On Diffusion
Processes, Lévy
Noises, and
Confusion
Cécile Penland
NOAA/Earth Systems
Research Laboratory
How do we model unresolved variability?
•Diffusion approximation?
•Lévy-driven processes?
Better representation of tails?
•Does our experience with
diffusion processes transfer
to (white) Lévy-driven
processes?
The diffusion approximation:
dx/dt = e2 G(x,t) + e F(x,t)
e2 G(x,t) is slow
e F(x,t) is fast
Choose a scaling s = e2t:
dx G(x,s/ e 2) 1 F(x,s/ e 2)
.
e
ds
(*)
For simplicity, say
Fi (x, s/ e 2 )   F k (x, s)k (s/ e 2 ) and
i

k
Ckm    k (t)m (t ' t)  dt '  (T )km

Lim (*) -> dx  G(x,s) ds +  F k (x,s)k  dW
t->∞
e->0
k ,
(W is a Brownian motion; dW  N(0,dt)).
The Good:
• Have a systematic way of handling a lot of
multiscale processes.
• Rigorous connection between dynamics
(PDEs) and probabilistic description.
dx  G(x,s) ds +  F k (x,s)k  dW
k ,
implies a Stratonovich Fokker-Planck eqn.
The Bad:
• Existence of multiple calculi can make
numerical generation difficult.
• The difference between Ito and Stratonovich
integrals is physically-based; a thermometer
will not perform the Ito correction for you.
• Mathematically, it’s an issue of who speaks
first: the continuum limit or the white-noise
limit.
NWS operational GCM (1993 version)
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
The Bad:
• Existence of multiple calculi can make
numerical generation difficult.
• The difference between Ito and Stratonovich
integrals is physically-based; a thermometer
will not perform the Ito correction for you.
• Mathematically, it’s an issue of who speaks
first: the continuum limit or the white-noise
limit.
The Ugly:
• Sometimes, the approximation just doesn’t
hold.
• The white-noise limit and Gaussian limit in the
CLT are more or less taken simultaneously
when that doesn’t always happen in nature.
The new fashion: Lévy noises
P(X>x) ~ x-
02
xs for s>
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Also get Langevin
equations:
dX = f(X)dt+ (X) dL
Why do we care?
 Anomalous diffusion in hydrology (Hurst
1951)
 Paleoclimate models, particularly as
concerns intermittency in the ice core
record (e.g., Ditlevsen 1999)
 Atmospheric turbulence (Viecelli 1998)
Even if we object to some of these models, we
still have to know how to analyze them.

dX = f(X)dt+ (X) dL
Fractional Fokker-Planck equation is derived
in spectral form as the continuum limit of a
finite jump process.
t p(k,t) 

  dh[ikfˆ (h  k) 


ˆ
 (h  k)


| k | ]p(h,t)
Questions for the Mathematicians
 Theoretical justification
 Implementation
 When can we get away with treating a Lévy
process as a system driven by multiplicative
Brownian noise?
 Are there limit theorems for continuous
systems which converge to continuous Lévydriven processes?
 If there are limit theorems converging to Lévydriven processes, what are the requirements
for convergence?
 How forgiving are these limit theorems?
 Do Lévy processes ever approximate the
physical system when the white-noise limit is
pretty good but the Gaussian limit is not?
Under what circumstances?
 Are there classes of Lévy noises for which the
stochastic integral is not unique?
 If so, what does the Lévy equivalent of a noiseinduced drift look like? Does it involve a
fractional derivative? Are there transformations
between calculi?
 Are there really limit theorems that would give
combinations of Lévy and Brownian noises in
the same dynamical equation?
 Is there a recipe like the CLT that scientists can
use to get an approximate stochastic equation?
 Is there such a thing as a Lévy-Taylor
expansion?
Choose a scaling s = e2t:
dx G(x,s/ e 2) 1 F(x,s/ e 2)
.
e
ds
(*)
For simplicity, say
Fi (x, s/ e 2 )   F k (x, s)k (s/ e 2 ) and
i

k
Ckm    k (t)m (t ' t)  dt '  (T )km

Lim (*) -> dx  G(x,s) ds +  F k (x,s)k  dW
t->∞
e->0
k ,
(W is a Brownian motion; dW  N(0,dt)).
 Are there limit theorems that would give
combinations of Lévy and Brownian noises in
the same dynamical equation?
 Is there a recipe like the CLT that scientists can
use to get an approximate stochastic equation?
 Is there such a thing as a Lévy-Taylor
expansion?
 Is there such a thing as an Ito-Lévy-Taylor
expansion?
 What about the numerical generation of Lévy
noises? Do we use ( )1/R to update the noisy
increment?
 Since some models use combinations of Wiener
and Lévy noises, can we use the same
numerical scheme to update both terms?
 How do we handle multiple stochastic integrals
when some of the noises are Brownian and
others are Lévy?
 If we only care about the distribution of the
solution, are there weak and strong numerical
schemes for Lévy-driven processes like there
are for Wiener-driven processes?
 Does the accuracy of these schemes depend on
any particular calculus like it does for Wienerdriven schemes?
What else is out
there?