Scaling Limits of Continuous Time Random Walks

1. Scaling Limits of Continuous Time Random Walks Boris Baeumer University of Otago New Zealand bbaeumer@maths.otago.ac.nz http://www.maths.otago.ac.nz/?baeumer

2. Today’s Goal • Show how scaling limits can provide robust, parsimonious PDE’s that scale. • Show numerical implementation. • Highlight that in complex systems classical statistics is often meaningless or misleading. • Provide a fundamental tool to adequately model scaling over a range of magnitudes.

3. The classical diffusion model (Random walk, probabilistic) • Particles in a fixed small time interval undergo random motion (jumps). • Jumps are (basically) independent. • If Xi is the random jump vector of the ith jump, position after n jumps is at • Central Limit Theorem implies that the distribution in the scaling limit is Gaussian.

4. Particle Laboratory • 4000 particles start at origin and make random jumps. Observe scaling limit!

5. The Central Limit Theorem in 1D:

6. The advection-dispersion equation • Linear drift, spreads like t1/2. • Solutions decay like exp(-x2) away from the mode. • Level sets of the density are ellipses, main directions are perpendicular.

7. Problem: When is the last time you saw a symmetric plume? • Even though the law is universal, hardly any plumes look like that. • Most data sets show some skewness. • Many data sets show sub/super diffusion. • Many data sets show power law tailing.

8. Example: Solute transport in Groundwater

9. log(c)=a log(x) =>c=xa

10. Systemic Failure of the Classical Model! • Extraordinary events are happening frequently (Some particles move a lot faster than the average). • Model is too simple. • Space/time dependent parameters? • How do I determine a gazillion parameters? • Expensive and still doesn’t work! • Something is fundamentally wrong!

11. Back to the drawing board • How can we get a power-law from the random walk model? • Long-Range dependence? • What happens if I start with a power-law? • There are lots of mechanisms (self-organisation, chaotic dynamical systems, etc.) that can give you power-law jumps (see e.g., D. Sornette, Critical Phenomena in Natural Sciences, Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, 2004).

12. The Central Limit Theorem fails! • We need to have finite second moments!

13. Manifestation of infinite second moments =rand()/rand() • Sample variance always exists but grows with scale/size of sample! • Example: • Diffusion coefficient for classical advection-dispersion equation modelling transport in aquifers is strongly scale dependent. mean VAR

14. Example: Income distribution • Pareto (over 100 years ago): Wealth distribution is a power-law! • Test: Assume you know the combined income of two people (say 1,000,000). What is your best guess about the individual incomes? • If it is not the average, you need to be very careful about using statistical methods! • Use Median instead of mean, Entropy methods, etc.

15. Exercise • Assume Xi independent and What is the density of • Which other attributes except wealth can you think of with this type of behaviour?

16. Frequent Flyer Miles • Internet downloads (in MB) • Number of citations • Fame (number of people that heard of you) • … • Maths doesn’t have to be about people…

18. Tail parameter Skewness parameter The Central Limit Theorem (complete version, Lévy 1934) • Let Xi be i.i.d. random variables, slowly varying with index a≤2 (density decays like x-a-1). Then • L is called a stable random variable and has Fourier transform

19. Fat (power-law) tails

20. Skewness

21. The governing PDE

22. Exercise: • Show that the stable process is scaling; i.e. show that • What is H?

23. Problem: Everything in nature is bounded, therefore we have always finite variance !? Scaling only holds over a certain range • Jump distributions that are power law over several orders of magnitude behave like infinite variance distributions.

24. Fractional and Traditional ADE Compared Fractional ADE with constant coefficients (Benson et al.) versus 2nd-order ADE with v and D variable with time (determined to have best fit for mean and variance)

25. MADE site semilog

26. What are fractional derivatives? • Really old, discovered by co-inventor of calculus (Leibniz) • Work by L’Hopital, Riemann, Liouville, Weyl, Abel … • Really, really modern, Proof:

27. Convolution kernel • The fractional derivative can be expressed via a power law convolution kernel: • The non-local character is apparent.

28. Fourier transform • The fractional derivative is the fractional power of the derivative:

29. Approximation: Grunwald formula

30. The Grunwald Formula • Mass/positivity preserving • Nice numerical properties • Yields Eulerian interpretation (Schumer et al.)

31. Numerical Project: • Solve • Think about boundary conditions! • Different alphas need different sign for D!

32. Operator stables (CLT in Rd) • Different a’s in different directions • Main axes aren’t necessarily perpendicular • Coupling of jump size in different directions matters. • Sample paths are random fractals

34. II. Continuous Time Random Walks • Hydrologic pump test often show power-law breakthrough curves (conc. vs time). 10 3 100 Concentration (ppb) 10 1 Stream tracer data from Haggerty 2002 0.1 0.01 10 10 10 10 3 4 5 6 Time (s)

35. Random jump times ? • So far, we were on a discrete clock, jumps happened at a fixed time interval.

38. Continuous time random walk • Random walk in space-time q(t,n)=“number of jumps by time t ” p(n,x)=“position after n jumps” • q is the inverse of “time to make n jumps”, which is additive process (positively skewed).

39. q(u,t)

40. The governing equation • Compute inverse of completely skewed stable limits in Laplace space and deduce

41. Let q solve and p solve Then Subordinator Classical solution The solution (via subordination)

42. Particle Laboratory • 4000 particles start at origin and make random jumps at random times. Observe scaling limit!

43. Stream tracer test (Haggerty, 2002)

44. Coupled Continuous Time Random Walks • If the jumps size depends on the waiting time… • Scaling limit can be done, messy… • Special case • Subordinator is a scaled beta-distribution

45. Other subordinators • Mobile/Immobile decomposition • Power-law waiting times • Memory function • … Memory function

46. III. Applications to Ecology / Epidemiology • Dispersion is everywhere! • Additional term in the differential equation: • A dispersion operator, g represents a density dependent growth rate. • E.g. (logistic growth)

47. Classical Fisher equation • Goes through hole, not across region of low K. • Linear spread.

48. Dispersal kernels Bullock, J. M. & Clarke, R. T.Long distance seed dispersal by wind: measuring and modelling the tail of the curveOecologia, 2000, 124, 506-521

49. Where is George? Brockmann, D.; Hufnagel, L. & Geisel, T.The scaling laws of human travelNature, 2006, 439, 462

50. Monte-Carlo Simulation • Simulate organism that divides and moves randomly. What kind of jump distribution leads to colonies?