# Quenched invariance principle for the Knudsen stochastic billiard in a random tube

@article{Comets2008QuenchedIP, title={Quenched invariance principle for the Knudsen stochastic billiard in a random tube}, author={Francis Comets and Serguei Yu. Popov and Gunter M. Schutz and M. Vachkovskaia}, journal={arXiv: Probability}, year={2008} }

We consider a stochastic billiard in a random tube which stretches to infinity in the direction of the first coordinate. This random tube is stationary and ergodic, and also it is supposed to be in some sense well behaved. The stochastic billiard can be described as follows: when strictly inside the tube, the particle moves straight with constant speed. Upon hitting the boundary, it is reflected randomly, according to the cosine law: the density of the outgoing direction is proportional to the…

## 22 Citations

### Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards

- Mathematics
- 2012

We consider a random walk in a stationary ergodic environment in $\mathbb Z$, with unbounded jumps. In addition to uniform ellipticity and a bound on the tails of the possible jumps, we assume a…

### Knudsen Gas in a Finite Random Tube: Transport Diffusion and First Passage Properties

- Mathematics
- 2010

We consider transport diffusion in a stochastic billiard in a random tube which is elongated in the direction of the first coordinate (the tube axis). Inside the random tube, which is stationary and…

### Transport diffusion coefficient for a Knudsen gas in a random tube

- Mathematics
- 2008

We consider transport diffusion in a stochastic billiard in a random tube which is elongated in the direction of the first coordinate (the tube axis). Inside the random tube, which is stationary and…

### Conditional and uniform quenched CLTs for one-dimensional random walks among random conductances

- Mathematics
- 2010

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform…

### Random walks with unbounded jumps among random conductances I: Uniform quenched CLT

- Mathematics
- 2012

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform…

### Random walks with unbounded jumps among random conductances II: Conditional quenched CLT

- Mathematics
- 2013

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform…

### A Conditional Quenched CLT for Random Walks Among Random Conductances on Z(d)

- Mathematics
- 2014

Consider a random walk among random conductances on Z with d ≥ 2. We study the quenched limit law under the usual diffusive scaling of the random walk conditioned to have its first coordinate…

### A conditional quenched CLT for random walks among random conductances on $\mathbb{Z}^d$

- Mathematics
- 2011

Consider a random walk among random conductances on $\mathbb{Z}^d$ with $d\geq 2$. We study the quenched limit law under the usual diffusive scaling of the random walk conditioned to have its first…

### Random Reflections in a High-Dimensional Tube

- Mathematics, Physics
- 2016

We consider light ray reflections in a d-dimensional semi-infinite tube, for $$d\ge 3$$d≥3, made of Lambertian material. The source of light is placed far away from the exit, and the light ray is…

### Knudsen gas in flat tire

- PhysicsThe Annals of Applied Probability
- 2019

We consider random reflections (according to the Lambertian distribution) of a light ray in a thin variable width (but almost circular) tube. As the width of the tube goes to zero, properly rescaled…

## References

SHOWING 1-10 OF 22 REFERENCES

### Transport diffusion coefficient for a Knudsen gas in a random tube

- Mathematics
- 2008

We consider transport diffusion in a stochastic billiard in a random tube which is elongated in the direction of the first coordinate (the tube axis). Inside the random tube, which is stationary and…

### Billiards in a General Domain with Random Reflections

- Mathematics
- 2006

We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain $$\fancyscript{D}\subset {\mathbb{R}}^d$$ until it hits the boundary and…

### Mott Law as Lower Bound for a Random Walk in a Random Environment

- Mathematics
- 2005

We consider a random walk on the support of an ergodic stationary simple point process on ℝd, d≥2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density…

### Asymptotic Behaviour of Randomly Reflecting Billiards in Unbounded Tubular Domains

- Mathematics
- 2008

We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary.…

### An invariance principle for reversible Markov processes. Applications to random motions in random environments

- Mathematics
- 1989

We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random…

### Functional CLT for Random Walk Among Bounded Random Conductances

- Mathematics
- 2007

We consider the nearest-neighbor simple random walk on $Z^d$, $d\ge2$, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$. Apart from the requirement that the…

### Quenched invariance principle for simple random walk on percolation clusters

- Mathematics
- 2006

We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in ℤd with d≥2. We prove that, for almost every percolation configuration, the path distribution…

### Random walks on supercritical percolation clusters

- Mathematics
- 2003

We obtain Gaussian upper and lower bounds on the transition density qt(x;y) of the continuous time simple random walk on a supercritical percolation cluster C1 in the Euclidean lattice. The bounds,…

### On symmetric random walks with random conductances on ℤd

- Mathematics
- 2004

We study models of continuous time, symmetric, ℤd-valued random walks in random environments. One of our aims is to derive estimates on the decay of transition probabilities in a case where a uniform…