Stochastic PDEs: Interdependence of Local and Long-term Behaviors, and Representation

Purdue University, West Lafayette IN

Investigators

Abstract

0204999 Viens The PI's research program on the behavior of parabolic stochastic partial differential equations (SPDEs) focuses on four topics: local behavior, long-term behavior, particle representation, and fractional Brownian motion (fBm). The goals are to exhibit a strong interdependence between local and long-term phenomena and to develop an implementable particle representation. The PI considers stochastic perturbations of the heat equation, with additive or multiplicative noise that depends on time and space. The dependence in time is of white-noise type, while the dependence in space can span a wide range of behaviors. The differential operator is the standard Laplacian on the real line or on the circle (or sometimes in higher dimensions), with a small diffusion parameter "kappa" included. In the multiplicative noise case, a nonlinearity may be included. The basic point of view is simple: an SPDE with a unique solution is an input-output system, and as such the solution's behavior must be inherited from the equation's coefficients. The local behavior to be considered is the modulus of continuity of the solution in the space variable. The long term behavior of interest, specific to the linear multiplicative case, is the large-time exponential asymptotics of the solution (Lyapunov "exponent" question). The PI characterizes both of these behaviors via the spatial modulus of continuity of the potential itself, extends some of the results to equations driven by fBm, and designs a new numerical method for the solution using a system of interacting particles. It is shown that the solution's Lyapunov exponent is of the order of kappa to the power of alpha/(1+alpha) where alpha is the almost-sure spatial Holder exponent of the solution. It is also shown that the solution has spatial Holder exponent alpha if and only if the same holds for the potential's antiderivative. An investigation of how these results change when one uses fractional noise instead of white noise is also considered. For the nonlinear equation, a branching and interacting particle system is constructed as the basis for a numerical method for simulating the solution. It is proved that the numerical scheme, when properly mollified, converges to the solution in the Skorohod space of continuous-function-valued "cadlag" processes. The particles interact because, as they branch at evenly spaced short intervals, their mean number of offspring is a random variable whose mean is calculated relative to a certain "fitness" of all the other particles, and depends explicitly on the non-linearity of the potential. The main physical meaning of the research is that a complex space-time-dependent system with significant random perturbations can exhibit an exponentially strong increase of size or energy in the long term, and that the exponential rate of increase can be predicted precisely by looking at the short-range spatial variability of the physical environment. This has potentially important applications in hydrodynamics and turbulence, including a local characterization of the so-called "fast dynamo effect." It would say that a magnetic field in a magnetic fluid can be enhanced precisely by controlling the fluid's mezoscopic level of turbulence. The research has important educational effects. Ph.D. students and, under the PI's supervision, undergraduates and MS students, conduct computer simulations in connection with the particle methods, with an emphasis on designing efficient algorithms. The simulations are being integrated as class projects in a new curriculum emphasis in computational finance and other applications, preparing the students for the technology-dominated job market and workplace. From the purely theoretical point of view, the research brings together several areas of great current interest in probability theory: infinite dimensional stochastic analysis, branching and interacting particle systems and numerical methods, Gaussian regularity theory, Lyapunov exponents.