On the existence of L1(R) solutions to wiener-Hopf type equations

By means of the Fourier transforms of distributions we find necessary and sufficient conditions for the existence of L¹(R) solutions to Wiener-Hopf type integral equations. Thus we establish general criteria for the existence of L¹(R) filters operating on the observed signal to best approximate the true signal. The theorems apply to wide sense stationary stochastic processes.     

A Continuous Super-Brownian Motion in a Super-Brownian Medium

A continuous super-Brownian motion is constructed in which branching occurs only in the presence of catalysts which evolve themselves as a continuous super-Brownian motion . More precisely, the collision local time (in the sense of Barlow et al. ⁽¹⁾) of an underlying Brownian motion path W with the catalytic mass process goerns the branching (in the sense of Dynkin's additive functional approach). In the one-dimensional case, a new type of limit behavior is encountered: The total mass process converges to a limit without loss of expectation mass (persistence) and with a nonzero limiting variance, whereas starting with a Lebesgue measure , stochastic convergence to occurs.