Some suggested projects for STT 962 (links are hot). You will
present to the class using the blackboard, or computer if you
prefer.
Project idea #1: Read a
paper and present it to the class. You need not understand every
detail, just tell us the main ideas, and maybe show an interesting
computation or proof from the paper. Here are some suggested
papers (links are hot).
M.M. Meerschaert and P. Straka, Inverse
stable subordinators, Mathematical
Modeling of Natural Phenomena, Vol. 8 (2013), No. 2, pp. 1–16.
Click here to download R codes
used in this paper. See also Remark 2.12 in our textbook.
M.M. Meerschaert, E. Nane and Y. Xiao, Fractal
dimensions for continuous time random walk limits, Statistics
and Probability Letters, Vol. 83 (2013), No. 4, pp. 1083–1093.
M.M. Meerschaert, P. Straka, Y. Zhou, and R.J. McGough, Stochastic solution to a time-fractional
attenuated wave equation, Nonlinear
Dynamics, Vol. 70 (2012), No. 2, pp. 1273–1281.
A. Kumar, M.M. Meerschaert, and P. Vellaisamy, Fractional
Normal Inverse Gaussian Diffusion,
Statistics and Probability
Letters, Vol. 81 (2011)
pp. 146–152.
M.M.
Meerschaert, E. Nane, and P. Vellaisamy,
The fractional Poisson process and the inverse
stable subordinator, Electronic
Journal of Probability, Vol. 16 (2011), Paper no. 59, pp.
1600–1620.
A.
Chakrabarty and M. M. Meerschaert, Tempered
stable laws as random walk limits,
Statistics and Probability
Letters, Vol. 81
(2011), No. 8, pp. 989-997.
S. Cohen, M. M. Meerschaert and J. Rosiński, Modeling
and simulation with operator scaling, Stochastic Processes and Their Applications,
Vol.120 (2010) pp. 2390–2411. Click here
for an extended version with
additional examples and details. Click here to download MAPLE software used to compute the
model parameters, and here for MATLAB
software to implement the sample path simulations.
K. M. Hill, L. DellAngelo, and M. M. Meerschaert, Heavy
tailed travel distance in gravel bed transport: An exploratory enquiry,
Journal of Geophysical Research,
Vol. 115 (2010), p. F00A14, doi:10.1029/2009JF001276. See
especially Appendix A.
B. Baeumer, M. M. Meerschaert and M. Naber,
Stochastic
models for relativistic diffusion, Physical Review E, Vol. 82 (2010),
pp. 011132.
M.M. Meerschaert, E. Nane, P. Vellaisamy, Fractional Cauchy problems on bounded domains, The Annals of Probability, Vol.
37 (2009), No. 3, pp. 979–1007. Click here to read the accompanying "Members' Discovery" article from the July 2009
IMS Bulletin.
B. Baeumer, M.M. Meerschaert and E. Nane, Space-time
duality for fractional diffusion, Journal
of Applied Probability, Vol. 46 (2009), pp. 1100-1115.
M.M. Meerschaert, E. Nane, Y. Xiao, Correlated continuous time random walks, Statistics and Probability Letters,
Vol. 79 (2009), pp. 1194-1202.
B. Baeumer, M.M. Meerschaert, E. Nane, Brownian subordinators and fractional Cauchy problems,
Transactions of the American Mathematical Society, Vol. 361
(2009), No. 7, pp. 3915–3930.
M.M. Meerschaert, S. Stoev, Extremal limit theorems
for observations separated by random power law waiting times, Journal of Statistical
Planning and Inference, Vol.139 (2009), pp. 2175-2188.
M.M. Meerschaert, S.W. Wheatcraft, Fractional
Conservation of Mass, Advances in Water Resources, Vol.
31 (2008), pp. 1377–1381.
B. Baeumer, M. Kovács, M.M. Meerschaert, Fractional reproduction-dispersal
equations and heavy tail dispersal kernels, Bulletin of Mathematical Biology,
Vol. 69 (2007), pp. 2281-2297.
T.J. Kozubowski, M.M. Meerschaert, K. Podgórski, Fractional Laplace Motion, Advances in
Applied Probability, Vol. 38 (2006), No. 2, pp. 451-464.
M.M. Meerschaert, H.P. Scheffler, Stochastic
model for ultraslow diffusion, Stochastic
Processes and Their Applications, Vol. 116 (2006), No. 9, pp.
1215-1235.
B. Baeumer, M.M. Meerschaert, and S. Kurita, Inhomogeneous
fractional diffusion equations, Fractional Calculus and Applied
Analysis, Vol. 8, No 4 (2005), pp. 371-386.
M.M. Meerschaert and C. Tadjeran, Finite
difference approximations for fractional advection-dispersion flow
equations Journal of Computational and Applied Mathematics,
Vol. 172 (2004), No. 1, pp. 65-77.
Project idea #2:
Carefully go step by step through the derivation of the characteristic
function for a stable law in Section 3.2, including the dominated
convergence arguments.
Project idea #3:
Carefully go through the regular variation arguments in the
"details" in Section 4.1. Explain why nP(a_nW>x) converging
to some G(x) implies that a_n is regularly varying and V(x)=P(W>x)
is regularly varying and G(x) is a power law.
Project idea #4: Write an
R simulation to compute and plot solutions to the space and time
fractional diffusion equation. Present to the class and discuss
features of the solution curves. What are their scaling
properties? What is their tail behavior?