223,460 research outputs found
The inverse problem for rough controlled differential equations
We provide a necessary and sufficient condition for a rough control driving a
differential equation to be reconstructable, to some order, from observing the
resulting controlled evolution. Physical examples and applications in
stochastic filtering and statistics demonstrate the practical relevance of our
result.Comment: added section on rough path theor
Stochastic analysis of different rough surfaces
This paper shows in detail the application of a new stochastic approach for
the characterization of surface height profiles, which is based on the theory
of Markov processes. With this analysis we achieve a characterization of the
scale dependent complexity of surface roughness by means of a Fokker-Planck or
Langevin equation, providing the complete stochastic information of multiscale
joint probabilities. The method is applied to several surfaces with different
properties, for the purpose of showing the utility of this method in more
details. In particular we show the evidence of Markov properties, and we
estimate the parameters of the Fokker-Planck equation by pure, parameter-free
data analysis. The resulting Fokker-Planck equations are verified by numerical
reconstruction of conditional probability density functions. The results are
compared with those from the analysis of multi-affine and extended multi-affine
scaling properties which is often used for surface topographies. The different
surface structures analysed here show in details advantages and disadvantages
of these methods.Comment: Minor text changes to be identical with the published versio
A tree approach to -variation and to integration
We consider a real-valued path; it is possible to associate a tree to this
path, and we explore the relations between the tree, the properties of
-variation of the path, and integration with respect to the path. In
particular, the fractal dimension of the tree is estimated from the variations
of the path, and Young integrals with respect to the path, as well as integrals
from the rough paths theory, are written as integrals on the tree. Examples
include some stochastic paths such as martingales, L\'evy processes and
fractional Brownian motions (for which an estimator of the Hurst parameter is
given)
Rough interfaces, accurate predictions: The necessity of capillary modes in a minimal model of nanoscale hydrophobic solvation
Modern theories of the hydrophobic effect highlight its dependence on length
scale, emphasizing in particular the importance of interfaces that emerge in
the vicinity of sizable hydrophobes. We recently showed that a faithful
treatment of such nanoscale interfaces requires careful attention to the
statistics of capillary waves, with significant quantitative implications for
the calculation of solvation thermodynamics. Here we show that a coarse-grained
lattice model in the spirit of those pioneered by Chandler and coworkers, when
informed by this understanding, can capture a broad range of hydrophobic
behaviors with striking accuracy. Specifically, we calculate probability
distributions for microscopic density fluctuations that agree very well with
results of atomistic simulations, even many standard deviations from the mean,
and even for probe volumes in highly heterogeneous environments. This accuracy
is achieved without adjustment of free parameters, as the model is fully
specified by well-known properties of liquid water. As illustrative examples of
its utility, we characterize the free energy profile for a solute crossing the
air-water interface, and compute the thermodynamic cost of evacuating the space
between extended nanoscale surfaces. Together, these calculations suggest that
a highly reduced model for aqueous solvation can serve as the basis for
efficient multiscale modeling of spatial organization driven by hydrophobic and
interfacial forces.Comment: 14 pages, 7 figure
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