Related papers: Rigidity for Markovian Maximal Couplings of Ellipt…
A maximal coupling of two diffusion processes makes two diffusion particles meet as early as possible. We study the uniqueness of maximal couplings under a sort of "reflection structure" which ensures the existence of such couplings. In…
Markovian maximal couplings of Markov processes are characterized by an equality of total variation and a distance of Wasserstein type. If a Markovian maximal coupling is a Feller process, the generator can be calculated, e.g. for…
We develop an approach to constructing non-Markovian, non-co-adapted couplings for sub-Riemannian Brownian motions in sub-Riemannian manifolds with large symmetry groups by treating the specific cases of the three-dimensional Heisenberg…
The well-known reflection coupling gives a maximal coupling of two one-dimensional Brownian motions with different starting points. Nevertheless, the reflection coupling does not generalize to more than two Brownian motions. In this paper,…
This is a case study concerning the rate at which probabilistic coupling occurs for nilpotent diffusions. We focus on the simplest case of Kolmogorov diffusion (Brownian motion together with its time integral, or, slightly more generally,…
This paper answers a question of \'{E}mery [In S\'{e}minaire de Probabilit\'{e}s XLII (2009) 383-396 Springer] by constructing an explicit coupling of two copies of the Bene\v{s} et al. [In Applied Stochastic Analysis (1991) 121-156 Gordon…
We construct a non-Markovian coupling for hypoelliptic diffusions which are Brownian motions in the three-dimensional Heisenberg group. We then derive properties of this coupling such as estimates on the coupling rate, and upper and lower…
In this paper we introduce three Markovian couplings of Brownian motions on smooth Riemannian manifolds without boundary which sit at the crossroad of two concepts. The first concept is the one of shy coupling put forward in…
We raise a question on whether a dynamical system driven by Markov process is Markovian, for which we are able to propose a criterion and examples of positive case. This investigation leads us to develop (i) a general construction of…
The notion of a successful coupling of Markov processes, based on the idea that both components of the coupled system ``intersect'' in finite time with probability one, is extended to cover situations when the coupling is unnecessarily…
We study optimal Markovian couplings of Markov processes, where the optimality is understood in terms of minimization of concave transport costs between the time-marginal distributions of the coupled processes. We provide explicit…
This article presents a review of some old and new results on the long time behavior of reflected diffusions. First, we present a summary of prior results on construction, ergodicity and geometric ergodicity of reflected diffusions in the…
We study Brownian flows on manifolds for which the associated Markov process is strongly mixing with respect to an invariant probability measure and for which the distance process for each pair of trajectories is a diffusion $r$. We provide…
Motivated by queues with many servers, we study Brownian steady-state approximations for continuous time Markov chains (CTMCs). Our approximations are based on diffusion models (rather than a diffusion limit) whose steady-state, we prove,…
Consider all the possible ways of coupling together two Brownian motions with the same starting position but with different drifts onto the same probability space. It is known that there exist couplings which make these processes agree for…
We consider different Markovian embedding schemes of non-Markovian stochastic processes that are described by generalized Langevin equations (GLE) and obey thermal detailed balance under equilibrium conditions. At thermal equilibrium…
We develop a new simulation method for multidimensional diffusions with sticky boundaries. The challenge comes from simulating the sticky boundary behavior, for which standard methods like the Euler scheme fail. We approximate the sticky…
For every bounded planar domain $D$ with a smooth boundary, we define a `Lyapunov exponent' $\Lambda(D)$ using a fairly explicit formula. We consider two reflected Brownian motions in $D$, driven by the same Brownian motion (i.e., a…
We present a novel approach of coupling two multidimensional and non-degenerate It\^o processes $(X_t)$ and $(Y_t)$ which follow dynamics with different drifts. Our coupling is sticky in the sense that there is a stochastic process $(r_t)$,…
Our object is to formulate and analyze a physically plausible and mathematically sound model to better understand the phenomenon of clumping in colloid dispersions. Our model is stochastic but rigorously derived from a deterministic setup…