# 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 stochastic processes and Feller semigroups , in particular jump processes which are generated by non-local pseudo-differential operators . Other topics I am working on are function spaces in probability theory, functional analytic aspects of Markov semigroups (in L p spaces), their 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 .

 ì ì MORE on PROBABILITY

If we restrict ourselves to Feller semigroups and Feller processes we know that A is, in fact, a pseudo-differential operator:

A u ( x )=  - p ( x ,D)=  - (2 p ) - n /2   ò R n   p ( x , x ) û ( x ) e i x × x   d x ,     " u Î S ( R n )

( û indicates the Fourier transform of the function u in the space S ( R n ) of rapidly decreasing functions.) The function p ( x , x ), x , x Î R n , is the so-called symbol of the process and it is given by the Lévy-Khinchine formula (with variable, i.e. x -dependent, "coefficients")

p ( x , x )=i a ( x ) ×x + å 1 £ j,k £ 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 ( · ), ( q jk ( · )) jk , N ( · , dy ) ) is the Lévy triplet (or characteristics ) of  p ( x , x ).

## 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   through its symbol ?

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 Hausdorff dimensions of the range of the paths , their " smoothness " (appropriately measured in terms of function spaces) and their long-term and short-time asymptotics etc. The general philosophy is to capture probabilistic phenomena by ( a priori ) purely deterministic, analytic expressions involving one single function: the symbol. A non-technical survey is given in my joint paper

N. Jacob, R. Schilling: Lévy-type processes and pseudo differential operators . In: O. Barndorff-Nielsen, T. Mikosch and S. Resnick (eds.): Lévy processes: theory and applications , Birkhäuser, Boston 2001, 139 -167  (get it here as .dvi file, .ps file or .pdf file ).

Detailed information can be found in my papers, see my list of publications and recent preprints .

Stand:
Autor: René Schilling