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Oct 24, 2013 (3 years and 7 months ago)

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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.