2014/01/25 by Le Chen, Chen, Le, Robert C. Dalang +1 · 1 citation
Computer Science · Economics, Econometrics and Finance · Mathematics · #35R60 #60G60 #60H15 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1401.6506
openalex publication_date 2014/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We exhibit a class of properties of an spde that guarantees existence, uniqueness and bounds on moments of the solution. These moment bounds are expressed in terms of quantities related to the associated deterministic homogeneous p.d.e. With these, we can, for instance, obtain solutions to the stochastic heat equation on the real line for initial data that falls in a certain class of Schwartz distributions, but our main focus is the stochastic wave equation on the real line with irregular initial data. We give bounds on higher moments, and for the hyperbolic Anderson model, explicit formulas for second moments. We establish weak intermittency and obtain sharp bounds on exponential growth indices for certain classes of initial conditions with unbounded support. Finally, we relate Hölder-continuity properties of the stochastic integral part of the solution to the stochastic wave equation to integrability properties of the initial data, obtaining the optimal Hölder exponent.