LMS Scheme 3 Meeting on Noncommutative Harmonic Analysis and Probability
School of Mathematics and Statistics, University of Sheffield
Tuesday 28th May 2013
ABSTRACTS
(All lectures to take place in Hicks Building, Lecture Theatre 10)
Pseudo-differential Operators on R^n x Z^m and Corresponding Stochastic
Processes
N.Jacob (Swansea)
Markov chains are stochastic processes with state space Z^m and their generators
(the Q-matrix) have a representation as a pseudo-differential operator acting on
functions defined on Z^m, hence having a symbol defined Z^m x T^m, and this
operator satisfies the positive maximum principle. Of course the analogous result is
well known and much studied for Feller processes with state space R^n. Jointly with
K.Evans and PhD-students I started to look at operators defined on the product
space. The aim is not so much to construct new processes, but to get more
examples to test certain ideas about the geometry behind transition functions.
Stabilization and exponential stability of SPDE with jumps.
C.Yuan (Swansea)
Sufficient conditions under which the perturbed systems decay with general rate
functions are provided and some examples are constructed to demonstrate the
applications of our theory.
Some constructions and approximations of Markov processes using pseudodifferential operators.
A.Potrykus (Swansea)
This talk will cover three result concerning the approximation/construction of Markov
processes using pseudo-differential operators:
1. approximation using the Yosida approximation of the symbol of the generator;
2. construction using 'non-smooth' symbols;
3. Feynman's time slicing approximation.
Semigroups of Measures on Lie Groups
V.Fischer (Imperial)
In 1934, Paul Levy characterised the weakly continuous semi-groups of measures
on Rn. In this talk, we will review similar results, not only by Hunt in 1956 but also
Faraut 1970-72 and Duflo 1978 for instance. We will also present some applications
of this, especially in relation with heat semi-groups associated with sub-Laplacians or
more generally with Rockland operators.
Probabilistic coupling and Nilpotent Diffusions
W.Kendall (Warwick)
Modern probability theory makes considerable use of the technique of probabilistic
coupling. The idea is, to analyse a given random process by constructing two interdependent copies of it, defined on the same probability space, and related in such a
way as to facilitate analysis. Applications include: establishing monotonicity in nonobvious situations, developing quantitative approximations to distributions of random
variables, constructing gradient estimates, and even producing exact simulation
algorithms for Markov chains.
However the thematic question, which has driven much of the theory of probabilistic
coupling, concerns whether or not one can construct the two coupled random
processes so that they almost surely meet ("couple") at some future random time,
and if so then whether one can construct a maximal coupling, for which the random
time is smallest possible? The question is sharpened if we require the coupling to be
co-adapted (also: immersed, or Markovian); this is an additional requirement that the
coupling respect the underlying causal structure of the random processes, and can
be viewed as implying that the coupling is easily constructable in some general
sense.
There is a considerable body of theory describing how to build successful coadapted couplings for elliptic diffusions, all building on the basic reflection coupling
for simple random walks or Brownian motion (very simply, the random jumps of the
coupled process are arranged so far as possible to be the opposites of the random
jumps of the original process). It is conjectured that successful co-adapted couplings
can be built for all hypoelliptic diffusions (diffusions in d dimensions with fewer than d
"directions of randomness").
In this talk I will survey the general theory of coupling, and describe the known
results for co-adapted couplings of hypoelliptic diffusions (in fact, Brownian motions
on nilpotent Lie groups).
The Levy-Khintchine formula on semi-simple Lie groups and
symmetric spaces
Tony Dooley (Bath)
The classical Levy-Khintchine formula calculates the characteristic function (=
Fourier transform) of a Levy process, and is a cornerstone of many applications. In 1964, Gangolli proved a version of the L-K formula for symmetric
spaces G=K, where G is a semi-simple Lie group and K is a maximal compact subgroup, where he calculated the spherical transform of a K-invariant
Levy process.
In recent joint work with David Applebaum, we have extended Gangolli's formula to arbitrary Levy processes on symmetric spaces, without
the restriction of K-invariance. Our formula is obtained from a version of
the L-K formula for the semi-simple group G.
On some operator ideals on compact Lie groups
Julio Delgado (Imperial College)
In this talk we first present a characterisation of invariant operators in Schatten
classes on compact Lie groups. Secondly, we establish sufficient conditions for $r$nuclearity on Lebesgue spaces as well as the corresponding trace formulas and
distribution of eigenvalues.
Gevrey Functions and ultradistributions on compact Lie groups and homogeneous
spaces
Aparajita Dasgupta (Imperial College)
In this talk the global characterisations of Gevrey-Roumieu and GevreyBeurling spaces of ultradifferentiable functions on compact Lie groups in terms of
the representation theory of the group and the spectrum of the Laplace-Beltrami
operator will be presented. Furthermore, the characterisation of their duals, the
spaces of corresponding ultradistributions will be discussed. For the latter, the
proof is based on first obtaining the characterisation of their α-duals in the sense
of K¨the and the theory of sequence spaces. Finally the corresponding characterisations on compact homogeneous spaces will be presented. (This is a joint work
with Professor Michael Ruzhansky)