Linear response theory and transient fluctuation theorems for diffusion processes: a backward point of view

On the basis of perturbed Kolmogorov backward equations and path integral representation, we unify the derivations of the linear response theory and transient fluctuation theorems for continuous diffusion processes from a backward point of view. We find that a variety of transient fluctuation theorems could be interpreted as a consequence of a generalized Chapman-Kolmogorov equation, which intrinsically arises from the Markovian characteristic of diffusion processes

Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems

We prove that the Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in $L_2$ for small complex $L_\infty$ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for $k$-forms are well posed for small perturbations of block matrices.Comment: Some changes made in the introduction of the pape

Work, heat and entropy production along quantum trajectories

Quantum open systems evolve according to completely positive, trace preserving maps acting on the density operator, which can equivalently be unraveled in term of so-called quantum trajectories. These stochastic sequences of pure states correspond to the actual dynamics of the quantum system during single realizations of an experiment in which the system's environment is monitored. In this chapter, we present an extension of stochastic thermodynamics to the case of open quantum systems, which builds on the analogy between the quantum trajectories and the trajectories in phase space of classical stochastic thermodynamics. We analyze entropy production, work and heat exchanges at the trajectory level, identifying genuinely quantum contributions due to decoherence induced by the environment. We present three examples: the thermalization of a quantum system, the fluorescence of a driven qubit and the continuous monitoring of a qubit's observable.Comment: Book chapter in 'Thermodynamics in the quantum regime - Recent Progress and Outlook

Limit theorems for Markov processes indexed by continuous time Galton--Watson trees

We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton--Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time t. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. The latter has the same generator as the Markov process along the branches plus additional jumps, associated with branching events of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time t favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching L\'{e}vy processes.Comment: Published in at http://dx.doi.org/10.1214/10-AAP757 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Tree-valued Feller diffusion

We consider the evolution of the genealogy of the population currently alive in a Feller branching diffusion model. In contrast to the approach via labeled trees in the continuum random tree world, the genealogies are modeled as equivalence classes of ultrametric measure spaces, the elements of the space $\mathbb{U}$. This space is Polish and has a rich semigroup structure for the genealogy. We focus on the evolution of the genealogy in time and the large time asymptotics conditioned both on survival up to present time and on survival forever. We prove existence, uniqueness and Feller property of solutions of the martingale problem for this genealogy valued, i.e., $\mathbb{U}$-valued Feller diffusion. We give the precise relation to the time-inhomogeneous $\mathbb{U}_1$-valued Fleming-Viot process. The uniqueness is shown via Feynman-Kac duality with the distance matrix augmented Kingman coalescent. Using a semigroup operation on $\mathbb{U}$, called concatenation, together with the branching property we obtain a L{\'e}vy-Khintchine formula for $\mathbb{U}$-valued Feller diffusion and we determine explicitly the L{\'e}vy measure on $\mathbb{U}\setminus\{0\}$. From this we obtain for $h>0$ the decomposition into depth-$h$ subfamilies, a representation of the process as concatenation of a Cox point process of genealogies of single ancestor subfamilies. Furthermore, we will identify the $\mathbb{U}$-valued process conditioned to survive until a finite time $T$. We study long time asymptotics, such as generalized quasi-equilibrium and Kolmogorov-Yaglom limit law on the level of ultrametric measure spaces. We also obtain various representations of the long time limits.Comment: 93 pages, replaced by revised versio