# Inverting Ray-Knight identity

@article{Sabot2013InvertingRI, title={Inverting Ray-Knight identity}, author={Christophe Sabot and Pierre Tarres}, journal={Probability Theory and Related Fields}, year={2013}, volume={165}, pages={559-580} }

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.

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## References

SHOWING 1-10 OF 23 REFERENCES

### An isomorphism theorem for random interlacements

- Mathematics
- 2011

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…

### Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model

- Mathematics
- 2011

Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process that takes values in the vertex set of a graph G, which is more likely to cross edges it has…

### A Ray-Knight theorem for symmetric Markov processes

- Mathematics
- 2000

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 ∈ S. For a fixed element 0 in the state space S, let τ(t):= inf{s: L s 0 > t}…

### ON THE TRANSIENCE OF PROCESSES DEFINED ON GALTON-WATSON TREES

- Mathematics
- 2006

We introduce a simple technique for proving the transience of certain processes defined on the random tree $\mathcal{G}$ generated by a supercritical branching process. We prove the transience for…

### Continuous-time vertex reinforced jump processes on Galton-Watson trees

- Mathematics
- 2010

We consider a continuous-time vertex reinforced jump process on a supercritical Galton-Watson tree. This process takes values in the set of vertices of the tree and jumps to a neighboring vertex with…

### Limit theorems for vertex-reinforced jump processes on regular trees

- Mathematics
- 2009

Consider a vertex-reinforced jump process defined on a regular tree, where each vertex has exactly $b$ children, with $b \geq 3$. We prove the strong law of large numbers and the central limit…

### Random interlacements and the Gaussian free field

- Mathematics
- 2012

We consider continuous time random interlacements on Zd, d≥3, and characterize the distribution of the corresponding stationary random field of occupation times. When d=3, we relate this random field…

### Vertex-reinforced jump processes on trees and finite graphs

- Mathematics
- 2004

We study the continuous time process on the vertices of the b-ary tree which jumps to each nearest neighbor vertex at the rate of the time already spent at that vertex times δ, plus 1, where δ is a…

### Random walks and a sojourn density process of Brownian motion

- Mathematics
- 1963

The sojourn times for the Brownian motion process in 1 dimension have often been investigated by considering the distribution of the length of time spent in a fixed set during a fixed time interval.…

### Continuous time vertex-reinforced jump processes

- Mathematics
- 2001

Abstract. We study the continuous time integer valued process , which jumps to each of its two nearest neighbors at the rate of one plus the total time the process has previously spent at that…