Stochastics
Cluster
Research interests include:
Functional inequalities and
applications
Stochastic partial differential
equations and applications to fluid
mechanics (in particular, stochastic
Burgers equation and turbulence),
to engineering and financial
mathematics
Pseudo

differential operators and
jump

type Markov processes.
Analysis related to multi

parameter
processes
Stochastic dynamics of infinite
systems of interacting particles in
continuum and their scaling limits.
Particle densities of the quasi

free
representations of CCR and CAR
Stochastic processes applied to
problems of astrophysics, in
particular, statistical analysis of
cosmic microwave background
radiation in search for relic
gravitational waves
Stochastic computational modelling
in engineering
Feynman integrals and functional
integrals in mathematical physics
Lévy

type processes and their
generators, geometry associated to
Lévy

type processes
Information theory: capacity of
discrete and continuous time
channels, entropy power
inequalities and around it
Stochastic differential equations
with
Markovian
switching and their
numerical solution
Non

commutative (quantum)
probability, in particular, free
probability, non

commutative
Lévy
processes and non

commutative
Markov chains
Stochastic modelling of
monofractal
and
multifractal
multiscale
systems
and related topics
Quantum open systems and
quantum control
Large deviations of heavy

tailed
random variables and exit times for
Markov processes
Numerical simulation of stochastic
processes
Szego

type limit theorems
, the
asymptotic theory of integrals and
quadratic forms of
spatio

temporal
random fields and their applications
to statistical inference
Infinite dimensional stochastic
analysis, including analysis on
Riemannian path spaces, point
processes, measure

valued
processes
Dirichlet
forms on infinite
dimensional spaces and related
Markov processes
Jump processes in engineering
A. Potrykus
Cluster Events
Future Activities Involving PhD Students
Poster Presenters: Dr A
Neate
, Dr A
Potrykus
I.
Functional inequalities and applications

Construct
successful
couplings
for
sub

elliptic
diffusion
processes,
such
as
degenerate
SDEs
on
manifolds
and
Hilbert
spaces,
so
that
the
convergence
rate
and
regularity
properties
of
sub

elliptic
diffusion
semigoups
can
be
described
.

Develop
stochasti c
anal ysi s
on
the
path
space
over
manifolds
with
boundary
.
Investigate
functional
inequalities
for
SDEs
with
jumps
.

Dimension

free
Harnack
inequalities
and
applications
for
SDEs
,
SPDEs
and
SDDEs
.
[
Feng

Yu Wang]
II.
Markov

modulated stochastic delay equations with reflection
A
Markov

modulated
system
is
a
hybrid
system
with
a
state
vector
that
has
two
components,
where
one
is
continuous
(the
state)
and
one
is
discrete
(the
mode)
.
In
its
operation,
the
system
will
switch
from
one
mode
to
another
in
a
random
way,
based
on
a
Markov
chain
with
a
finite
state
space
.
We
shall
bring
delay,
reflection,
noise
and
Markov
chain
together,
and
investigate
a
Markov

modulated
stochastic
differential
delay
equations
with
reflection
(MMSDDER)
and
discuss
the
following
problems
:

The
existence,
uniqueness,
and
invariant
measures
of
solutions
of
MMSDDER
.

Numerical
approximations
of
MMSDDER
.

We
shall
apply
this
dynamical
system
to
population
dynamics
.
[
Chenggui
Yuan]
III.
Stochastic dynamics of infinite particle systems in continuum

Construction
and
study
of
equilibrium
and
non

equilibrium
stochastic
dynamics
of
binary
jumps
.
Uniqueness
of
the
equilibrium
dynamics
(essential
self

adjointness
of
the
generator)
.
Diffusion
approximation
for
such
a
dynamics
.
A
mean

field
type
scaling
limit
leading
to
a
birth

and

death
process
in
continuum
.
A
spectral
gap
for
the
limiting
generator
.
Vlasov

type
and
hydrodynamic
scaling
limits
of
non

equilibrium
dynamics
.

Develop
a
theory
of
determinantal
point
processes
whose
correlation
kernel
is
J

Hermitian
,
i
.
e
.
,
Hermitian
in
an
indefinite
scalar
product
.
[Eugene
Lytvynov]
IV.
Nonlinear
SPDEs
of Burgers type
and their interacting particle approximation
The
equation
is
of
stochastic
parabolic
type
in
h i g h er
s p a c e
d i men s i on s,
i n v ol v i n g
a
Ma r kov
g en er a t or
of
a
(symmetric)
stable

like
process
and
with
a
Levy
space

time
noise
force
.
Besides
the
existence
and
uniqueness
problems,
it
is
planned
to
study
the
absolute
continuity
of
the
laws
of
the
solutions
with
respect
to
(spatial)
Lebesgue
measure
and
the
(stochastic)
smoothness
property
of
the
solution
.
A
further
study
is
to
establish
the
infinite
particle
approximation
for
the
initial
value
problem,
and
hence
to
derive
a
statistical
physics
picture
of
the
non

linear
SPDEs
.
[
Jian

Lung
Wu]
Mathematical Feynman Path Integrals, 18

19 Jan 2010
Swansea hosted a meeting on the rigorous theory of Feynman path integrals and their applications on 18

19 January 2010. The
event was
organised
by A Truman and A
Neate
with the support of WIMCS. This workshop coincided with the visit to the
department of O.
Smolyanov
(Moscow State) who has a long

standing collaboration with A. Truman. The workshop focused on
recent developments in several different rigorous approaches to the Feynman path integral together with their applications
to
areas such as the theory of quantum open systems and quantum control. Speakers included:
N. Kumano

Go (
Kogakuin
), S.
Mazzucchi
(Trento) , M. Morrow (Nottingham), O.
Smolyanov
(Moscow State) , L.
Cattaneo
(Imperial), M.
Grothaus
(Kaiserslautern), V.
Kolokoltsov
(Warwick)
Sefydliad Gwyddorau Cyfrifiadurol a Mathemategol Cymru
SGCMC
WIMCS
Wales Institute of Mathematical and Computational Sciences
Stochastic Processes at the Quantum Level, 21

22 Oct 2009
Aberystwyth
hosted a meeting on recent developments and applications of classical and non

commutative probability in
modelling
the quantum world. There has been considerable interest in the quantum
generalisations
of open systems, statistics,
stochastic processes, measurement and filtering theory, however, these pioneering
endeavours
are now finding direct
applications in emergent technologies based on the prospect of quantum control. The workshop brought together experts in
mathematical and theoretical physics to discuss topics in quantum stochastic processes, quantum filtering based optimal contr
ol
and coherent control, quantum feedback networks, and quantum statistics and independence. Speakers included:
V.
Belavkin
(Nottingham), A. Belton (Lancaster), M.
Guta
(Nottingham), R. Hudson (
Loughborough
), M.
Mirrahimi
(SISYPHE

INRIA

Rocquencourt
), H.
Nurdin
(ANU
–
Canberra)
Motivations and Aim
In
the
computational
modeling
of
real

life
civil,
mechanical
and
aerospace
engineering
systems,
uncertainties
have
to
be
taken
into
account
and
need
to
be
specified
.
Most
existing
models
are
based
on
Gaussian
white
noise
or,
more
generally,
on
stationary
processes
.
More
realistic
models
are
possible
when
allowing
the
noise
of
the
system
to
belong
to
a
wider
class
of
stochastic
processes,
in
this
case
to
the
class
of
Markov
processes
.
One
method
of
constructing
and
approximating
time

and
space

inhomogeneous
Markov
processes
is
via
pseudo

differential
operators
.
The
advantage
of
this
approach
is
that
it
relies
only
on
analytical
tools
.
The
aim
is
to
•
Formulate
engineering
problems
using
time

and
space

inhomogeneous
Markov
processes
•
Derive
explicit
formulae
for
statistical
properties
using
stochastic
calculus
for
jump
processes
•
Compare
models
driven
by
different
types
of
jump
processes
•
Construct
and
approximate
time

and
space

inhomogeneous
Markov
processes
using
pseudo

differential
operators
Methods
•
Stochastic
calculus
for
jump
processes
•
Pseudo

differential
operator
calculus
for
the
construction
of
jump
processes
•
Numerical
approximation
of
jump
processes
•
Numerical
and
explicit
solutions
of
SDEs
Outcomes
•
Development
of
a
model
for
the
vibration
of
wind
turbines
using
time

inhomogeneous
jump
processes
•
Explicit
formula
for
the
autocorrelation
function
of
the
displacement
of
wind
turbines
driven
by
jump
processes
•
Modeling
of
the
displacement
and
voltage
ouput
of
an
energy
harvesting
device
using
(non

)linear
coupled
SDEs
Selected publications
1.
A.
POTRYKUS
,
A symbolic calculus and a
parametrix
construction for
pseudodifferential
operators with non

smooth negative definite symbols.
Rev. Mat.
Complut
. 22 (2009), no. 1, 187

207.
2.
A.
POTRYKUS
,
AND
S. ADHIKARI
,
Dynamical response of damped structural
systems driven by jump processes
.
Probab
. Eng. Mech. 25 (2010), no. 3, 305


314
.
3.
A.
POTRYKUS
,
Pseudodifferential
operators with rough negative definite
symbols.
Integr
.
Equ
.
Oper
. Theory (2010), no. 66, 441

461
.
4.
A. POTRYKUS, AND S. ADHIKARI,
Response Statistics of Linear Oscillators
With
Ito

Levy Noise, submitted.
Next steps
•
Study
of
SPDEs
such
as
the
beam
or
plate
equation
driven
by
jump
processes
•
Develop
a
theory
of
multi

parameter
processes
generated
by
pseudo

differential
operators
•
Investigation
of
structural
health
monitoring
devices
on
bridges
where
the
vibrations
of
the
bridge
are
modeled
using
time

inhomogeneous
Markov
processes
Voltage of an energy
harvester driven by a jump
process
Phase portrait of energy
harvester driven by a jump
process
Functional Integration and
SPDEs
A.
Neate
Motivations and Aim
Elworthy
and
Truman
showed
that
the
asymptotic
properties
of
the
heat
equation
in
the
limit
as
the
diffusion
parameter
tends
to
zero
can
be
investigated
using
a
functional
integral
representation
based
on
the
Feynman

Kac
formula
together
with
the
underlying
classical
mechanical
system
and
the
related
Hamilton

Jacobi
theory
.
This
work
has
been
extended
with
Davies,
Zhao
and
others
to
consider
stochastic
heat
equations,
Burgers
equations
and
KPP
equations
.
In
particular
it
has
been
used
to
derive
an
explicit
asymptotic
series
solution
for
the
stochastic
Burgers
equation
and
in
investigating
the
singularities
in
the
inviscid
limit
of
the
Burgers
velocity
field
.
The
aim
here
was
to
:
•
Further
investigate
the
singularity
structure
and
form
models
of
turbulent
behaviour
.
•
Extend
methods
to
consider
a
stochastic
Burgers
equation
with
vorticity
.
•
Extend
the
configuration
space
approaches
used
previously
to
classical
mechanics
in
phase
space
.
•
Develop
applications
to
astrophysics
.
Methods
•
Theory
of
stochastic
flows
of
diffeomorphisms
(
Kunita
,
Elworthy
)
.
•
Hamilton

Jacobi
theory
of
Elworthy
,
Truman,
Zhao
for
changing
measures
in
Feynman

Kac
formulae
.
•
Stochastic
perturbations
of
classical
mechanical
systems
and
Nelson’s
stochastic
mechanics
.
Outcomes
•
Derivation
of
asymptotic
series
solutions
for
Hamilton

Jacobi
equations
based
on
continuous
semi

martingale
Hamiltonians
.
•
Asymptotic
series
expansions
for
stochastic
Burgers
equations
with
random
vorticity
.
•
Geometric
discussion
of
caustics
and
Maxwell
set
singularities
in
terms
of
classical
mechanics
and
Schilder
type
asymptotic
expansions
.
•
Models
for
the
advent
and
intermittence
of
turbulence
based
on
the
geometric
behaviour
of
singularity
structures
.
•
Stationary
state
solutions
for
the
stochastic
Burgers
equation
with
vorticity
under
the
influence
of
a
Coulomb
potential
.
Selected publications
1.
A. NEATE, AND A. TRUMAN,
The stochastic Burgers equation with
vorticity
:
semiclassical
asymptotic series solutions with applications
,
Submitted to J. Math. Phys. (2010).
2.
A. NEATE, AND A. TRUMAN,
Hamilton

Jacobi Theory and the stochastic
elementary formula
,
SU Preprint (2010).
3.
A. NEATE, AND A. TRUMAN,
A Burgers
Zeldovich
model for the formation
of
planetesimals
via Nelson’s stochastic mechanics
,
SU Preprint (2010).
4.
R. DURRAN, A. NEATE, A. TRUMAN, AND (F.Y. WANG)
The Divine
clockwork, On the Divine Clockwork
, J. Math. Phys. 49,
032102
&
102103
(2008)
5.
A.NEATE, AND A. TRUMAN,
A one

dimensional analysis of singularities and
turbulence for the stochastic Burgers equation in
d
dimensions.
Seminar on
Stochastic Analysis, Random Fields and Applications V (2008)
Next steps
•
Applications of phase space stochastic flows to Feynman path integrals
•
Extension of comparison results of Truman & Williams to
SDEs
with
singular potentials.
An example of a swallowtail caustic
and Maxwell set for a Burgers fluid
A trajectory of the stationary state
solution for the SBE with
vorticity
under a Coulomb potential.
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