Sefydliad Gwyddorau Cyfrifiadurol a Mathemategol Cymru
SGCMC
WIMCS
Wales Institute of Mathematical and Computational Sciences
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
Functional Integration and SPDEs
A. Potrykus
A. Neate
Motivations and Aim
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
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
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
• 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.
Outcomes
• Development of a model for the vibration of wind turbines using timeinhomogeneous 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.
2.
3.
4.
Voltage of an energy
harvester driven by a jump
process
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.
A. POTRYKUS, AND S. ADHIKARI, Dynamical response of damped structural
systems driven by jump processes. Probab. Eng. Mech. 25 (2010), no. 3, 305-314.
A. POTRYKUS, Pseudodifferential operators with rough negative definite
symbols. Integr. Equ. Oper. Theory (2010), no. 66, 441--461.
A. POTRYKUS, AND S. ADHIKARI, Response Statistics of Linear Oscillators
With Ito-Levy Noise, submitted.
An example of a swallowtail caustic
and Maxwell set for a Burgers fluid
Selected publications
1.
2.
3.
4.
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 pseudodifferential operators
• Investigation of structural health monitoring devices on bridges where the
vibrations of the bridge are modeled using time-inhomogeneous Markov
processes
5.
Phase portrait of energy
harvester driven by a jump
process
A trajectory of the stationary state
solution for the SBE with vorticity
under a Coulomb potential.
A. NEATE, AND A. TRUMAN, The stochastic Burgers equation with
vorticity: semiclassical asymptotic series solutions with applications,
Submitted to J. Math. Phys. (2010).
A. NEATE, AND A. TRUMAN, Hamilton-Jacobi Theory and the stochastic
elementary formula, SU Preprint (2010).
A. NEATE, AND A. TRUMAN, A Burgers Zeldovich model for the formation
of planetesimals via Nelson’s stochastic mechanics, SU Preprint (2010).
R. DURRAN, A. NEATE, A. TRUMAN, AND (F.Y. WANG) The Divine
clockwork, On the Divine Clockwork, J. Math. Phys. 49, 032102 & 102103
(2008)
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.
Cluster Events
Mathematical Feynman Path Integrals, 18-19 Jan 2010
Stochastic Processes at the Quantum Level, 21-22 Oct 2009
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:
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 control
and coherent control, quantum feedback networks, and quantum statistics and independence. 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)
V. Belavkin (Nottingham), A. Belton (Lancaster), M. Guta (Nottingham), R. Hudson (Loughborough), M. Mirrahimi (SISYPHEINRIA - Rocquencourt), H. Nurdin (ANU – Canberra)
Future Activities Involving PhD Students
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 stochastic analysis 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 meanfield type scaling limit leading to a birth-and-death process in continuum. A spectral gap for the limiting generator. Vlasovtype 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 higher space dimensions, involving a Markov generator 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]
Poster Presenters: Dr A Neate, Dr A Potrykus