9,897 research outputs found
Distributional properties of exponential functionals of Levy processes
We study the distribution of the exponential functional
I(\xi,\eta)=\int_0^{\infty} \exp(\xi_{t-}) \d \eta_t, where and
are independent L\'evy processes. In the general setting using the theories of
Markov processes and Schwartz distributions we prove that the law of this
exponential functional satisfies an integral equation, which generalizes
Proposition 2.1 in Carmona et al "On the distribution and asymptotic results
for exponential functionals of Levy processes". In the special case when
is a Brownian motion with drift we show that this integral equation leads to an
important functional equation for the Mellin transform of , which
proves to be a very useful tool for studying the distributional properties of
this random variable. For general L\'evy process ( being Brownian
motion with drift) we prove that the exponential functional has a smooth
density on , but surprisingly the second derivative at zero
may fail to exist. Under the additional assumption that has some positive
exponential moments we establish an asymptotic behaviour of \p(I(\xi,\eta)>x)
as , and under similar assumptions on the negative exponential
moments of we obtain a precise asympotic expansion of the density of
as . Under further assumptions on the L\'evy process
one is able to prove much stronger results about the density of the
exponential functional and we illustrate some of the ideas and techniques for
the case when has hyper-exponential jumps.Comment: In this version we added a remark after Theorem 1 about extra
conditions required for validity of equation (2.3
Electric double layer effect on observable characteristics of the tunnel current through a bridged electrochemical contact
The Exact Electron Propagator in a Magnetic Field as the Sum over Landau Levels on a Basis of the Dirac Equation Exact Solutions
The exact propagator for an electron in a constant uniform magnetic field as
the sum over Landau levels is obtained by the direct derivation by standard
methods of quantum field theory from exact solutions of the Dirac equation in
the magnetic field. The result can be useful for further development of the
calculation technique of quantum processes in an external active medium,
particularly in the conditions of moderately large field strengths when it is
insufficient to take into account only the ground Landau level contribution.Comment: 9 pages, LaTeX; v2: 3 misprints corrected, a note and 1 reference
added; to appear in Int. J. Mod. Phys.
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