Noise Prevents Singularities in Linear Transport Equations
A stochastic linear transport equation with multiplicative noise is considered and the question of no-blow-up is investigated. The drift is assumed only integrable to a certain power. Opposite to the deterministic case where smooth initial conditions may develop discontinuities, we prove that a certain Sobolev degree of regularity is maintained, which implies H\"older continuity of solutions. The proof is based on a careful analysis of the associated stochastic flow of characteristics.
We study a new technique for optimal data compression subject to conditions of causality and different types of memory. The technique is based on the assumption that some information about compressed data can be obtained from a solution of the associated problem without constraints of causality and memory. This allows us to consider two separate problem related to compression and de- compression subject to those constraints. Their solutions are given and the analysis of the associated errors is provided. Keywords—stochastic signals, optimization problems in signal processing.