Ray Knight Research Articles

Population Dynamics and Random Genealogies

We start with stochastic models of genealogies in discrete time, distinguishing models where the population size is fixed (models of Cannings, Wright–Fisher, and Moran; compositions of bridges) with models where the population size fluctuates randomly (processes of Bienaymé–Galton–Watson, Jirina; birth–death processes, logistic branching process). Scaling limits of these models can be seen as genealogies of continuous populations. The continuous analogue of models with a fixed size is the stochastic flow of bridges of Bertoin and Le Gall. That of branching models is the continuous-state branching process. Both processes have diffusion versions called, respectively, the Fisher–Wright diffusion and the Feller diffusion. Connections between the two kinds of models are also studied, and special attention is given to extinction/fixation (probability, expected time, conditioning). When (sub)populations are bound to extinction, the distribution of their size conditional on extinction, either not to have occurred yet or to occur in the distant future, converges in many situations. The limits are called quasi-stationary distributions and provide a rigorous notion for the dynamics of populations which seem stable at least on the human time-scale. We display quasi-stationarity for all models introduced previously, tackling the problem of uniqueness of quasi-stationary distributions, and comparing different conditionings (Q-process). In a slightly different setting, we prove that the contour process of splitting trees with general lifetime is a killed Lévy process, and use this observation to derive a certain number of properties of these trees, including connections between Lévy processes and branching processes (one by Le Gall and Le Jan and another one by Bertoin and Le Gall), and a generalization of the coalescent point process of Aldous and Popovic. These two kinds of results open up to more general problems such as Ray–Knight type theorems on the one hand, and coalescent processes on the other hand.

Continuum random trees and branching processes with immigration

We study a genealogical model for continuous-state branching processes with immigration with a (sub)critical branching mechanism. This model allows the immigrants to be on the same line of descent. The corresponding family tree is an ordered rooted continuum random tree with a single infinite end defined by two continuous processes denoted by ( H ⃖ t ; t ≥ 0 ) and ( H ⃗ t ; t ≥ 0 ) that code the parts at resp. the left and the right hand side of the infinite line of descent of the tree. These processes are called the left and the right height processes. We define their local time processes via an approximation procedure and we prove that they enjoy a Ray–Knight property. We also discuss the important special case corresponding to the size-biased Galton–Watson tree in a continuous setting. In the last part of the paper we give a convergence result under general assumptions for rescaled discrete left and right contour processes of sequences of Galton–Watson trees with immigration. We also provide a strong invariance principle for a sequence of rescaled Galton–Watson processes with immigration that also holds in the supercritical case.

Joint density for the local times of continuous-time Markov chains

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 local times, and (3) we derive large deviation upper bounds for continuous-time simple random walk on large subboxes of ℤd tending to ℤd as time diverges. We finally discuss the relation of our density formula to the Ray–Knight theorem for continuous-time simple random walk on ℤ, which is analogous to the well-known Ray–Knight description of Brownian local times.

A trivariate law for certain processes related to perturbed Brownian motions

D. Williams' path decomposition and Pitman's representation theorem for BES(3) are expressions of some deep relations between reflecting Brownian motion and the 3-dimensional Bessel process. In [Ph. Carmona et al., Stochastic Process. Appl. 7 (1999) 323–333], we presented an attempt to relate better reflecting Brownian motion and the 2-dimensional Bessel process, using space and time changes related to the Ray–Knight theorems on local times, in the manner of Jeulin [Lect. Notes Math., vol. 1118, Springer, Berlin, 1985] and Biane–Yor [Bull. Sci. Math. 2ème Sér. 111 (1987) 23–101]. Here, we characterize the law of a triplet linked to the perturbed Brownian motion which naturally arises in [Ph. Carmona et al., Stochastic Proc. Appl. 7 (1999) 323–333], and we point out its relations with Bessel processes of several dimensions. The results provide some new understanding of the generalizations of Lévy's arc sine law for perturbed Brownian motions previously obtained by the second author.

Local time flow related to skew brownian motion

We define a local time flow of skew Brownian motions ,that is, a family of solutions to the stochastic differential equation defining the skew Brownian motion, starting from different points but driven by the same Brownian motion. We prove several results on distributional and path properties of the flow. Our main result is a version of the Ray–Knight theorem on local times. In our case, however, the local time process viewed as a function of the spatial variable is a pure jump Markov process rather than a diffusion.

A Ray-Knight theorem for symmetric Markov processes

Let $X$ be a strongly symmetric recurrent Markov process with state space $S$ and let $L_t^x$ denote the local time of $X$ at $X \in S$. For a fixed element 0 in the state space S, let $$ \tau(t) := \inf \{s: L^0_s > t \}. $$ The 0-potential density, $u_{\{0\}}(x, y)$, of the process $X$ killed at $T_0 = \inf \{s:X_s =0\}$ is symmetric and positive definite. Let $\eta = \{\eta_x; x \in S \}$ be a mean-zero Gaussian process with covariance $$ E_\eta (\eta_x \eta_y ) = u_{\{0\}}(x, y). $$ The main result of this paper is the following generalization of the classical second Ray–Knight theorem: for any $b \in R$ and $t > 0$, $$ \{L^x_{\tau(t)} + \frac{1}{2} (\eta_x + b)^2; x \in S \} = \{ \frac{1}{2} ( \eta_x + \sqrt{2+ b^2})^2 ; x \in S \} \text{ in law}. $$ A version of this theorem is also given when $X$ is transient.

Ray–Knight theorems related to a stochastic flow

We study a stochastic flow of C 1 -homeomorphisms of R . At certain stopping times, the spatial derivative of the flow is a diffusion in the space variable and its generator is given. This answers several questions posed in a previous study by Bass and Burdzy (1999, Ann. Probab. 27, 50–108).

The most visited sites of symmetric stable processes

Let X be a symmetric stable process of index α∈ (1,2] and let L x t denote the local time at time t and position x. Let V(t) be such that L t V(t) = sup x∈ ℝ L t x . We call V(t) the most visited site of X up to time t. We prove the transience of V, that is, lim t →∞ |V(t)| = ∞ almost surely. An estimate is given concerning the rate of escape of V. The result extends a well-known theorem of Bass and Griffin for Brownian motion. Our approach is based upon an extension of the Ray–Knight theorem for symmetric Markov processes, and relates stable local times to fractional Brownian motion and further to the winding problem for planar Brownian motion.

A transition function expansion for a diffusion model with selection

Using duality, an expansion is found for the transition function of the reversible $K$-allele diffusion model in population genetics. In the neutral case, the expansion is explicit but already known. When selection is present, it depends on the distribution at time $t$ of a specified $K$-type birth-and-death process starting at “infinity.” The latter process is constructed by means of a coupling argument and characterized as the Ray process corresponding to the Ray–Knight compactification of the $K$-dimensional nonnegative-integer lattice.

Stochastic Bifurcation Models

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.

Branching processes in Lévy processes: the exploration process

The main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure. The exploration process appears as a simple local time functional of a Levy process with no negative jumps, whose Laplace exponent coincides with the branching mechanism function. This new relation between spectrally positive Levy processes and continuous branching processes provides a unified perspective on both theories. In particular, we derive the adequate formulation of the classical Ray–Knight theorem for such Levy processes. As a consequence of this theorem, we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process.

Some Brownian functionals and their laws

We develop some topics about Brownian motion with a particular emphasis on the study of principal values of Brownian local times. We show some links between principal values and Doob’s $h$-transforms of Brownian motion, for nonpositive harmonic functions $h$. We also give a survey and complement some martingale approaches to Ray–Knight theorems for local times.

Cyclically stationary Brownian local time processes

Local time processes parameterized by a circle, defined by the occupation density up to time T of Brownian motion with constant drift on the circle, are studied for various random times T. While such processes are typically non-Markovian, their Laplace functionals are expressed by series formulae related to similar formulae for the Markovian local time processes subject to the Ray–Knight theorems for BM on the line, and for squares of Bessel processes and their bridges. For T the time that BM on the circle first returns to its starting point after a complete loop around the circle, the local time process is cyclically stationary, with same two-dimensional distributions, but not the same three-dimensional distributions, as the sum of squares of two i.i.d. cyclically stationary Gaussian processes. This local time process is the infinitely divisible sum of a Poisson point process of local time processes derived from Brownian excursions. The corresponding intensity measure on path space, and similar Levy measures derived from squares of Bessel processes, are described in terms of a 4-dimensional Bessel bridge by Williams’ decomposition of Ito’s law of Brownian excursions.

Processus de Markov associ� a une forme de Dirichlet non sym�trique

The purpose of the present paper is the probabilistic interpretation of potential theory associated with a Dirichlet form on a functional space. We generalise the works of M. Fukushima to unsymmetric forms using a new method based on the utilisation of quasi continuous representation of functions belonging to the Dirichlet space and a Ray Knight compactification.