## Probability Theory & Analysis- Stochastic analysis
- Stochastic Processes (in particular Lévy and jump processes
- Feller semigroups
- Pseudo differential operators
- Subordination (in the sense of S. Bochner) and functional calculi
## MSC 2000 Classification- 60 (E,F,G,H,J), 62 (E,M,P), 28, 31, 35S, 46, 47
My research interests span probability theory,
analysis and the interplay of these fields. A central
topic are subordinates , their
generators, related functional calculi and
ramifications into the theory of Dirichlet forms
. Recently, I have been studying applications of
jump processes and stochastic calculus to
solutions of random Hamiltonian systems in
mathematical physics. Stochastic processes are often used to model random evolutions. They can be as simple as a random walk (imagine a drunkard stumbling through Manhattan) but they are also used to describe physical phenomena like pollen movement (that's how Brownian motion got its name), random clumping of galaxies or, quite popular nowadays, movements of stocks, currencies or interest rates. Through (abstract) semigroup theory ¾ the transition mechanism for stochastic processes is described by sub-Markovian operator semigroups ¾ these probabilistic objects are intimately related with analysis, the most important object being the infinitesimal generator of the semigroup or the process .
If we restrict ourselves to Feller semigroups and Feller processes we know that A is, in fact, a pseudo-differential operator: A ^{ x }
d x ,
" u
Î S
( R ^{ n } ) ( q _{ jk
} ( x ) x _{
j } x _{
k } + ò _{
R n \{0} }
( 1 - e ^{
i y ×
x }
^{ }
- i y ×
x /(1 + |
x | ^{
2 } ) )
N ( x , dy ) . where ( a ( · ), ( ## The two main problems are:- Can we associate to each symbol
*p*(*x*, x ) a (unique) Feller process*X**X*(*t*, w ) : t ³ 0 } ¾ and*vice versa*? - Can we characterize stochastic properties, e.g.,
the behaviour of the trajectories
, of
*X*
Both questions can be answered in the affirmative
and I have been working on these questions for some
time. (Notice that positive answers to both questions
allow us to construct tailor-made stochastic processes
which can be used in real-world applications.) I
looked, in particular, at the sample path
behaviour such as N. Jacob, R. Schilling: Detailed information can be found in my papers, see my list of publications and recent preprints . |