We use commutator techniques and calculations in solvable Lie groups to investigate certain evolution Partial Differential Equations (PDEs for short) that arise in the study of stochastic volatility models, pricing contingent claims on risky assets. In particular we derive the exact kernel of the fundamental solution of a degenerate PDE, which corresponds to a singular small-diffusion limit of ...

We demonstrate that the kernel of the Lippmann-Schwinger equation, associated with interactions consisting of a sum of the Coulomb plus a short range nuclear potential, below threshold becomes degenerate. Taking advantage of this fact, we present a simple method of calculating the effective range function for negative energies. This may be useful in practice since the effective range expansion ...

Recursive algorithms for the linear least mean-squared one-stage prediction, filtering and fixed-point smoothing problems are obtained in distributed parameter systems with uncertain observations, when only the information on the first and second-order moments of the signal and observation noise is available. It is assumed that the variables describing the uncertainty in the observations are no...

The task to construct parametrices of elliptic differential operators on a manifold with edges requires a calculus of operators with a two-component principal symbolic hierarchy, consisting of (edge-degenerate) interior and (operator-valued) edge symbols. This so-called edge-algebra can be interpreted as a generalisation of the (pseudo-differential) algebra of boundary value problems without th...

By means of Riemann boundary value problems and of certain convenient systems of linear algebraic equations, this paper deals with the solvability of a class of singular integral equations with rotations and degenerate kernel within the case of a coefficient vanishing on the unit circle. All the possibilities about the index of the coefficients in the corresponding equations are considered and ...

We devise a general result on the consistency of model-based bootstrap methods for U and V -statistics under easily verifiable conditions. For that purpose we derive the limit distribution of degree-2 degenerate U and V -statistics for weakly dependent Rd-valued random variables first. To this end, only some moment conditions and smoothness assumptions concerning the kernel are required. Based ...

We report an unexpected systematic degeneracy between different multiplets in an inversion symmetric system of two coupled Gaudin models with homogeneous couplings, as occurring for example in the context of solid state quantum information processing. We construct the full degenerate subspace (being of macroscopic dimension), which turns out to lie in the kernel of the commutator between the tw...

We prove existence of a bounded weak solution to degenerate quasilinear subdiffusion problem with measurable coefficients that may explicitly depend on time. The kernel in the involved integro-differential operator w.r.t. time belongs large class PC kernels. In particular, case fractional derivative order less than 1 is included. A key ingredient proof new compactness criterion Aubin-Lions type...

We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for degenerate Langevin dynamics with multiplicative noise, studying longtime behaviour strongly continuous contraction semigroup solving abstract Cauchy problem associated backward Kolmogorov operator. Hypocoercivity constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in correspon...