182 research outputs found
A partial Fourier transform method for a class of hypoelliptic Kolmogorov equations
We consider hypoelliptic Kolmogorov equations in spatial dimensions,
with , where the differential operator in the first spatial
variables featuring in the equation is second-order elliptic, and with respect
to the st spatial variable the equation contains a pure transport term
only and is therefore first-order hyperbolic. If the two differential
operators, in the first and in the st co-ordinate directions, do not
commute, we benefit from hypoelliptic regularization in time, and the solution
for is smooth even for a Dirac initial datum prescribed at . We
study specifically the case where the coefficients depend only on the first
variables. In that case, a Fourier transform in the last variable and standard
central finite difference approximation in the other variables can be applied
for the numerical solution. We prove second-order convergence in the spatial
mesh size for the model hypoelliptic equation subject to
the initial condition , with and , proposed by Kolmogorov, and for an
extension with . We also demonstrate exponential convergence of an
approximation of the inverse Fourier transform based on the trapezium rule.
Lastly, we apply the method to a PDE arising in mathematical finance, which
models the distribution of the hedging error under a mis-specified derivative
pricing model
Wave equations on graded groups and hypoelliptic Gevrey spaces
The overall goal of this dissertation is to investigate certain classical
results from harmonic analysis, replacing the Euclidean setting, the abelian
structure and the elliptic Laplace operator with a non-commutative environment
and hypoelliptic operators.
More specifically, we consider wave equations for hypoelliptic homogeneous
left-invariant operators on graded Lie groups with time-dependent non-negative
propagation speeds that are H\"older-regular or even more so. The corresponding
Euclidean problem has been extensively studied in the `80s and some additional
results have been recently obtained by Garetto and Ruzhansky in the case of a
compact Lie group. We establish sharp well-posedness results in the spirit of
the classical result by Colombini, De Giorgi and Spagnolo. In this
investigation, a structure reminiscent of Gevrey regularity appears, inspiring
deeper investigation of certain classes of functions and a comparison with
Gevrey classes.
In the latter part of this thesis we discuss such Gevrey spaces associated to
the sums of squares of vector fields satisfying the H\"ormander condition on
manifolds. This provides a deeper understanding of the Gevrey hypoellipticity
of sub-Laplacians. It is known that if is a Laplacian on a closed
manifold then the standard Gevrey space on defined in local
coordinates can be characterised by the condition that . The aim in this part
is to discuss the conjecture that a similar characterisation holds true if
is H\"ormander's sum of squares of vector fields, with a
sub-Laplacian version of the Gevrey spaces involving these vector fields only.
We prove this conjecture in one direction, while in the other we show it holds
for sub-Laplacians on and on the Heisenberg group .Comment: 163 pages, dissertation, Imperial College (Oct 2017
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