Narrow Results

We study scalar quantum field theory on a compact manifold. The free theory is defined in terms of functional integrals. For positive mass it is shown to have the Markov property in the sense of Nelson. This property is used to establish a reflection positivity result when the manifold has a reflection symmetry. In dimension d=2 we use the Markov property to establish a sewing operation for manifolds with boundary circles. Also in d=2 the Markov property is proved for interacting fields.

For some distinguished properties of semigroups, such as having idempotents, regularity, hopficity, etc., we study the following: whether that property or its negation is a Markov property, whether it is decidable for finitely presented semigroups and for one-relation semigroups and monoids. All the results and open problems are summarized in a table.

We study the optimal martingale transport problem under an additional constraint imposing the underlying process to be Markovian. This formulation results in a modified transportation problem in which the solutions correspond to robust price bounds for exotic derivatives within the class of calibrated martingale models exhibiting the Markov property. We investigate the arising consequences which comprise a dual perspective of the transport problem in terms of liquid replication strategies. Eventually an empirical investigation illustrates the influence of the Markov property on robust price bounds for financial derivatives.

In this note we study the 2D stochastic quasi-geostrophic equation in 𝕋² for general parameter α ∈ (0, 1) and multiplicative noise. We prove the existence of martingale solutions and pathwise uniqueness under some condition in the general case, i.e. for all α ∈ (0, 1). In the subcritical case α > 1/2, we prove existence and uniqueness of (probabilistically) strong solutions and construct a Markov family of solutions. In particular, it is uniquely ergodic for α > 2/3 provided the noise is non-degenerate. In this case, the convergence to the (unique) invariant measure is exponentially fast. In the general case, we prove the existence of Markov selections.

It is shown that if a Quantum Markov semigroup is “multiplicatively” perturbed under suitable conditions, the resulting minimal semigroup remains Markov (or conservative).

We prove the existence and uniqueness of solutions to Wiener–Poisson type multivalued stochastic evolution equations in abstract spaces. We also prove that the solution has the Markov property. Moreover, applications to stochastic ordinary differential equations and stochastic partial differential equations are presented.

Some models of probabilities are described by generalised stochastic equations. These models lead to the resolution of boundary problems for random distributions (generalized equations). We are interested in the equation Lx = f in S ⊂ IR^d where L is a linear operator, f is a random distribution and to the class of boundary conditions on the frontier Γ = ∂S in order to define for the corresponding boundary conditions. The resolution of boundary problems for random distributions lead to the Markov property for the solution of these equations.

Some problems concerning the Markov processes are discussed shortly. Among other things, it is presented an extended Markov property. Also, two variants of the strong Markov property, as they are synthetized by Kiyosi Itô, are discussed.