Introductory Chapter: Stochastic Processes – Theoretical Advances and Applications in Complex Systems

1. Role of randomness in complex systems

Randomness is ubiquitous in complex systems. Many systems that we encounter in day-to-day life, whether in macroscopic or microscopic scales, exhibit random fluctuations caused by often interwoven internal and external causes. In the theory of biological evolution [1], random variation plays a pivotal role in selection. Dynamic systems are often modeled by deterministic differential equations, and these equations sometimes exhibit bifurcations with the time evolution of a critical parameter. Bifurcation points of the parameter divide the time evolution of the system along different branches [2]. Fluctuations in parameters near critical points can lead systems to bifurcations [3].

The concept of stability is extended to understand the dynamic nature of system behaviors when the parameters are perturbed under internal and external fluctuations. Self-organization of dissipative structures in irreversible reactions is a prime example of order when the variables are not in static states but in dynamic states. These self-organized behaviors are evident in open, far-from-thermodynamic equilibrium systems [4]. Beyond a certain threshold of fluctuations, chaos may exist. Hence to understand complex behaviors under fluctuation regimes, we need mathematical tools, theories, and frameworks. This is where a vast body of literature on stochastic processes based on probability theory can be very useful.

2. Stochastic processes

Mathematics of stochastic processes has been developed over almost 100 years with rigors of characterization of probability theory, and even though most of the stochastic processes are domains of mathematicians, there are many applications in engineering, physics, economics, and biology.

Markov processes are one of the most applied branches of stochastic processes that rely on the assumption (Markov property) that the current state of a system only depends on the previous state. This implies a state transition dynamism and the transition probabilities to guide the transition. The Markov property reduces the evolution of a complex system to a tractable mathematical framework. In other words, the Markov property is memoryless, limiting the memory only to the previous state. But many behaviors of complex systems such as weather/climate systems are based on many past events converging onto the present moment. Therefore, non-Markovian processes also should play a significant role in applications.

It is often important to understand the role of noise in system variables. Theories of stochastic differential equations (SDEs) may be applied in certain systems to explore the impact of noise on dependent variables. To this end, Ito calculus is a significant addition [5] to the theory. Many applications now exist using SDEs as well as Stochastic Partial Differential Equations (SPDEs). SPDEs are used in space-time problems with irregular properties in the medium in which the modeled phenomenon exists. The flow of solutes in porous media is a classic example [6].

Large-scale computations and numerical methodologies developed in recent decades have improved the applicability of SDEs/SPDEs on a more practical level. Indeed, this advancement has demonstrated the relevance of these equations to a wide range of technological disciplines, including reliability and control theory, data-driven computer prediction and design, decision-making, and risk analysis in the environmental and financial sectors. In the presence of inherent and extrinsic noise, SDEs/SPDEs are being utilized as a mathematical tool to describe numerous physical, biological, and economic systems. Modeling uncertainties, inbuilt parameters, and aspects of the applied theory such as open-end boundary conditions are all instances of intrinsic noise. In contrast, extrinsic noise can be caused by environmental factors, geographic disparity, or random user input. Another reason for using SPDEs relates to the need to find a controlled or adjustable solution to complicated physical systems. The advent of noise frequently leads to the emergence of a new phenomenon, both mathematically and phenomenologically.

There are literally hundreds of new advances in stochastic processes during the last several decades. Hence, readers are encouraged to understand the depth and breadth through a thorough literature review of the subject.