An SDE approximation for stochastic differential delay equations with colored state-dependent noise
Austin McDaniel, Ozer Duman, Giovanni Volpe & Jan Wehr
Markov Processes and Related Fields 22(3), 595-628 (2016)
We consider a general multidimensional stochastic differential delay equation (SDDE) with colored state-dependent noises. We approxi-mate it by a stochastic differential equation (SDE) system and calcu- late its limit as the time delays and the correlation times of the noises go to zero. The main result is proven using a theorem of convergence of stochastic integrals developed by Kurtz and Protter. The result formalizes and extends a method that has been used in the analysis of a noisy electrical circuit with delayed state-dependent noise, and may be further used as a working SDE approximation of an SDDE system modeling a real system, where noises are correlated in time and whose response to stimuli is delayed.
Engineering sensorial delay to control phototaxis and emergent collective behaviors
Mite Mijalkov, Austin McDaniel, Jan Wehr & Giovanni Volpe
Physical Review X 6(1), 011008 (2016)
Collective motions emerging from the interaction of autonomous mobile individuals play a key role in many phenomena, from the growth of bacterial colonies to the coordination of robotic swarms. For these collective behaviors to take hold, the individuals must be able to emit, sense, and react to signals. When dealing with simple organisms and robots, these signals are necessarily very elementary; e.g., a cell might signal its presence by releasing chemicals and a robot by shining light. An additional challenge arises because the motion of the individuals is often noisy; e.g., the orientation of cells can be altered by Brownian motion and that of robots by an uneven terrain. Therefore, the emphasis is on achieving complex and tunable behaviors from simple autonomous agents communicating with each other in robust ways. Here, we show that the delay between sensing and reacting to a signal can determine the individual and collective long-term behavior of autonomous agents whose motion is intrinsically noisy. We experimentally demonstrate that the collective behavior of a group of phototactic robots capable of emitting a radially decaying light field can be tuned from segregation to aggregation and clustering by controlling the delay with which they change their propulsion speed in response to the light intensity they measure. We track this transition to the underlying dynamics of this system, in particular, to the ratio between the robots’ sensorial delay time and the characteristic time of the robots’ random reorientation. Supported by numerics, we discuss how the same mechanism can be applied to control active agents, e.g., airborne drones, moving in a three-dimensional space. Given the simplicity of this mechanism, the engineering of sensorial delay provides a potentially powerful tool to engineer and dynamically tune the behavior of large ensembles of autonomous mobile agents; furthermore, this mechanism might already be at work within living organisms such as chemotactic cells.
The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction
Scott Hottovy, Austin McDaniel, Giovanni Volpe & Jan Wehr
Communications in Mathematical Physics 336(3), 1259—1283 (2015)
We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation and, in particular, the additional drift term that appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein–Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient.
Stratonovich-to-Itô transition in noisy systems with multiplicative feedback
Giuseppe Pesce, Austin McDaniel, Scott Hottovy, Jan Wehr & Giovanni Volpe
Nature Communications 4, 2733 (2013)
Intrinsically noisy mechanisms drive most physical, biological and economic phenomena. Frequently, the system’s state influences the driving noise intensity (multiplicative feedback). These phenomena are often modelled using stochastic differential equations, which can be interpreted according to various conventions (for example, Itô calculus and Stratonovich calculus), leading to qualitatively different solutions. Thus, a stochastic differential equation–convention pair must be determined from the available experimental data before being able to predict the system’s behaviour under new conditions. Here we experimentally demonstrate that the convention for a given system may vary with the operational conditions: we show that a noisy electric circuit shifts from obeying Stratonovich calculus to obeying Itô calculus. We track such a transition to the underlying dynamics of the system and, in particular, to the ratio between the driving noise correlation time and the feedback delay time. We discuss possible implications of our conclusions, supported by numerics, for biology and economics.