Brownian Motion and Diffusion ProcessesLucas Capoia, John Haug, Zhaopeng Liu, Zhe Hu, Professor Jing Wang

Illinois Geometry LabIGL Open House, May 5th, 2016

Background InformationBrownian motion is the random motion of particlessuspended in a fluid (a liquid or a gas) resulting fromtheir collision with the quickly moving atoms ormolecules in the fluid.This transport phenomenon is named after thebotanist Robert Brown.The mathematical model of Brownian motion hasnumerous real-world applications, such as stockmarket analysis.

Definition1-D Brownian Motion

B(0) = 0Bt ∼N (0, t), t ∈R≥0

B(t + ∆t)−B(t) isindependent of B(s) forall s ≤ t ;

2-D Brownian MotionCreated based on 1-DBrownian MotionB(t ,s) = (Bt,Bs)

Bt ∼N (0, t), t ∈R≥0Bs ∼N (0,s), s ∈R≥0

Kolmogorov ProcessA Kolmogorov process is a collection of randomvariables {Kt|t ∈R≥0} such that Kt = (Bt,

∫ t0 Bsds)

where Bt is a one-dimensional Brownian motion.

Figure: Kolmogorov process sample paths

Roto Transition{(V ,θ1)} 7→ {(X ,Y ,θ2)}V ∈R,θ1 ∈ (−π,−π), (X ,Y ) ∈RN,θ2 ∈ (−π,−π)

Give initial state, could find a function whoseparameter is velocity and direction, which representthe process from initial state to ending state at aspecific location and angel.Application: parallel parking

Brownian sheetA 1-D Brownian Sheet is a 2-Parameter, centeredGaussian process B = {B(s, t);s, t ≥ 0}E {B(s, t)B(s′, t ′)} = min(s,s′) ×min(t , t ′)

Our WorkSimulationWe began by creating simulations of Brownianmotion using Python and Matlab. Our simulationsof Brownian motion are quite simple: we start aparticle at the origin, and after each increment oftime, we sample a normal distribution of mean = 0and variance = 1√

total steps, and add it to the

previous location of the particle. Every step isindependent, and we see a true Brownian motionfrom the simulation. We often chose to have 1000steps per simulation, as it is big enough to see itsrandom nature, but small enough to be computedefficiently.

Extended Use of SimulationWe can extend these methods to simulate othertopics. We can create 2-D Brownian motion byhaving two separate and independent simulationsof Brownian Motion on each axis at the sametime. We can observe the Kolmogorov process aswell, where one dimension is standard Brownianmotion, while the other dimension records theintegral of that Brownian motion. We can extendour code further to explore roto-translationalmotion, where a Brownian motion determines aparticle’s movement forward or backward, whileanother random process allows it to turn left orright. This process models the possible motion ofa car.

Gallery

(a) 1-D Brownian Motion (b) 2-D Brownian Motion

(c) Many sample paths of Kolmogorov process (blue) and anoptimal path (red) (d) Sample paths of roto-translation

Results and Further WorkWe have created simulations of 1-D and 2-DBrownian motions, the Kolmogorov process, and thediffusion process on the roto-translation group.After the simulations were completed, we turned ourattention to the study of optimal paths in each of thediffusion processes mentioned above. Using anapproach reminiscent of the Monte Carlo method,we ran our simulations multiple times, fixing theendpoints of our sample paths. By observing wherethe sample paths were concentrated, we couldsuccessfully infer the approximate form of optimalpaths.In our study of optimal paths, we also observedphenomena associated with Hörmander’s condition.That is, when we reduced the time variable in oursimulations, we noticed that the Brownian motionand the diffusion process on the roto-translationgroup had a converging behavior, and the samplepaths became more concentrated around theoptimal path. For the Kolmogorov process, however,constraining the time variable resulted in an entirelynew optimal path.At the moment, we are working to create simulationsof other diffusion processes, such as the Browniansheet and the diffusion process on the Heisenberggroup. Moreover, we plan to review our code andfind ways to improve the computational efficiency ofour simulations, since at the moment they are verycostly and can take hours to run.

ReferencesCAPOGNA, L., DANIELLI, D., PAULS, S., andTYSON, J. An Introduction to the Heisenberg Groupand the Sub-Riemannian Isoperimetric Problem.Birkhäuser Basel, 2007.MÖRTERS, P., and PERES, Y. Brownian Motion[draft], 2008.

These posters are made with the support of University of Illinois at Urbana-Champaign Public Engagement Office

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