Let $(X,d_X)$, $(\Omega ,d_{\Omega })$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_p$ the p-Wasserstein metric with some p ∈ [1,∞), and by ${}_p\left(X\right)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f:\Omega {\to }_p\left(X\right)$, $\omega \mapsto f_{\omega }$, be a Borel map such that f is a contraction from $(\Omega ,d_{\Omega })$ into ${(}_p\left(X\right),W_p)$. Let ν₁,ν₂ be probability measures on Ω with $W_p(\nu ₁,\nu ₂)$ finite. On X we consider the subordinated measures ${\mu }_i={\int }_{\Omega }{f}_{\omega }d{\nu }_i\left(\omega \right)$. Then $W_p(\mu ₁,\mu ₂)\le W_p(\nu ₁,\nu ₂)$. As an application we show that the solution measures ${\varrho }_{\alpha }\left(t\right)$ to the partial...

Let WF⁎ be the wave front set with respect to $C^{\infty }$, quasi analyticity or analyticity, and let K be the kernel of a positive operator from $C{₀}^{\infty }$ to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution $u{\ast }_B\phi \left(x\right)=\int u(x-y)\phi \left(y\right)B(x,y)dy$, where $B\in C^{\infty }$ is appropriate, and prove that if $(u{\ast }_B\phi ,\phi )\ge 0$ for every $\phi \in C{₀}^{\infty }$ and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.

2000 Mathematics Subject Classification: Primary: 34L25; secondary: 47A40, 81Q10.In this article we prove that the wave operators describing the direct scattering of the defocusing matrix Zakharov-Shabat system with potentials having distinct nonzero values with the same modulus at ± ∞ exist, are asymptotically complete, and lead to a unitary scattering operator. We also prove that the free Hamiltonian operator is absolutely continuous.

For a class of anisotropic integrodifferential operators $ℬ$ arising as semigroup generators of Markov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equations $ℬ$ u = f on [0,1]n with possibly large n. Under certain conditions on $ℬ$, the scheme is of essentially optimal and dimension independent complexity $𝒪$(h-1| log h |2(n-1)) without corrupting the convergence or smoothness requirements...

This paper examines the pricing of two-asset European options under the Merton model represented by a nonstationary integro-differential equation with two state variables. For its numerical solution, the wavelet-Galerkin method combined with the Crank-Nicolson scheme is used. A drawback of most classical methods is the full structure of discretization matrices. In comparison, the wavelet method enables the approximation of discretization matrices with sparse matrices. Sparsity is essential for the...

It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on $L_1$ is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex $L_1$ such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let τ be an invertible...

2000 Mathematics Subject Classification: 54H25, 47H10.The concept of semi compatibility is given in probabilistic metric space and it has been applied to prove the existence of unique common fixed point of four self-maps with weak compatibility satisfying an implicit relation. At the end we provide examples in support of the result.Authors thank to MPCOST, Bhopal for financial support through the project M-19/2006.