, are then addressed as particular examples of quasi…
In this paper free harmonic analysis tools are used to study parabolic iteration in the complex upper half-plane. The main result here is a complete characterization for the norming constants in the monotonic central limit theorem. This allows us to construct a new class of conservative and ergodic measure-preserving transformations on the real line with Lebesgue measure. Among all, we mention that the generalized Boole transformation with infinitely many poles is shown to be conservative, as long as its residues are summable.
