Transformed Path Integral Based Approaches for Stochastic Dynamical Systems: Prediction, Filtering, and Optimal Control

The study of stochastic systems, their characterization, prediction, and control are of great importance to many fields in science and engineering. This often involves obtaining accurate estimates of quantities of interest such as the system state distribution and/or the expected cost in nonlinear dynamical systems subjected to random forces. The computational prediction and control of such systems are often challenging (and involve large computational costs) due to the presence of nonlinearities, model and measurement uncertainties. Novel path integral–based frameworks for efficient solutions to problems in prediction, nonlinear filtering, and optimal control of stochastic dynamical systems are presented to address several key challenges. The presented frameworks are as follows: (1) the transformed path integral (TPI) approach for solution of the Fokker-Planck equation in stochastic dynamical systems with a full rank diffusion coefficient matrix, (2) the generalized transformed path integral (GTPI) approach—a non-trivial extension of the TPI to stochastic dynamical systems with rank deficient diffusion coefficient matrices, (3) the generalized transformed path integral filter (GTPIF) for solution of nonlinear filtering problems, and (4) the generalized transformed path integral control (GTPIC) for solution of a large class of stochastic optimal control problems are presented. The proposed frameworks are based on the underlying short-time propagators and dynamic transformations of the state variables that ensure the appropriate distributions in the transformed space (state distributions in TPI and GTPI; and corresponding conditional distributions in GTPIF and GTPIC) always have zero mean and identity covariance. In systems where the dynamics are linear with respect to the state variables and initial distribution is Gaussian, the appropriate distributions in the transformed space remain invariant with a standard normal distribution as expected. The frameworks thus allow for the underlying distributions necessary for evaluating the quantities of interest to be accurately represented and evolved in a transformed computational domain. Compared to conventional fixed grid approaches and Monte-Carlo simulations, the challenges in dynamical systems with large drift, diffusion, and concentration of PDF can be addressed more efficiently using the proposed frameworks. In addition, straightforward error bounds for the underlying distributions in the transformed space can be established via Chebyshev's inequality

Logarithmic Gradient Transformation and Chaos Expansion of Ito Processes

Since the seminal work of Wiener, the chaos expansion has evolved to a powerful methodology for studying a broad range of stochastic differential equations. Yet its complexity for systems subject to the white noise remains significant. The issue appears due to the fact that the random increments generated by the Brownian motion, result in a growing set of random variables with respect to which the process could be measured. In order to cope with this high dimensionality, we present a novel transformation of stochastic processes driven by the white noise. In particular, we show that under suitable assumptions, the diffusion arising from white noise can be cast into a logarithmic gradient induced by the measure of the process. Through this transformation, the resulting equation describes a stochastic process whose randomness depends only upon the initial condition. Therefore the stochasticity of the transformed system lives in the initial condition and thereby it can be treated conveniently with the chaos expansion tools

Generalized Stochastic Quantization of Yang-Mills Theory

We perform the stochastic quantization of Yang-Mills theory in configuration space and derive the Faddeev-Popov path integral density. Based on a generalization of the stochastic gauge fixing scheme and its geometrical interpretation this result is obtained as the exact equilibrium solution of the associated Fokker--Planck equation. Included in our discussion is the precise range of validity of our approach.Comment: 19 pages, Late

On the Aggregation of Inertial Particles in Random Flows

We describe a criterion for particles suspended in a randomly moving fluid to aggregate. Aggregation occurs when the expectation value of a random variable is negative. This random variable evolves under a stochastic differential equation. We analyse this equation in detail in the limit where the correlation time of the velocity field of the fluid is very short, such that the stochastic differential equation is a Langevin equation.Comment: 16 pages, 2 figure

A Nonlocal Approach to The Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry

The Accardi-Boukas quantum Black-Scholes equation can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus. In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.Comment: 19 page

Stochastic gauge fixing for N = 1 supersymmetric Yang-Mills theory

The gauge fixing procedure for N=1 supersymmetric Yang-Mills theory (SYM) is proposed in the context of the stochastic quantization method (SQM). The stochastic gauge fixing, which was formulated by Zwanziger for Yang-Mills theory, is extended to SYM_4 in the superfield formalism by introducing a chiral and an anti-chiral superfield as the gauge fixing functions. It is shown that SQM with the stochastic gauge fixing reproduces the probability distribution of SYM_4, defined by the Faddeev-Popov prescription, in the equilibrium limit with an appropriate choice of the stochastic gauge fixing functions. We also show that the BRST symmetry of the corresponding stochastic action and the power counting argument in the superfield formalism ensure the renormalizability of SYM_4 in this context.Comment: 35 pages, no figures, published version in Prog. Theor. Phy