Ali Süleyman Üstünel

Stochastic Analysis and Related Topics

Presents recent research papers, all related to stochastic analysis, motivated by stochastic partial differential equations, Markov fields, the Malliavin calculus and Feynman path integrals. Topics covered include: super processes; Dirichlet forms; and anticipative stochastic calculus.

Random rotations of the Wiener path

SummaryLet (W, H, μ) be an abstract Wiener space and letR(w) be a strongly measurable random variable with values in the set of isometries onH. Suppose that ∇Rh is smooth in the Sobolev sense and that it is a quasi-nilpotent operator onH for everyh∈H. It is shown that δ(R(w)h) is again a Gaussian (0, |h|H2)-random variable. Consequently, if (ei,i∈ℕ)⊂W* is a complete, orthonormal basis ofH, then

Transformation of the Wiener measure under non-invertible shifts

\tilde w = \sum\nolimits_i {(\delta R(w)e_i )e_i }

Transportation Cost Inequalities for Diffusions Under Uniform Distance

defines a measure preserving transformation, a “rotation”, onW. It is also shown that if for some strongly measurable, operator valued (onH) random variableR, δ(R(w+k)h) is (0, |h|H2)-Gaussian for allk, h∈H, thenR is an isometry and ∇Rh is quasi-nilpotent for allH∈H. The relation between the stochastic calculi for these Wiener pathsw and

The Composition of Wiener Functionals with Non Absolutely Continuous Shifts

\tilde w

Ergodicité des rotations sur l'espace de Wiener

, as well as the conditions of the inverbibility of the map