Change of measure in the lookdown particle system

We perform various changes of measure in the lookdown particle system of Donnelly and Kurtz. The first example is a product type h-transform related to conditioning a Generalized Fleming Viot process without mutation on coexistence of some genetic types in remote time. We give a pathwise construction of this h-transform by just "forgetting" some reproduction events in the lookdown particle system. We also provide an intertwining relationship for the Wright Fisher diffusion and explicit the associated pathwise decomposition. The second example, called the linear or additive h-transform, concerns a wider class of measure valued processes with spatial motion. Applications include: -a simple description of the additive h-transform of the Generalized Fleming Viot process, which confirms a suggestion of Overbeck for the usual Fleming Viot process -an immortal particle representation for the additive h-transform of the Dawson Watanabe process.Comment: 24 page

On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity

In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operators. We show that these operators extend to generators of ergodic Markov semigroups with unique invariant probability measures and study their spectral and convergence properties. In particular, we derive a series expansion of the semigroup in terms of explicitly defined polynomials, which generalize the classical Jacobi orthogonal polynomials. In addition, we give a complete characterization of the spectrum of the non-self-adjoint generator and semigroup. We show that the variance decay of the semigroup is hypocoercive with explicit constants, which provides a natural generalization of the spectral gap estimate. After a random warm-up time, the semigroup also decays exponentially in entropy and is both hypercontractive and ultracontractive. Our proofs hinge on the development of commutation identities, known as intertwining relations, between local and non-local Jacobi operators and semigroups, with the local objects serving as reference points for transferring properties from the local to the non-local case

Approximate filtering via discrete dual processes

We consider the task of filtering a dynamic parameter evolving as a diffusion process, given data collected at discrete times from a likelihood which is conjugate to the marginal law of the diffusion, when a generic dual process on a discrete state space is available. Recently, it was shown that duality with respect to a death-like process implies that the filtering distributions are finite mixtures, making exact filtering and smoothing feasible through recursive algorithms with polynomial complexity in the number of observations. Here we provide general results for the case of duality between the diffusion and a regular jump continuous-time Markov chain on a discrete state space, which typically leads to filtering distribution given by countable mixtures indexed by the dual process state space. We investigate the performance of several approximation strategies on two hidden Markov models driven by Cox-Ingersoll-Ross and Wright-Fisher diffusions, which admit duals of birth-and-death type, and compare them with the available exact strategies based on death-type duals and with bootstrap particle filtering on the diffusion state space as a general benchmark

Spectral gap of the symmetric inclusion process

We consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on the same graph. In the regime in which the gamma-like reversible measures of the particle systems are log-concave, our bounds match, yielding a version for the symmetric inclusion process of the celebrated Aldous' spectral gap conjecture originally formulated for the interchange process. Finally, by means of duality techniques, we draw analogous conclusions for an interacting diffusion-like unbounded conservative spin system known as Brownian energy process.Comment: 16 page

The Aldous diffusion: a stationary evolution of the Brownian CRT

Motivated by a down-up Markov chain on cladograms, David Aldous conjectured in 1999 that there exists a "diffusion on continuum trees" whose mass partitions at any finite number of branch points evolve as Wright-Fisher diffusions with some negative mutation rates, until some branch point disappears. Building on previous work on interval-partition-valued processes, we construct this conjectured process via a consistent system of stationary evolutions of binary trees with k labeled leaves and edges decorated with interval partitions. The interval partitions are scaled Poisson-Dirichlet interval partitions whose interval lengths record subtree masses. They also possess a diversity property that captures certain distances in the continuum tree. Continuously evolving diversities give access to continuously evolving continuum tree distances. The pathwise construction allows us to study this "Aldous diffusion" in the Gromov-Hausdorff-Prokhorov space of rooted, weighted R-trees. We establish the simple Markov property and path-continuity. The Aldous diffusion is stationary with the distribution of the Brownian continuum random tree. While the Brownian CRT is a.s. binary, we show that there is a dense null set of exceptional times when the Aldous diffusion has a ternary branch point, including stopping times at which the strong Markov property fails. Our construction relates to the two-parameter Chinese restaurant process, branching processes, and stable L\'evy processes, among other connections. Wright-Fisher diffusions and the aforementioned processes of Poisson-Dirichlet interval partitions arise as interesting projections of the Aldous diffusion. Finally, one can embed Aldous's stationary down-up Markov chain on cladograms in the Aldous diffusion and hence address a related conjecture by David Aldous by establishing a scaling limit theorem.Comment: 193+vi pages, 26 figure