36,576 research outputs found
Reduction and reconstruction of stochastic differential equations via symmetries
An algorithmic method to exploit a general class of infinitesimal symmetries
for reducing stochastic differential equations is presented and a natural
definition of reconstruction, inspired by the classical reconstruction by
quadratures, is proposed. As a side result the well-known solution formula for
linear one-dimensional stochastic differential equations is obtained within
this symmetry approach. The complete procedure is applied to several examples
with both theoretical and applied relevance
Quantum control theory and applications: A survey
This paper presents a survey on quantum control theory and applications from
a control systems perspective. Some of the basic concepts and main developments
(including open-loop control and closed-loop control) in quantum control theory
are reviewed. In the area of open-loop quantum control, the paper surveys the
notion of controllability for quantum systems and presents several control
design strategies including optimal control, Lyapunov-based methodologies,
variable structure control and quantum incoherent control. In the area of
closed-loop quantum control, the paper reviews closed-loop learning control and
several important issues related to quantum feedback control including quantum
filtering, feedback stabilization, LQG control and robust quantum control.Comment: 38 pages, invited survey paper from a control systems perspective,
some references are added, published versio
The Einstein Relation on Metric Measure Spaces
This note is based on F. Burghart's master thesis at Stuttgart university
from July 2018, supervised by Prof. Freiberg.
We review the Einstein relation, which connects the Hausdorff, local walk and
spectral dimensions on a space, in the abstract setting of a metric measure
space equipped with a suitable operator. This requires some twists compared to
the usual definitions from fractal geometry. The main result establishes the
invariance of the three involved notions of fractal dimension under
bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more
generally, how the transport of the analytic and stochastic structure behind
the Einstein relation works. While any homeomorphism suffices for this
transport of structure, non-Lipschitz maps distort the Hausdorff and the local
walk dimension in different ways. To illustrate this, we take a look at
H\"older regular transformations and how they influence the local walk
dimension and prove some partial results concerning the Einstein relation on
graphs of fractional Brownian motions. We conclude by giving a short list of
further questions that may help building a general theory of the Einstein
relation.Comment: 28 pages, 3 figure
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