Potential Theory and Right Processes (Mathematics and Its Applications)

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Potential Theory of Lévy Processes

Name of contact person: Mgr. Degree Courses Bachelor study BSc. Fields of Study and Study Programmes Mathematics Insurance and Financial Mathematics Applied probability, actuarial mathematics, theory of finance, banking and insurance, accounting, mathematical modelling and computational methods, financial management, risk theory, pensions. Probability, Mathematical Statistics, Econometry Probability: Application of probability theory to problems in natural sciences, technology and economy.

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Hypothesis i of Hunt. Smooth measures and sub-Markovian resolvents. Measure perturbation of sub-Markovian resolvents. Probabilistic interpretations: Positive left additive functionals. Weak duality hypothesis. Natural potential kernels and the Revuz correspondence. Smooth and cosmooth measures. Subordinate resolvents in weak duality. Semi-Dirichlet forms. Weak duality induced by a semi-Dirichlet form. Probabilistic interpretations: Multiplicative functionals in weak duality. Appendix: A. Complements on measure theory, kernels, Choquet boundary and capacity.

Complements on right processes. Cones of potentials and H-cones. Basics on coercive closed bilinear forms. Ta kontakt med Kundesenteret. Avbryt Send e-post. Les mer. Om boka Potential Theory and Right Processes The developmentsin the recent yearsof the potential theoryemphasized a classof functions larger than that of excessive functions i. It turns out that a positive Borel function will be strongly supermedian if and only if it is the in? In the following we shall present a version which is sufficiently general for our purposes.

In the case of probability measures, the converse is also true. The compatibility condition 2 is a direct consequence of Eq. From a rigorous mathematical point of view, the set of transition probabilities is represented by a semigroup of Markov kernels , i. Smoluchowski see, e. Albeverio For a detailed description of the Wiener process see, e. In the following we present some suggestive probabilistic representations of solutions of partial differential equations. For an overview of the scope of functional integration tecniques, in particular when applied to quantum or statistical physics, see, e.

Simon , Kac and Path integral. Kac who was actually inspired by a lecture by R. It turns out see, e. An important tool for the construction of probability measures on vector spaces or, more generally, on locally compact abelian groups, is harmonic analysis.

From this result, one can infer that on an infinite dimensional Hilbert space there cannot exists a Gaussian measure having the identity as covariance operator, as the latter is not trace class. Da Prato The corresponding Gaussian measure is called the Brownian bridge process.

Mathematical Sciences

The computation of concrete Gaussian integrals permits to study the interplay with certain problems in differential and integral equations. A collection of concrete computations of integrals involving Brownian motion and related processes appears in Borodin and Salminen For computations involving other types of infinite dimensional measures associated, e. Important work for applications involves controlling certain transformations on the infinite-dimensional space where the measures have their support.

In fact, this is a special case of a theorem by Feldman and Hajek see, e. The case of a finite number of local non-degenerate minima can be reduced by a smooth partition of unity to a sum of terms of this form, corresponding to expansions around each minimum. In the case of infinitely many local non degenerate minima the same holds, provided the sum over the contributions of the single minima converges. Combet In this case a smooth, not necessarily Gaussian reference measure has been used, see Albeverio and Steblovskaya Other types of dependence on the parameters have been studied by other methods, in relation to the heat equation with potentials or stochastic differential equations.

Application of probabilistic path integrals are numerous and spread out over many different areas, from mathematics and natural sciences to engineering and technical sciences, as well as to socio-economical sciences. First of all, let us mention some applications within mathematics to areas other than stochastics , starting from analysis on Eucliedean spaces or manifolds. We already saw that solutions of some parabolic PDEs, like the heat equation, allow for representations in terms of Wiener integrals. Such representations have also been applied to many problems of differential geometry and related areas, where heat kernels of semigroups associated to second order elliptic or hypoelliptic operators of geometric relevance play an important role.

These extend to much more general context of certain second order differential operators on Riemannian manifolds, where the closed geodesics are replaced by periodic orbits of an underlying classical Hamiltonian system. In this case the formulas are to be understood as asymptotic expansions with respect to a suitable parameter, the expansion being obtained by a Laplace method applied to the infinite dimensional Wiener type integral yielding the probabilistic representation for the semigroup. Schuss and Ikeda and Matsumoto For related applications to homotopy theory see, e.

Probabilistic infinite dimensional integrals are also a powerful tool for providing upper and lower bounds for heat kernels, in the case of manifolds with singularities resp. Related small time, resp.