The Galois ring GR$(4^\Delta)$ is the residue $Z_4[x]/(h(x))$, where $h(x)$ a basic primitive polynomial of degree $\Delta$ over $Z_4$. For any odd larger than $1$, we construct partition GR$(4^\Delta) \backslash \{0\}$ into $6$-subsets type $\{a,b,-a-b,-a,-b,a+b\}$ and $3$-subsets $\{c,-c,2c\}$ such that invariant under multiplication by nonzero element Teichmuller set in and, if not multiple ...

We consider diffusion processes xt on the unit interval. Doobtransformation techniques consist of a selection of xt−paths procedure. The law of the transformed process is the one of a branching diffusion system of particles, each diffusing like a new process x̃t, superposing an additional drift to the one of xt. Killing and/or branching of x̃t−particles occur at some space-dependent rate λ. For t...

We consider some time-changed diffusion processes obtained by applying the Doob transformation rule to a Brownian motion. The time-change is via inverse of an α-stable subordinator. These are specified in terms Gauss-Markov and fractional diffusions. A pseudo-Fokker-Planck equation for such given. investigate their first passage time densities providing generalized integral they satisfy rules. ...

We derive a general scheme to obtain quantum fluctuation relations for dynamical observables in open systems. For concreteness, we consider Markovian nonunitary dynamics that is unraveled terms of jump trajectories and exploit techniques from the theory large deviations like tilted ensemble Doob transform. Our results here generalize system previously obtained classical systems add vast literat...

The large deviation (LD) statistics of dynamical observables is encoded in the spectral properties deformed Markov generators. Recent works have shown that tensor network methods are well suited to compute relevant leading eigenvalues and eigenvectors accurately. However, efficient generation corresponding rare trajectories a harder task. Here we show how exploit MPS approximation dominant eige...

We study the long-time behaviour of a non conservative piecewise deterministic measure-valued Markov process modelling proliferation an age-and-size structured population, which generalises “adder” model bacterial growth. Firstly, we prove existence eigenelements associated infinitesimal generator, are used to bring ourselves back using Doob $h$ -transform. Finally, obtain exponential ergodicit...

The large deviations at Level 2.5 are applied to Markov processes with absorbing states in order obtain the explicit extinction rate of metastable quasi-stationary terms their empirical time-averaged density and flows over a time-window $T$. standard spectral problem for slowest relaxation mode can be recovered from full optimization all these observables equivalence understood via Doob generat...