60J25 Continuous-time Markov processes on general state spaces
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- Chemical master equation (1)
- Conformation Dynamics; Markov State Models; Molecular Kinetics; Transition Rates (1)
- Generalized gradient structure (1)
- Markov processes (1)
- continuous-time Markov process (1)
- convergence to steady state (1)
- energy-dissipation principle (1)
- evolutionary Γ-convergence (1)
- explicit solution formula (1)
- gradient system (1)
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures.
A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials.
A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary -limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of difusion over a high barrier, which gives a reaction-diffusion system.
Supercomputers can simulate complex molecular systems. However, there is a very large gap between the fastest oscillations of covalent bonds of a molecule and the time-scale of the dominant processes. In order to extract the dominant time-scales and to identify the dominant processes, a clustering of information is needed. This thesis shows that only the subspace-based Robust Perron Cluster Analysis (PCCA+) can solve this problem correctly by the construction of a Markov State Model. PCCA+ allows for time-extrapolation in molecular kinetics. This thesis shows the difference between molecular dynamics and molecular kinetics. Only in the molecular kinetics framework a definition of transition rates is possible. In this context, the existence of an infinitesimal generator of the dynamical processes is discussed. If the existence is assumed, the Theorem of Gauß can be applied in order to compute transition rates efficiently. Molecular dynamics, however, is not able to provide a suitable statistical basis for the determination of the transition pattern.
Expected suprema of a function f observed along the paths of a nice Markov process define an excessive function, and in
fact a potential if f vanishes at the boundary. Conversely, we show under mild regularity conditions that any
potential admits a representation in terms of expected suprema. Moreover, we identify the maximal and the minimal
representing function in terms of probabilistic potential theory. Our results are motivated by the work of El Karoui and
Meziou on the max-plus decomposition of supermartingales, and they provide a singular analogue to
the non-linear Riesz representation in El Karoui and Föllmer.
The stochastic dynamics of a well-stirred mixture of molecular species
interacting through different biochemical reactions can be
accurately modelled by the chemical master equation (CME). Research in
the biology and scientific computing community has
concentrated mostly on the development of numerical techniques to
approximate the solution of the CME via many realizations of the associated
Markov jump process. The domain of exact and/or efficient methods for
directly solving the CME is still widely open, which is due to its
large dimension that grows exponentially with the number of molecular
species involved. In this article, we present an exact solution
formula of the CME for arbitrary initial conditions in the case where
the underlying system is governed by monomolecular reactions. The
solution can be expressed in terms of the convolution of multinomial
and product Poisson distributions with time-dependent parameters
evolving according to the traditional reaction-rate equations. This
very structured representation allows to deduce any property of the
solution. The model class includes many interesting examples and may
also be used as the starting point for the design of new numerical
methods for the CME of more complex reaction systems.