position=current page.north, angle=0, nodeanchor=north, vshift=14mm,
opacity=0.5, scale=1, contents=  PUBLISHED AS: Hoffmann, M. (2022), Body
schema or the body as its own best model, in G. Šejnová and M. Vavrečka and J.
Hvorecký, ed., ‘Kognice a umělý život XX (Cognition and Artificial Life XX)’,
CTU in Prague, pp. 45-51.

# Body schema or the body as its own best model

Matej Hoffmann
Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical
University in Prague
Email<EMAIL_ADDRESS>

Abstract

Rodney Brooks (1991) put forth the idea that during an agent’s interaction
with its environment, representations of the world often stand in the way.
Instead, using the world as its own best model, i.e. interacting with it
directly without making models, often leads to better and more natural
behavior. The same perspective can be applied to representations of the
agent’s body. I analyze different examples from biology—octopus and humans in
particular—and compare them with robots and their body models. At one end of
the spectrum, the octopus, a highly intelligent animal, largely relies on the
mechanical properties of its arms and peripheral nervous system. No central
representations or maps of its body were found in its central nervous system.
Primate brains do contain areas dedicated to processing body-related
information and different body maps were found. Yet, these representations are
still largely implicit and distributed and some functionality is also
offloaded to the periphery. Robots, on the other hand, rely almost exclusively
on their body models when planning and executing behaviors. I analyze the pros
and cons of these different approaches and propose what may be the best
solution for robots of the future.

##  1 Introduction

In artificial intelligence and robotics, models of the world have been and
largely still are the key means of realizing interaction of a mechanism with
its environment. This position was attacked by Brooks (1990) stating: “The key
observation is that the world is its own best model. It is always exactly up
to date. It always contains every detail there is to be known. The trick is to
sense it appropriately and often enough.” Brooks (1991) added “When we examine
very simple level intelligence we find that explicit representations and
models of the world simply get in the way.”

If this were the case for the world, how about for the body of an agent—human,
animal, or robot? Our body seems to be even more “always there” than our
environment. The representationalist stance typical of robotics and (good old-
fashioned) artificial intelligence (Haugeland, 1985) is also applied to the
body. Indeed, traditional robots heavily rely on internal models of their
bodies. These are in particular the models of their kinematics—joints and
links, their dimensions and orientations—and their dynamics which deal with
masses and forces needed to generate motion (see 10.2.3 in (Hoffmann, 2021)
for more details). With traditional robots, the interaction with the world is
mediated by these models. In cognitive science, this approximately corresponds
to the “body in the brain” approach which emphasizes representations of the
body in the cerebral cortex—see for example Section 5.1 in (De Vignemont,
2018). This contrasts with the “brain in the body” or “body in the world”
perspective, also called the sensorimotor approach that is in line with
Brooks’ perspective (see Section 4.2 in (De Vignemont, 2018) or a discussion
in the Introduction to (Ataria et al., 2021)).

This work draws on (Hoffmann, 2021, 2022; Hoffmann and Müller, 2017).

##  2 Biology – from octopus to humans

The octopus constitutes an interesting case. Belonging to cephalopods, highly
derived molluscs, it is the most intelligent among them and with the largest
nervous system. Cephalopods, the most advanced invertebrate class, feature, on
one hand, the highest centralization of the nervous system. On the other hand,
next to the central nervous system (CNS) composed of the brain and two optic
lobes, there is a large peripheral nervous system (PNS) of the body and the
arms. Despite the high level of centralization and in contrast to vertebrate
and insect brains, there is no obvious somatotopic arrangement in either motor
or sensory areas (see (Zullo and Hochner, 2011) for more details). The octopus
also has a unique embodiment—a flexible body and eight arms with virtually
infinite degrees of freedom. From an engineering perspective, modeling and
controlling such a body (plant in engineering jargon) using inverse kinematics
and dynamics would be a nightmare. However, Yekutieli et al. (2005) speculate
that the octopus reaches toward a target using the following strategy: (1)
Initiating a bend in the arm so that the suckers point outward. (2) Orienting
the base of the arm in the direction of the target or just above it. (3)
Propagating the bend along the arm at the desired speed by a wave of muscle
activation that equally activates all muscles along the arm. (4) Terminating
the reaching movement when the suckers touch the target by stopping the bend
propagation and thus catching the target. A big part of the complexity is thus
“off-loaded” from to the peripheral nervous system and the body itself.

In humans, central representations of the body in the cerebral cortex
certainly exist. There has been more than a century of empirical observations
and theorizing, leading to concepts like body image (system of perceptions,
attitudes, and beliefs pertaining to one’s own body) and body schema (system
of sensory-motor capacities that function without awareness or the necessity
of perceptual monitoring) (definitions taken from the Introduction to (Ataria
et al., 2021)). The most well-known body maps are the somatotopic
representations (the “homunculi”) in the primary motor and somatosensory
cortices (Leyton and Sherrington, 1917; Penfield and Boldrey, 1937). Yet, the
somatosensory homunculi are only an “entry point” or “relay station” to
downstream cortical processing rather than accurate representations or models
of the body (e.g., (Longo and Haggard, 2010)). Downstream areas in the
posterior parietal cortex (like Brodmann area 5) are thought to be involved in
higher-level more integrated representations related to the configuration of
the body in space, for example, but detailed understanding is still missing.
Reaching in primates bears some similarity to that in the octopus. A reaching
movement has some high-level characteristics like the direction of a hand’s
movement in space, the extent of the movement (amplitude), the overall
duration (movement time), and other parameters such as anticipated level of
resistance to the movement (Schöner et al., 2018). Also, movement generation
involves cooperation between the CNS and PNS. The exact mechanisms of motor
control in humans and other primates are still debated. Compared to
invertebrates, motor control in vertebrates, specifically mammals and in
particular primates, becomes more “cortical” and the motor cortex has the
possibility of more direct control over the details of a particular movement,
which is likely correlated with the need for dexterous manipulation (see
10.2.2 in Hoffmann (2021) for more details).

##  3 Body models for controlling movements

Body models can be classified according to different characteristics, such as
fixed vs. adaptive, amodal vs. modal, explicit vs. implicit, serial vs.
parallel, modular vs. holistic, or centralized vs. distributed (Hoffmann,
2021). For this article, we focus on the dimensions shown in Fig. 1.

Fig. 1: Body model characteristics. Upper row: examples from biology. Lower
row: examples from robotics. (A) iCub humanoid robot and its models. (B)
Hybrid model of the ANYmal robot (Hwangbo et al., 2019). (C) Robot
manipulators learning to grasp end-to-end (Levine et al., 2018). (D) Human and
schematic illustration of brain areas important for body representations.
Brain areas involved in body image representation after Berlucchi and Aglioti
(2010). (E) Octopus and schematic of its nervous system.
Credit: A – iCub cartoon: Laura Taverna, Italian Institute of Technology.
Credit: D – Walking human: Public domain
(https://commons.wikimedia.org/wiki/File:BSicon_WALK.svg). Credit D – Brain
image source: Hugh Guiney / Attribution-ShareAlike 3.0 Unported (CC BY- SA
3.0). Credit: E – Common octopus - albert kok / CC BY-SA
(https://creativecommons.org/licenses/by-sa/3.0). Credit: E – Octopus nervous
system - Jean-Pierre Bellier / CC BY-SA
(https://creativecommons.org/licenses/by-sa/4.0).

###  3.1 Explicit and veridical versus implicit and action-oriented

Traditional robot body models are explicit; it is clear what in the model
corresponds to what in the body (e.g., a certain parameter to the length of
the left forearm). They are also objective and veridical; the parameters
should be the true physical values of the quantities (lengths, angles, masses,
etc.). This is illustrated by the iCub humanoid robot (Metta et al., 2010) and
its models positioned at the far left in Fig. 1 A. In the biological realm,
representations in general are not like that and this should hold for
representations of the body as well. “What the nervous system needs to do, in
general, is to transform the input into the right action” (Webb, 2006)—hence
the implicit and action-oriented character of the representations. The
octopus—with no known map of its body in its central nervous system—is
positioned at the opposite end of the spectrum (Fig. 1 E). Successful action
is also the only criterion for the “quality” of what is represented about the
animal’s body in its brain; there is no need for any objective or veridical
representation. Similar arguments hold for primate brains, but to a lesser
extent. Numerous sites dedicated to representing the body were found (e.g.,
Kanayama and Hiromitsu (2021) for a review). Compared to the octopus, much
more of the body seems more explicitly represented. Longo (2015) considers the
implicit–explicit axis within human body representations and draws a line
roughly between the body schema and the body image. In tasks more related to
action and where humans do not consciously represent their body, the body
models seem more implicit and also less accurate. These representations may
also be dominated by somatosensation and inherit some of the distortions
typical of the somatosensory homunculi. Conversely, tasks that relate to
conscious perception of our body seem to draw on more explicit representations
that are also more accurate/veridical (e.g., image of our hand). This is
schematically illustrated in Fig. 1, D.

###  3.2 Centralized, universal, modular versus distributed, specialized, end-
to-end

Robot models are normally centralized—exist only in one place in the robot
software. On the other hand, neural representations are known to be
distributed. Whereas this “spatial aspect” may be also related to the
computational substrate (computers versus neurons), more important is a
functional division. Albeit centralized, robot body models are highly modular.
For the iCub (Fig. 1 A), there would normally be a single model of its
kinematics and another one of its dynamics (mass distribution etc.). Then,
there are distinct modules like forward/inverse kinematics and dynamics that
may draw from the same robot model and be recruited for different purposes
like state estimation, movement planning etc. There would be typically only
one module of every kind (imagine a software library) providing this
functionality upon request. The representations/modules will thus be universal
and not overlapping. For deep learning applied to robotics, this is not the
case. Levine et al. (2018) specialize on a single task (grasping); a different
task will likely need a different network. The representations are thus end-
to-end, task-specific and in case of multiple tasks also overlapping. In
nervous systems, there may also be complete sensorimotor loops specialized on
individual tasks, partially overlapping or redundant. However, this approach
does not scale well. In primates, the posterior parietal cortex is regarded as
a site where information about the body from different modalities converges.
Specific areas related to representations of body parts or reaching targets in
different reference frames have been found. These are recruited in different
tasks or contexts and hence, there is certain universality and
modularity—again more for the body image than body schema Fig. 1, D.

##  4 Use the body directly

While body representations can take very different forms, one should at the
same time consider the radical possibility of using the body directly rather
than through an internal model. Again, as Brooks (1990) put it: “The key
observation is that the world is its own best model. It is always exactly up
to date. It always contains every detail there is to be known. The trick is to
sense it appropriately and often enough.” In fact, for the case of the body,
one can do even without sensing its state. First, there are examples how a
completely passive body, “pure physics”, can generate useful behavior. In the
biological realm, this is for example the body of a trout. Liao (2004) shows
that, paradoxically, under specific circumstances, a dead trout body can
exploit vortices in the water to the extent that it swims upstream. In
robotics, a similar well-known example are the passive dynamics walkers McGeer
(1990)—carefully designed mechanical devices that walk down a ramp without any
motors, sensors, or controllers. Second, in case there is actuation, we have
privileged access to the body current or future state (using forward models /
efference copies (Webb, 2004)) and hence, it may be unnecessary to sense it.

Different positions on the imaginary landscape ranging from model-based
control to direct use of the body are illustrated in Fig. 2. The passive
dynamic walker is positioned at the far right of the schematics. As discussed
above, the octopus is able to reach for visual targets, but it may not
know—and may not need to know—how long its arm is or where it is exactly in
space. Orienting the base of the arm and propagating the bend until contact is
detected by the suckers may well suffice. The need to represent the body, its
state, and the complex inverse kinematics and dynamics has been largely
offloaded to embodiment—the properties of the octopus arm, supported by the
peripheral nervous system and low-dimensional inputs from the central nervous
system. Human reaching, Fig. 2 D, is probably less embodied compared to the
octopus, but still sharing some important characteristics. Cisek and Kalaska
(2003) highlight the importance of online, dynamically generated character of
movement generation in primates. At the same time, they also point out that
due to conduction delays inherent to the sensorimotor system, purely feedback
control is limited, or at least slow. Thus, feed-forward commands and local
neural reflex loops have to work in concert. Robots, on the other hand,
typically heavily rely on models. Importantly, this is the case also for the
solutions employing deep learning. In (Levine et al., 2018), Fig. 2 C, the
embodiment of the robot arm or the gripper is not significantly exploited.

Fig. 2: Model-based control or direct use of the body. (A) iCub humanoid robot
and its models. (B) Model of the ANYmal robot (Hwangbo et al., 2019). (C)
Robot manipulators learning to grasp end-to-end (Levine et al., 2018). (D)
Infant reaching. (E) Octopus and schematic of its nervous system. (F) The
Cornell passive biped with arms Collins et al. (2005).
Credit: A, E – see Fig. 1. Credit F: H. Morgan.

##  5 Robots: with or without a model?

Mechanical engineers naturally think in terms of how to make the best design
of a machine for a task. However, control engineers have a strong preference
for model-based control. Moreover, solutions for nonlinear systems are much
more difficult to obtain, and they often involve a linearization of the system
of some sort. Thus, complex (highly dimensional, dynamic, nonlinear,
compliant, deformable, ‘soft’) robot bodies are avoided as they cannot be
modeled and controlled with the available methods. Many robot engineers then
simply take the body as fixed and seek to exploit to the maximum what can be
done at the “software level”.

Including the parameters of the body into the design considerations may give
rise to better performance of the whole system; these may be solutions
involving a simpler controller, but also solutions that were previously
unattainable when the body was fixed. Following the dynamical systems’
perspective, Füchslin et al. (2013) provide an illustration of the possible
goals of the design process: (1) To design the physical dynamical system such
that desired regions of the state space have attracting properties. Then it is
sufficient to use a simple control signal that will bring the system to the
basins of attraction of individual stable points that correspond to target
behaviors. (2) More complicated behavior can be achieved if the attractor
landscape can be manipulated by the control signal.

If a mathematical formulation of the controller and the plant is available,
this design methodology can be directly applied. The first part is
demonstrated by on the passive dynamic walker (McGeer, 1990): The influence of
scale, foot radius, leg inertia, height of center of mass, hip mass and
damping, mass offset, and leg mismatch is evaluated. In addition, the
stability of the walker is calculated. Jerrold Marsden and his coworkers
presented a method that allows for co-optimization of the controller and plant
by combining an inner loop (with discrete mechanics and optimal control) and
an outer loop (multiscale trend optimization). They applied it to a model of a
walker and obtained the best position of the knee joints ((Pekarek, 2010) –
Ch. 5). However, typical real-world agents are more complex than simple
walkers. Holmes et al. (2006) provide an excellent dynamical systems analysis
of the locomotion of rapidly running insects and derive implications for the
design of the RHex robot. Yet, they conclude that “a gulf remains between the
performance we can elicit empirically and what mathematical analyses or
numerical simulations can explain. Modeling is still too crude to offer
detailed design insights for dynamically stable autonomous machines in
physically interesting settings.” Modeling and optimization of more
complicated morphologies—like compliant structures—is nevertheless an active
research topic (e.g., (Wang, 2009)). The second point of Füchslin et al.
(2013)—achieving “morphological programmability” by constructing a dynamical
system with a parametrized attractor landscape—remains even more challenging
though.

One of the merits of exploiting the contributions of body morphology should be
that the physical processes do not need to be modeled, but can be used
directly. However, without a model of the body at hand, several body designs
need to be produced and—together with the controller—tested in the respective
task setting. The design space of the joint controller-body system blows up
and we may be facing a curse of dimensionality. This is presumably the
strategy adopted by the evolution of biological organisms that could cope with
the enormous dimensionality of the space. In robotics, this has been taken up
by evolutionary robotics (Nolfi and Floreano, 2000). The simulated agents of
Sims (1994) demonstrate that co-evolving brains and bodies together can give
rise to unexpected solution to problems. Bongard (2011) showed that
morphological change indeed accelerates the evolution of robust behavior in
such a brain-body co-evolution setting. With the advent of rapid prototyping
technologies, physics-based simulation could be complemented by testing in
real hardware Lipson and Pollack (2000), but this reintroduces the modeling
through the back door: the phenotypes in the simulator now become models and
they need to sufficiently match their real counterparts. Yet, a “reality gap”
(Jakobi et al., 1995; Koos et al., 2013) always remains between simulated and
real physics. The only alternative is to optimize in hardware directly, which
is in general slow and costly. Brodbeck et al. (2015) provide an interesting
illustration how locomoting cube-like creatures can be evolved in a model-free
fashion through automated manufacturing and testing. However, in summary, the
design decisions—which parameters to optimize—are based on heuristics and a
clear methodology is still missing. Furthermore, with the absence of an
analytical model of the controller and plant, no guarantees on the system’s
performance can be given.

##  6 Conclusion and outlook

Rich properties of complex bodies (highly dimensional, dynamic, nonlinear,
compliant and deformable) have been mostly overlooked or deliberately
suppressed by classical mechatronic designs, as they are largely incompatible
with traditional control frameworks, where linear plants are preferred. This
is definitely a missed opportunity. On the other hand, while complex bodies
carry a lot of “self-control” potential, this property does not come for free.
It has to be said that the exploitation of truly complex bodies to accomplish
tasks is still mostly at a “proof-of-concept” stage. A closely connected issue
is the one of modeling of these systems—complex, or for example soft, bodies
are notoriously difficult to model. The model may not be necessary for the
system to perform the task; however, without a model, the understanding and
design is more complicated and performance guarantees are limited. The field,
which has been dominated by heuristics so far, needs to embrace more
systematic approaches that allow to navigate in this complex landscape.

The area of soft robotics and morphological computation/morphological
control/morphology facilitating control (Füchslin et al., 2013; Müller and
Hoffmann, 2017) is rife with different trading spaces (Pfeifer et al., 2013).
As we move from the traditional engineering framework with a central
controller that commands a “dumb” body toward delegating more functionality to
the physical morphology, some convenient properties will be lost. In
particular, the solutions may not be portable to other platforms anymore, as
they will become dependent on the particular morphology and environment (the
passive dynamic walker is the extreme case). The versatility of the solutions
is likely to drop as well. To some extent, the morphology itself can be used
to alleviate these issues—if it becomes adaptive. Online changes of morphology
(like changes of stiffness or shape) thus constitute another tough
technological challenge. Completely new, distributed control algorithms that
rely on self-organizing properties of complex bodies and local distributed
control units will need to be developed (McEvoy and Correll, 2015; Rieffel et
al., 2010).

In summary, computer scientists, roboticists, and control engineers impose a
representationalist perspective on designing machines and their behaviors.
This is similar to traditional cognitive science (cognitivism). It is
sometimes acknowledged that the representations—world or body—should be
embodied. However, rather than “embodied body models”, it seems more natural
to think of the “brain in the body” or “body in the world” (cf. discussion in
(Alsmith and De Vignemont, 2012; De Vignemont, 2018; Ataria et al., 2021), and
more direct use of the body wherever possible. For engineers, this will be a
major challenge though.

Acknowledgement
This work was supported by the Czech Science Foundation (GA CR), project no.
20-24186X. I would like to thank to Rolf Pfeifer for discussions along these
lines.

## References

  * Alsmith and De Vignemont (2012) Alsmith, A. J. T. and De Vignemont, F. (2012).  Embodying the mind and representing the body.  Review of Philosophy and Psychology, 3(1):1–13.
  * Ataria et al. (2021) Ataria, Y., Tanaka, S., and Gallagher, S. (2021).  Body Schema and Body Image: New Directions.  Oxford University Press.
  * Berlucchi and Aglioti (2010) Berlucchi, G. and Aglioti, S. M. (2010).  The body in the brain revisited.  Experimental brain research, 200(1):25–35.
  * Bongard (2011) Bongard, J. (2011).  Morphological change in machines accelerates the evolution of robust behavior.  Proceedings of the National Academy of Sciences, 108(4):1234–1239.
  * Brodbeck et al. (2015) Brodbeck, L., Hauser, S., and Iida, F. (2015).  Morphological evolution of physical robots through model-free phenotype development.  PLoS ONE, 10(6):e0128444.
  * Brooks (1990) Brooks, R. A. (1990).  Elephants don’t play chess.  Robotics and autonomous systems, 6(1-2):3–15.
  * Brooks (1991) Brooks, R. A. (1991).  Intelligence without representation.  Artificial intelligence, 47(1-3):139–159.
  * Cisek and Kalaska (2003) Cisek, P. and Kalaska, J. (2003).  Reaching movements: implications for computational models.  In Handbook of Brain Theory and Neural Networks, pages 945–948. MIT Press.
  * Collins et al. (2005) Collins, S., Ruina, A., Tedrake, R., and Wisse, M. (2005).  Efficient bipedal robots based on passive dynamic walkers.  Science, 307:1082–1085.
  * De Vignemont (2018) De Vignemont, F. (2018).  Mind the body: An exploration of bodily self-awareness.  Oxford University Press.
  * Füchslin et al. (2013) Füchslin, R., Dzyakanchuk, A., Flumini, D., Hauser, H., Hunt, K., Luchsinger, R., Reller, B., Scheidegger, S., and Walker, R. (2013).  Morphological computation and morphological control: steps towards a formal theory and applications.  Artificial Life, 19(1):9–34.
  * Haugeland (1985) Haugeland, J. (1985).  Artificial intelligence: the very idea.
  * Hoffmann (2021) Hoffmann, M. (2021).  Body models in humans, animals, and robots: mechanisms and plasticity.  In Ataria, Y., Tanaka, S., and Gallagher, S., editors, Body Schema and Body Image: New Directions, pages 152–180. Oxford University Press.
  * Hoffmann (2022) Hoffmann, M. (2022).  Biologically inspired robot body models and self-calibration.  In Ang, M. H., Khatib, O., and Siciliano, B., editors, Encyclopedia of Robotics, pages 1–14. Springer Berlin Heidelberg, Berlin, Heidelberg.
  * Hoffmann and Müller (2017) Hoffmann, M. and Müller, V. C. (2017).  Simple or complex bodies? trade-offs in exploiting body morphology for control.  In Dodig-Crnkovic, G. and Giovagnoli, R., editors, Representation and Reality in Humans, Other Living Organisms and Intelligent Machines, Studies in Applied Philosophy, Epistemology, and Rational Ethics (SAPERE), pages 335–345. Springer.
  * Holmes et al. (2006) Holmes, P., Full, R. J., Koditschek, D., and Guckenheimer, J. (2006).  The dynamics of legged locomotion: Models, analyses and challenges.  SIAM Review, 48(2):207–304.
  * Hwangbo et al. (2019) Hwangbo, J., Lee, J., Dosovitskiy, A., Bellicoso, D., Tsounis, V., Koltun, V., and Hutter, M. (2019).  Learning agile and dynamic motor skills for legged robots.  Science Robotics, 4(26).
  * Jakobi et al. (1995) Jakobi, N., Husbands, P., and Harvey, I. (1995).  Noise and the reality gap: The use of simulation in evolutionary robotics.  In Advances in artificial life, pages 704–720. Springer.
  * Kanayama and Hiromitsu (2021) Kanayama, N. and Hiromitsu, K. (2021).  Triadic body representations in the human cerebral cortex and peripheral nerves.  In Ataria, Y., Tanaka, S., and Gallagher, S., editors, Body Schema and Body Image: New Directions, pages 133–151. Oxford University Press.
  * Koos et al. (2013) Koos, S., Mouret, J.-B., and Doncieux, S. (2013).  The transferability approach: Crossing the reality gap in evolutionary robotics.  Evolutionary Computation, IEEE Transactions on, 17(1):122–145.
  * Levine et al. (2018) Levine, S., Pastor, P., Krizhevsky, A., Ibarz, J., and Quillen, D. (2018).  Learning hand-eye coordination for robotic grasping with deep learning and large-scale data collection.  The International Journal of Robotics Research, 37(4-5):421–436.
  * Leyton and Sherrington (1917) Leyton, A. S. and Sherrington, C. S. (1917).  Observations on the excitable cortex of the chimpanzee, orang-utan, and gorilla.  Quarterly Journal of Experimental Physiology: Translation and Integration, 11(2):135–222.
  * Liao (2004) Liao, J. C. (2004).  Neuromuscular control of trout swimming in a vortex street: implications for energy economy during the karman gait.  Journal of Experimental Biology, 207(20):3495–3506.
  * Lipson (2013) Lipson, H. (2013).  Challenges and opportunities for design, simulation, and fabrication of soft robots.  Soft Robotics, 1:21–27.
  * Lipson and Pollack (2000) Lipson, H. and Pollack, J. (2000).  Automatic design and manufacture of robotic lifeforms.  Nature, 406(6799):974–978.
  * Longo (2015) Longo, M. R. (2015).  Implicit and explicit body representations.  European Psychologist, 20(1):6–15.
  * Longo and Haggard (2010) Longo, M. R. and Haggard, P. (2010).  An implicit body representation underlying human position sense.  Proceedings of the National Academy of Sciences, 107(26):11727–11732.
  * McEvoy and Correll (2015) McEvoy, M. and Correll, N. (2015).  Materials that couple sensing, actuation, computation, and communication.  Science, 347(6228):1261689.
  * McGeer (1990) McGeer, T. (1990).  Passive dynamic walking.  The International Journal of Robotics Research, 9(2):62–82.
  * Metta et al. (2010) Metta, G., Natale, L., Nori, F., Sandini, G., Vernon, D., Fadiga, L., Von Hofsten, C., Rosander, K., Lopes, M., Santos-Victor, J., et al. (2010).  The iCub humanoid robot: An open-systems platform for research in cognitive development.  Neural networks, 23(8-9):1125–1134.
  * Müller and Hoffmann (2017) Müller, V. and Hoffmann, M. (2017).  What is morphological computation? on how the body contributes to cognition and control.  Artificial Life, 23(1):1–24.
  * Nolfi and Floreano (2000) Nolfi, S. and Floreano, D. (2000).  Evolutionary robotics: The biology, intelligence, and technology of self-organizing machines, volume 26.  MIT press Cambridge.
  * Pekarek (2010) Pekarek, D. (2010).  Variational methods for control and design of bipedal robot models.  PhD thesis, California Institute of Technology.
  * Penfield and Boldrey (1937) Penfield, W. and Boldrey, E. (1937).  Somatic motor and sensory representation in the cerebral cortex of man as studied by electrical stimulation.  Brain, 37:389–443.
  * Pfeifer et al. (2013) Pfeifer, R., Marques, H., and Iida, F. (2013).  Soft robotics: the next generation of intelligent machines.  In Proc. 23rd Int. Joint Conf. on Artificial Intelligence, pages 5–11. AAAI Press.
  * Rieffel et al. (2010) Rieffel, J., Valero-Cuevas, F., and Lipson, H. (2010).  Morphological communication: exploiting coupled dynamics in a complex mechanical structure to achieve locomotion.  Journal of the Royal Society Interface, 7(45):613–621.
  * Schöner et al. (2018) Schöner, G., Tekülve, J., and Zibner, S. (2018).  Reaching for objects: a neural process account in a developmental perspective.  In Reach-to-Grasp Behavior, pages 281–318. Routledge.
  * Sims (1994) Sims, K. (1994).  Evolving 3D morphology and behavior by competition.  Artificial Life, 1(4):353–372.
  * Wang (2009) Wang, M. Y. (2009).  A kinetoelastic formulation of compliant mechanism optimization.  Journal of Mechanisms and Robotics, 1(2):021011.
  * Webb (2004) Webb, B. (2004).  Neural mechanisms for prediction: do insects have forward models?  Trends in Neurosciences, 27:278–282.
  * Webb (2006) Webb, B. (2006).  Transformation, encoding and representation.  Current Biology, 16(6):R184–R185.
  * Yekutieli et al. (2005) Yekutieli, Y., Sagiv-Zohar, R., Hochner, B., and Flash, T. (2005).  Dynamic model of the octopus arm. II. control of reaching movements.  Journal of neurophysiology, 94(2):1459–1468.
  * Zullo and Hochner (2011) Zullo, L. and Hochner, B. (2011).  A new perspective on the organization of an invertebrate brain.  Communicative & integrative biology, 4(1):26–29.


#  FLUCTUATIONS OF MULTI-DIMENSIONAL
KINGMAN-LÉVY PROCESSES

Thu Nguyen  Department of Mathematics; International University, HCM City;
No.6 Linh Trung ward, Thu Duc District, HCM City; Email<EMAIL_ADDRESS>
(Date: August 10, 2009)

###### Abstract.

In the recent paper [15] we have introduced a method of studying the multi-
dimensional Kingman convolutions and their associated stochastic processes by
embedding them into some multi-dimensional ordinary convolutions which allows
to study multi-dimensional Bessel processes in terms of the cooresponding
Brownian motions. Our further aim in this paper is to introduce k-dimensional
Kingman-Lévy (KL) processes and prove some of their fluctuation properties
which are analoguous to that of k-symmetric Lévy processes. In particular, the
Lévy-Itô decomposition and the series representation of Rosiński type for
k-dimensional KL-processes are obtained.

Keywords and phrases: Cartesian products of Kingman convolutions; Rayleigh
distributions

##  1\. Introduction. notations and prelimilaries

The purpose of this paper is to introduce and study the multivariate KL
processes defined in terms of multicariate Kingman convolutions. To begin with
we review the following information of the Kingman convolutions and their
Cartesian products.

Let $\mathcal{P}:=\mathcal{P}(\mathbb{R}^{+})$ denote the set of all
probability measures (p.m.’s) on the positive half-line $\mathbb{R}^{+}$. Put,
for each continuous bounded function f on $\mathbb{R}^{+}$,

(1)
$\int_{0}^{\infty}f(x)\mu\ast_{1,\delta}\nu(dx)=\frac{\Gamma(s+1)}{\sqrt{\pi}\Gamma(s+\frac{1}{2})}\\\
\int_{0}^{\infty}\int_{0}^{\infty}\int_{-1}^{1}f((x^{2}+2uxy+y^{2})^{1/2})(1-u^{2})^{s-1/2}\mu(dx)\nu(dy)du,$

where $\mu\mbox{ and }\nu\in\mathcal{P}\mbox{ and }\delta=2(s+1)\geq 1$ (cf.
Kingman[7] and Urbanik[19]). The convolution algebra
$(\mathcal{P},\ast_{1,\delta})$ is the most important example of Urbanik
convolution algebras (cf. Urbanik[19]). In language of the Urbanik convolution
algebras, the characteristic measure, say $\sigma_{s}$, of the Kingman
convolution has the Rayleigh density

(2)
$d\sigma_{s}(y)=\frac{2{(s+1)^{s+1}}}{\Gamma(s+1)}y^{2s+1}\exp{(-(s+1)y^{2})}dy$

with the characteristic exponent $\varkappa=2$ and the kernel $\Lambda_{s}$

(3) $\Lambda_{s}(x)=\Gamma(s+1)J_{s}(x)/(1/2x)^{s},$

where $J_{s}(x)$ denotes the Bessel function of the first kind,

(4)
$J_{s}(x):=\Sigma_{k=0}^{\infty}\frac{(-1)^{k}(x/2)^{\nu+2k}}{k!\Gamma(\nu+k+1)}.$

It is known (cf. Kingman [7], Theorem 1), that the kernel $\Lambda_{s}$ itself
is an ordinary characteristic function (ch.f.) of a symmetric p.m., say
$F_{s}$, defined on the interval [-1,1]. Thus, if $\theta_{s}$ denotes a
random variable (r.v.) with distribution $F_{s}$ then for each
$t\in\mathbb{R}^{+}$,

(5) $\Lambda_{s}(t)=E\exp{(it\theta_{s})}=\int_{-1}^{1}\cos{(tx)}dF_{s}(x).$

Suppose that $X$ is a nonnegative r.v. with distribution $\mu\in\mathcal{P}$
and $X$ is independent of $\theta_{s}$. The radial characteristic function
(rad.ch.f.) of $\mu$, denoted by $\hat{\mu}(t),$ is defined by

(6)
$\hat{\mu}(t)=E\exp{(itX\theta_{s})}=\int_{0}^{\infty}\Lambda_{s}(tx)\mu(dx),$

for every $t\in\mathbb{R}^{+}$. The characteristic measure of the Kingman
convolution $\ast_{1,\delta}$, denoted by $\sigma_{s}$, has the Maxwell
density function

(7)
$\frac{d\sigma_{s}(x)}{dx}=\frac{2(s+1)^{s+1}}{\Gamma(s+1)}x^{2s+1}exp\\{-(s+1)x^{2}\\},\quad(0<x<\infty).$

and the rad.ch.f.

(8) $\hat{\sigma}_{s}(t)=exp\\{-t^{2}/4(s+1)\\}.$

Let $\tilde{P}:=\tilde{\mathcal{P}}(\mathbb{R})$ denote the class of symmetric
p.m.’s on $\mathbb{R}.$ Putting, for every $G\in\mathcal{P}$,

$F_{s}(G)=\int_{0}^{\infty}F_{cs}G(dc),$

we get a continuous homeomorphism from the Kingman convolution algebra
$(\mathcal{P},\ast_{1,\delta})$ onto the ordinary convolution algebra
$(\tilde{\mathcal{P}},\ast)$ such that

(9) $\displaystyle F_{s}\\{G_{1}\ast_{1,\delta}G_{2}\\}$ $\displaystyle=$
$\displaystyle(F_{s}G_{1})\ast(F_{s}G_{2})\qquad(G_{1},G_{2}\in\mathcal{P})$
(10) $\displaystyle F_{s}\sigma_{s}$ $\displaystyle=$ $\displaystyle
N(0,2s+1)$

which shows that every Kingman convolution can be embedded into the ordinary
convolution $\ast$.

Denote by $\mathbb{R}^{+k},k=1,2,...$ the k-dimensional nonnegative cone of
$\mathbb{R}^{k}$ and $\mathcal{P}(\mathbb{\mathbb{R}}^{+k})$ the class of all
p.m.’s on $\mathbb{\mathbb{R}}^{+k}$ equipped with the weak convergence. In
the sequel, we will denote the multidimensional vectors and random vectors
(r.vec.’s) and their distributions by bold face letters.

For each point z of any set $A$ let $\delta_{z}$ denote the Dirac measure (the
unit mass) at the point z. In particular, if
$\mathbf{x}=(x_{1},x_{2},\cdots,x_{k})\in\mathbb{R}^{k+}$, then

(11)
$\delta_{\mathbf{x}}=\delta_{x_{1}}\times\delta_{x_{2}}\times\ldots\times\delta_{x_{k}},\quad(k\;times),$

where the sign $"\times"$ denotes the Cartesian product of measures. We put,
for $\mathbf{x}=(x_{1},\cdots,x_{k})\mbox{ and
}\mathbf{y}=(y_{1},y_{2},\cdots,y_{k})\in\mathbb{R}^{+k},$

(12)
$\delta_{\mathbf{x}}\bigcirc_{s,k}\delta_{\mathbf{y}}=\\{\delta_{x_{1}}\circ_{s}\delta_{y_{1}}\\}\times\\{\delta_{x_{2}}\circ_{s}\delta_{y_{2}}\\}\times\cdots\
\times\\{\delta_{x_{k}}\circ_{s}\delta_{y_{k}}\\},\quad(k\;times),$

here and somewhere below for the sake of simplicity we denote the Kingman
convolution operation $\ast_{1,\delta},\delta=2(s+1)\geq 1$ simply by
$\circ_{s},s\geq\frac{!}{2}.$ Since convex combinations of p.m.’s of the form
(11) are dense in $\mathcal{P}(\mathbb{R}^{+k})$ the relation (12) can be
extended to arbitrary p.m.’s $\mathbf{G}_{1}\mbox{ and
}\mathbf{G}_{2}\in\mathcal{P}(\mathbb{R}^{+k})$. Namely, we put

(13)
$\mathbf{G}_{1}\bigcirc_{s,k}\mathbf{G}_{2}=\iint\limits_{\mathbb{R}^{+k}}\delta_{\mathbf{x}}\bigcirc_{s,k}\delta_{\mathbf{y}}{\mathbf{G}}_{1}(d\mathbf{x}){\mathbf{G}}_{2}(d\mathbf{y})$

which means that for each continuous bounded function $\phi$ defined on
$\mathbb{R}^{+k}$

(14)
$\int\limits_{\mathbb{R}^{+k}}\phi({\mathbf{z}}){\mathbf{G}}_{1}\bigcirc_{s,k}{\mathbf{G}}_{2}(d{\mathbf{z}})=\iint\limits_{\mathbb{R}^{+k}}\big{\\{}\int\limits_{\mathbb{R}^{+k}}\phi({\mathbf{z}})\delta_{{\mathbf{x}}}\bigcirc_{s,k}\delta_{{\mathbf{y}}}(d{\mathbf{z}})\big{\\}}{\mathbf{G}}_{1}(d{\mathbf{x}}){\mathbf{G}}_{2}(d{\mathbf{y}}).$

In the sequel, the binary operation $\bigcirc_{s,k}$ will be called the
k-times Cartesian product of Kingman convolutions and the pair
$(\mathcal{P}(\mathbb{R}^{+k}),\bigcirc_{s,k})$ will be called the
k-dimensional Kingman convolution algebra. It is easy to show that the binary
operation $\bigcirc_{s,k}$ is continuous in the weak topology which together
with (1) and (13) implies the following theorem.

######  Theorem 1.

The pair $(\mathcal{P}{(\mathbb{R}^{+k})},\bigcirc_{s,k})$ is a commutative
topological semigroup with $\delta_{\mathbf{0}}$ as the unit element.
Moreover, the operation $\bigcirc_{s,k}$ is distributive w.r.t. convex
combinations of p.m.’s in $\mathcal{P}(\mathbb{R}^{+k})$.

For every ${\mathbf{G}}\in\mathcal{P}(\mathbb{R}^{+k})$ the k-dimensional
rad.ch.f. $\hat{{\mathbf{G}}}({\mathbf{t}}),{\mathbf{t}}=(t_{1},t_{2},\cdots
t_{k})\in\mathbb{R}^{+k},$ is defined by

(15)
$\hat{\mathbf{G}}(\mathbf{t})=\int\limits_{\mathbb{R}^{+k}}\prod_{j=1}^{k}\Lambda_{s}(t_{j}x_{j}){\mathbf{G}}(\mathbf{dx}),$

where $\mathbf{x}=(x_{1},x_{2},\cdots x_{k})\in\mathbb{R}^{+k}.$ Let
$\mathbf{\Theta_{s}}=\\{\theta_{s,1},\theta_{s,2},\cdots,\theta_{s,k}\\}$,
where $\theta_{s,j}$ are independent r.v.’s with the same distribution
$F_{s}$. Next, assume that ${\mathbf{X}}=\\{X_{1},X_{2},...,X_{k}\\}$ is a
k-dimensional r.vec. with distribution $\mathbf{G}$ and $\mathbf{X}$ is
independent of $\mathbf{\Theta}_{s}$. We put

(16)
$[{\mathbf{\Theta}}_{s},{\mathbf{X}}]=\\{{\theta_{s,1}X_{1},\theta_{s,2}X_{2},...,\theta_{s,k}X_{k}}\\}.$

Then, the following formula is equivalent to (15) (cf. [14])

(17)
$\widehat{\mathbf{G}}({\mathbf{t}})=Ee^{i<{\mathbf{t}},[{\mathbf{\Theta}_{s},\mathbf{X}}]>},\qquad({\mathbf{t}}\in\mathbb{R}^{+k}).$

The Reader is referred to Corollary 2.1, Theorems 2.3 & 2.4 [14] for the
principal properties of the above rad.ch.f. Given $s\geq-1/2$ define a map
$F_{s,k}:\mathcal{P}(\mathbb{R}^{+k})\rightarrow\mathcal{P}(\mathbb{R}^{k})$
by

(18)
$F_{s,k}({\mathbf{G}})=\int\limits_{\mathbb{R}^{+k}}(T_{c_{1}}F_{s})\times(T_{c_{2}}F_{s})\times\ldots\times(T_{c_{k}}F_{s}){\mathbf{G}}(d{\mathbf{c}}),$

here and in the sequel, for a distribution $\mathbf{G}$ of a r.vec.
$\mathbf{X}$ and a real number c we denote by $T_{c}{\mathbf{G}}$ the
distribution of $c{\mathbf{X}}$. Let us denote by
$\tilde{\mathcal{P}}_{s,k}(\mathbb{R}^{+k})$ the sub-class of
$\mathcal{P}(\mathbb{R}^{k})$ consisted of all p.m.’s defined by the right-
hand side of (18). By virtue of (15)-(18) one can prove the following theorem.

######  Theorem 2.

The set $\tilde{\mathcal{P}}_{s,k}(\mathbb{R}^{+k})$ is closed w.r.t. the weak
convergence and the ordinary convolution $\big{.}\ast$ and the following
equation holds

(19)
$\hat{\mathbf{G}}({\mathbf{t}})=\mathcal{F}(F_{s,k}({\mathbf{G}}))({\mathbf{t}})\qquad({\mathbf{t}}\in{\mathbb{R}^{+k}})$

where $\mathcal{F}({\mathbf{K}})$ denotes the ordinary characteristic function
(Fourier transform) of a p.m. ${\mathbf{K}}$. Therefore, for any
${\mathbf{G}}_{1}\mbox{ and }{\mathbf{G}}_{2}\in\mathbb{R}^{+k}$

(20) $F_{s,k}({\mathbf{G}}_{1})\big{.}\ast
F_{s,k}({\mathbf{G}}_{2})=F_{s,k}({\mathbf{G}}_{1}\bigcirc_{s,k}{\mathbf{G}}_{2})$

and the map $F_{s,k}$ commutes with convex combinations of distributions and
scale changes $T_{c},c>0$. Moreover,

(21) $F_{s,k}({\Sigma_{s,k}})=N({\mathbf{0}},2(s+1){\mathbf{I}})$

where $\Sigma_{s,k}$ denotes the k-dimensional Rayleigh distribution and
$N({\mathbf{0}},2(s+1){\mathbf{I}})$ is the symmetric normal distribution on
$\mathbb{R}^{k}\mbox{ with variance operator
}{\mathbf{R}}=2(s+1){\mathbf{I}},{\mathbf{I}}$ being the identity operator.
Consequently, Every Kingman convolution algebra
$\big{(}\mathcal{P}(\mathbb{R}^{+k}),\bigcirc_{s,k}\big{)}$ is embedded in the
ordinary convolution algebra
$\big{(}\mathcal{P}_{s,k}(\mathbb{R}^{+k}),\big{.}\star\big{)}$ and the map
$F_{s,k}$ stands for a homeomorphism.

Let us denote by
$\mathcal{E}=\\{{\mathbf{e}}=(e_{1},e_{2},\ldots,e_{k}),e_{j}=\pm,j=1,2,\ldots,k\\}$.
It is convenient to regard the elements of $\mathcal{E}$ as sign vectors.
Denote
$\mathbb{R}^{+k}_{\mathbf{e}}=\\{[{\mathbf{e}},{\mathbf{x}}]:{\mathbf{x}}\in\mathbb{R}^{+k}\\},\mbox{
where
}[{\mathbf{e}},{\mathbf{x}}]:=(e_{1}x_{1},e_{2}x_{2},\ldots,e_{k}x_{k}).$ Then
the family $\\{\mathbb{R}^{+k}_{\mathbf{e}},{\mathbf{e}}\in\mathcal{E}\\}$ is
a partition of the space $\mathbb{R}^{k}.$ If $\mathbf{X}$ is a k-dimensional
r.vec. with distribution $\mathbf{G},$ the k-symmetrization of $\mathbf{G}$,
denoted by $\tilde{\mathbf{G}},$ is defined by

(22)
$\tilde{\mathbf{G}}=\frac{1}{2^{k}}\sum_{\mathbf{e}\in\mathcal{E}}S_{\mathbf{e}}{\mathbf{G}},$

where the operator $S_{\mathbf{e}}$ is defined by

(23)
$S_{\mathbf{e}}({\mathbf{x}})=[{\mathbf{e}},{\mathbf{x}}]\qquad{\mathbf{x}}\in{\mathbb{R}^{k}}$

and the symbol $S_{\mathbf{e}}\tilde{\mathbf{G}}$ denotes the image of
$\mathbf{G}$ under $S_{\mathbf{e}}$.

######  Definition 1.

We say that a distribution $\mathbf{G}\in\mathcal{P}(\mathbb{R}^{k})$ is
k-symmetric, if the equation $\mathbf{G}=\tilde{\mathbf{G}}$ holds.

######  Definition 2.

A p.m. ${\mathbf{F}}\in\mathcal{P}(\mathbb{R}^{+k})$ is called
$\bigcirc_{s,k}-$infinitely divisible ($\bigcirc_{s,k}-$ID), if for every m=1,
2, …there exists $\mathbf{F}_{m}\in\mathbf{P}(\mathbb{R}^{+k})$ such that

(24)
${\mathbf{F}}={\mathbf{F}}_{m}\bigcirc_{s,k}{\mathbf{F}}_{m}\bigcirc_{s,k}\ldots\bigcirc_{s,k}{\mathbf{F}}_{m}\quad(m\;times).$

######  Definition 3.

$\mathbf{F}$ is called stable, if for any positive numbers a and b there
exists a positive number c such that

(25)
$T_{a}{\mathbf{F}}\;{\bigcirc_{s,k}}\;T_{b}{\mathbf{F}}=T_{c}{\mathbf{F}}$

By virtue of Theorem 2 it follows that the following theorem holds.

######  Theorem 3.

A p.m. $\mathbf{G}\mbox{ is }\bigcirc_{s,k}-ID$, resp. stable if and only if
$H_{s,k}({\mathbf{G}})$ is ID, resp. stable, in the usual sense.

The following theorem gives a representation of rad.ch.f.’s of
$\bigcirc_{s,k}-$ID distributions. The proof is a verbatim reprint of that for
([14], Theorem 2.6).

######  Theorem 4.

A p.m. $\mu\in ID(\bigcirc_{s,k})$ if and only if there exist a
$\sigma$-finite measure M (a Lévy’s measure) on $\mathbb{R}^{+k}$ with the
property that $M({\mathbf{0}})=0,{\mathbf{M}}$ is finite outside every
neighborhood of ${\mathbf{0}}$ and

(26)
$\int_{\mathbb{R}^{+k}}\frac{\|{\mathbf{x}}\|^{2}}{1+\|{\mathbf{x}}\|^{2}}{\mathbf{M}}(d{\mathbf{x}})<\infty$

and for each ${\mathbf{t}}=(t_{1},...,t_{k})\in\mathbb{R}^{+k}$

(27)
$-\log{\hat{\mu}({\mathbf{t}})}=\int_{\mathbb{R}^{+k}}\\{1-\prod_{j=1}^{k}\Lambda_{s}(t_{j}x_{j})\\}\frac{1+\|{\mathbf{x}}\|^{2}}{\|{\mathbf{x}}\|^{2}}M({\mathbf{dx}}),$

where, at the origin $\mathbf{0}$, the integrand on the right-hand side of
(27) is assumed to be

(28) $lim_{\|\mathbf{x}\|\rightarrow
0}\\{1-\prod_{j=1}^{k}\Lambda_{s}(t_{j}x_{j})\\}\frac{1+\|\mathbf{x}\|^{2}}{\|\mathbf{x}\|^{2}}=\Sigma_{j=1}^{k}\lambda^{2}_{j}t_{j}^{2}$

for nonnegative $\lambda_{j},j=1,2,...,k.$ In particular, if $M=0,\mbox{ then
}\mu$ becomes a Rayleighian distribution with the rad.ch.f (see definition 4)

(29)
$-\log{\hat{\mu}({\mathbf{t}})}=\frac{1}{2}\sum_{j=1}^{k}\lambda^{2}_{j}t_{j}^{2},\quad{\mathbf{t}}\in\mathbb{R}^{+k},$

for some nonnegative $\lambda_{j},j=1,...,k.$ Moreover, the representation
(27) is unique.

An immediate consequence of the above theorem is the following:

######  Corollary 1.

Each distribution $\mu\in ID(\bigcirc_{s,k})$ is uniquely determined by the
pair $[\mathbf{M},\boldsymbol{\lambda}]$, where $\mathbf{M}$ is a Levy’s
measure on $\mathbb{R}^{+k}$ such that $\mathbf{M}(\mathbf{0})=0,$
$\mathbf{M}$ is finite outsite every neighbourhood of $\mathbf{0}$ and the
condition (26) is satisfied and
$\boldsymbol{\lambda}:=\\{\lambda_{1},\lambda_{2},\cdots\lambda_{k}\\}\in\mathbb{R}^{+k}$
is a vector of nonnegative numbers appearing in (29). Consequently, one can
write $\mu\equiv[\mathbf{M},\boldsymbol{\lambda}].$
In particular, if $\mathbf{M}$ is zero measure then
$\mu=[\boldsymbol{\lambda}]$ becomes a Rayleighian p.m. on $\mathbb{R}^{+k}$
as defined as follows:

######  Definition 4.

A k-dimensional distribution, say $\boldsymbol{\mathbf{\Sigma}}_{s,k}$, is
called a Rayleigh distribution, if

(30)
$\boldsymbol{\mathbf{\Sigma}}_{s,k}=\sigma_{s}\times\sigma_{s}\times\cdots\times\sigma_{s}\quad(k\;times).$

Further, a distribution ${\mathbf{F}}\in\mathcal{P}(\mathbb{R}^{+k})$ is
called a Rayleighian distribution if there exist nonnegative numbers
$\lambda_{r},r=1,2\cdots k$ such that

(31)
${\mathbf{F}}=\\{T_{\lambda_{1}}\sigma_{s}\\}\times\\{T_{\lambda_{2}}\sigma_{s}\\}\times\ldots\times\\{T_{\lambda_{k}}\sigma_{s}\\}.$

It is evident that every Rayleigh distribution is a Rayleighian distribution.
Moreover, every Rayleighian distribution is $\bigcirc_{s,k}-$ID. By virtue of
(7 ) and (30) it follows that the k-dimensional Rayleigh density is given by

(32)
$g({\mathbf{x}})=\Pi_{j=1}^{k}\frac{2^{k}(s+1)^{k(s+1)}}{\Gamma^{k}(s+1)}x_{j}^{2s+1}exp\\{-(s+1)||{\mathbf{x}}||^{2}\\},$

where ${\mathbf{x}}=(x_{1},x_{2},\ldots,x_{k})\in\mathbb{R}^{+k}$ and the
corresponding rad.ch.f. is given by

(33)
$\hat{\Sigma}_{s,k}({\mathbf{t}})=Exp(-|{\mathbf{t}}|^{2}/4(s+1)),\quad{\mathbf{t}}\in\mathbb{R}^{+k}.$

Finally, the rad.ch.f. of a Rayleighian distribution $\mathbf{F}\mbox{ on
}\mathbb{R}^{+k}$ is given by

(34)
$\hat{\mathbf{F}}({\mathbf{t}})=Exp(-\frac{1}{2}\sum_{j=1}^{k}\lambda_{j}^{2}t_{j}^{2})$

where $\lambda_{j},j=1,2,\ldots,k$ are some nonnegative numbers.

##  2\. Multivariate Bessel processes

##  3\. Multivrariate Kingman-Lévy processes and their Lévy-Itô decomposition

Suppose that $\mu_{t},t\geq 0$ is continuous semigroup in
$ID(\bigcirc_{s,k})$, that is for any $t,s\geq 0$

(35) $\mu_{t}\bigcirc_{s,k}\mu_{s}=\mu_{t+s}$

and $\\{\mu_{t}\\}$ is continuous at 0 i.e.

$lim_{t\rightarrow 0}\mu_{t}=\delta_{\mbox{0}}.$

By virtue of Theorem 2 it follows that $\\{\mathcal{F}_{s,k}(\mu_{t})\\}$ is
an ordinary continuous convolution semigroup on $\mathbb{R}^{k}.$ Putting, for
each $\mathbf{x}\in\mathbb{R}^{k+}$ and for every Borel subset
$\mathcal{E}\mbox{ of }\mathbb{R}^{k+},$

(36)
$\mathbf{P}(t,\mathcal{E},\mathbf{x})=\mu_{t}\bigcirc_{s,k}\delta_{\mathbf{x}}(\mathcal{E})$

and using the rad.ch.f. it follows that the family
$\\{\mathbf{P}(t,\mathcal{E},\mathbf{x}),t\geq 0\\}$ satisfies the Chapman-
Kolmogorov equation and, consequently, the formula (36) defines transition
probabilities of a $\mathbb{R}^{k+}-$valued homogeneous strong Markov Feller
process $\\{\mathbf{X}^{\mathbf{x}}_{t},t\geq 0\\}$, say, such that it is
stochastically continuous and has a cadlag version (compare [11], Theorem
2.6).

######  Definition 5.

A $\mathbb{R}^{k+}$-valued stochastic process $\\{\mathbf{X}_{t},t\geq 0\\}$
is called a Kingman-Lévy process, if $\mathbf{X}_{t}=$

(i) $\mathbf{X}_{0}=\mathbf{0}\qquad(P.1);$

(ii) There exists a $\mathbb{R}^{k+}-$valued homogeneous strong Markov Feller
process having a cadlag version $\\{\mathbf{X}^{\mathbf{x}}_{t},t\geq 0\\}$
with transition probabilities defined by (36) and
$\mathbf{X}_{t}=\mathbf{X}^{\mathbf{0}}_{t},t\geq 0;$

##  4\. Fluctuations of Multidimensional Bessel Processes

######  Definition 6.

Let $(W_{t},t\geq 0)$ be a d-dimensional Brownian motion (d=1, 2, …). The
Euclidean norm of $(W_{t})$, denoted by $B_{t},t\geq 0$ is called a Bessel
process.

It has been proved that Bessel processes inherit the well-known
characteristics of Brownian motions: They are independent stationary
”increments” processes with continuous sample paths. The term ’increment’ is
defined as follows:

######  Definition 7.

For any $s>u$ the random variable $|W_{s}-W_{u}|$ is called an increments of
the Bessel process.

The following theorem gives a Lévy-Khinczyn representation of the Bessel
process in the sense of the Kingman convolution.

######  Theorem 5.

The radial characteristic function $\phi(x)$ of the Bessel process $(B_{t})$
is of the form

(37) $\phi(x)=exp\\{-\frac{tx^{2}}{4(s+1)}\\}\qquad x,t\geq 0$

where d=2(s+1).

Since for any $s>u$ the ’increment’ of the Bessel process $(B_{t})$ is
infinitely divisible in the ordinary convolution $\ast$ we have the following
representation of the Fourier transform of $B_{s-u}.$

(38) $\mathcal{F}_{B_{s-u}}(x)=exp(-(s-u)\psi(x))$

where $\psi(x)$ is a symmetric characteristic exponent

(39) $\psi(x)=\frac{1}{2}\sigma^{2}+\int_{0}^{\infty}(1-cos\,xv)\Pi(dv)$

where the measure $\Pi$ satisfies the condition
begin equation

(40) $\int_{0}^{\infty}(min(1,x^{2})\Pi(dx)<\infty.$

which implies the following Lévy-Itô decomposition.

######  Theorem 6.

(Lévy-Itô decomposition) There exists a Brownian motion $X^{(1)}_{t}$ and a
compound Poison process $X^{(2)}_{t}$ independent of $X^{(1)}_{t}$ such that

(41) $B_{t}=||W_{t}||\overset{d}{=}X^{(1)}_{t}+X^{(2)}_{t}\qquad(t\geq 0).$

Before stating the Wienner-Hopf factorization (WHf) theorem for Bessel
processes we introduce some concepts and notations. The importance of WHf is
that it gives us information of the ascending and descending ladder processes.
We begin by recalling that for $\alpha,\beta\geq 0$ the Laplace exponents
$\kappa(\alpha,\beta)\mbox{ and }\hat{\kappa}(\alpha,\beta)$ of the ascending
ladder process $(\hat{L}^{-1},\hat{H})$ and the descending ladder process
$(\hat{L}^{-1},\hat{H}).$ Further, we define

$\overset{-}{G}_{t}=sup\\{s<t:\overset{-}{X}_{s}=X_{s}\\}\mbox{ and
}\underset{-}{G_{t}}=sup\\{s<t:\underset{-}{X_{t}}=X_{s}.$

######  Theorem 7.

(Wienner-Hopf Factorization) Let $(B_{t},t\geq 0)$ be a Bessel process. Denote
by ${\mathbf{e}}_{p}$ an independent and exponentially distributed random
variable.

The pairs
$(\overset{-}{G}_{{\mathbf{e}}_{p}},\overset{-}{X}_{{\mathbf{e}}_{p}})\mbox{
and
}({\mathbf{e}}_{p}-\overset{-}{G}_{{\mathbf{e}}_{p}},\overset{-}{X}_{{\mathbf{e}}_{p}}-X_{{\mathbf{e}}_{p}})$
are independent and infinitely divisible, yielding the factorization

(42)
$\frac{p}{p-i\nu+\psi(\theta)}=\Psi^{+}(\nu,\theta).\Psi^{-}(\nu,\theta)\qquad\nu,\theta\in\mathbb{R},$

$\psi^{+},\psi^{-}$ being Fourier transforms and called the Wienner-Hopf
factors.

##  5\. Levy-Ito decomposition of Kingman-Levy processes

## References

  * [1] Bingham, N.H., Random walks on spheres, Z. Wahrscheinlichkeitstheorie Verw. Geb., 22, (1973), 169-172.
  * [2] Bingham, N.H., On a Theorem of Klosowska about generalized convolutions, Colloquium Math., 28 No. 1, (1984), 117-125.
  * [3] Cox, J.C., Ingersoll, J.E.Jr., and Ross, S.A., A theory of the term structure of interest rates. Econometrica, 53(2), (1985).
  * [4] Feller, W., An Introduction to probability Theory and Its Applications, John Wiley & Sons Inc., Vol.II, 2nd Ed., (1971).
  * [5] Ito, K., Mckean H.P., Jr., Diffusion processes and their sample paths, Berlin-Heidelberg-New York. Springer (1996).
  * [6] Kalenberg O., Random measures, 3rd ed. New York: Academic Press, (1983).
  * [7] Kingman, J.F.C., Random walks with spherical symmetry, Acta Math., 109, (1963), 11-53.
  * [8] Kyprianou, Andreas E., Introductory lectures on fluctuations of Lévy processes with applications,
  * [9] Levitan B.M., Generalized translation operators and some of their applications, Israel program for Scientific Translations, Jerusalem, (1962).
  * [10] Linnik Ju. V., Ostrovskii, I. V., Decomposition of random variables and vectors, Translation of Mathematical Monographs, vol. 48, American Mathematical Society, Providence R. L, 1977, ix+380 pp.,$38.80. (Translated from the Russian, 1972, by Israel Program for Scientific Translations).
  * [11] Nguyen V.T, Generalized independent increments processes, Nagoya Math. J.133, (1994), 155-175.
  * [12] Nguyen V.T., Generalized translation operators and Markov processes, Demonstratio Mathematica, 34 No 2, ,295-304.
  * [13] Nguyen T.V., OGAWA S., Yamazato M. A convolution Approach to Mutivariate Bessel Processes, Proceedings of the 6th Ritsumeikan International Symposium on ”Stochastic Processes and Applications to Mathematical Finance”, edt. J. Akahori, S. OGAWA and S. Watanabe, World Scientific, (2006) 233-244.
  * [14] Nguyen V. T., A Kingman convolution approach to Bessel processes, Probab. Math. Stat, Probab. Math. Stat. 29, fasc. 1(2009) 119-134.
  * [15] Nguyen V. T., An analogue of the Cramér-Lévy theorem for multi-dimensional Rayleigh distributions, arxiv.org/abs/0907.5035.
  * [16] Revuz, D. and Yor, M., Continuous martingals and Brownian motion. Springer-verlag Berlin Heidelberg, (1991).
  * [17] Sato K, Lévy processes and infinitely divisible distributions, Cambridge University of Press, (1999).
  * [18] Shiga T., Watanabe S., Bessel diffusions as a one-parameter family of diffusion processes, Z. Warscheinlichkeitstheorie Verw. geb. 27,(1973), 34-46.
  * [19] Urbanik K., Generalized convolutions, Studia math., 23 (1964), 217-245.
  * [20] Urbanik K., Cramér property of generalized convolutions,Bull. Polish Acad. Sci. Math.37 No 16 (1989), 213-218.
  * [21] Vólkovich, V. E., On symmetric stochastic convolutions, J. Theor. Prob. 5, No. 3(1992), 417-430.