Local and nonlocal diffusion
Brownian motion and Lévy flights
Jørgen Endal
Department of mathematical sciences
NTNU
13 December 2016
Informal seminar, Institut Mittag-Leffler, Stockholm
J. Endal
Local and nonlocal diffusion
Pollen grains in water
J. Endal
Local and nonlocal diffusion
What is diffusion (from a probabilistic viewpoint)?
J. Endal
Local and nonlocal diffusion
What is diffusion (from a probabilistic viewpoint)?
Heat equation
1
∂t u = 4u
2
A. Einstein. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung
von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik (in German), 322(8):
549–560, 1905.
J. Endal
Local and nonlocal diffusion
What is diffusion (from a probabilistic viewpoint)?
Heat equation
1
∂t u = 4u
2
A. Einstein. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung
von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik (in German), 322(8):
549–560, 1905.
Fractional heat equation
α
∂t u = −(−4) 2 u
with
α ∈ (0, 2)
E. Valdinoci. From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat.
Apl. SeMA, (49):33–44, 2009.
J. Endal
Local and nonlocal diffusion
Ten equivalent definitions of the fractional Laplace operator
Singluar integral definition:
ˆ
α
2
ψ(x+z)−ψ(x)−z·Dψ(x)1|z|≤1
−(−4) ψ := cN,α
|z|>0
J. Endal
Local and nonlocal diffusion
dz
.
|z|N+α
Ten equivalent definitions of the fractional Laplace operator
Singluar integral definition:
ˆ
α
2
ψ(x+z)−ψ(x)−z·Dψ(x)1|z|≤1
−(−4) ψ := cN,α
|z|>0
Fourier definition:
α
F{−(−4) 2 ψ} := −|ξ|α F{ψ}.
J. Endal
Local and nonlocal diffusion
dz
.
|z|N+α
Ten equivalent definitions of the fractional Laplace operator
Singluar integral definition:
ˆ
α
2
ψ(x+z)−ψ(x)−z·Dψ(x)1|z|≤1
−(−4) ψ := cN,α
|z|>0
dz
.
|z|N+α
Fourier definition:
α
F{−(−4) 2 ψ} := −|ξ|α F{ψ}.
Through harmonic extension:

2
2

4x w (x, y ) + α2 cαα y 2− α ∂y2 w (x, y ) = 0 for y > 0,

w (x, 0) = ψ(x),


∂ w (x, 0) =: −(−4) α2 ψ(x).
y
J. Endal
Local and nonlocal diffusion
Ten equivalent definitions of the fractional Laplace operator
Singluar integral definition:
ˆ
α
2
ψ(x+z)−ψ(x)−z·Dψ(x)1|z|≤1
−(−4) ψ := cN,α
|z|>0
dz
.
|z|N+α
Fourier definition:
α
F{−(−4) 2 ψ} := −|ξ|α F{ψ}.
Through harmonic extension:

2
2

4x w (x, y ) + α2 cαα y 2− α ∂y2 w (x, y ) = 0 for y > 0,

w (x, 0) = ψ(x),


∂ w (x, 0) =: −(−4) α2 ψ(x).
y
M. Kwaśnicki. Ten equivalent definitions of the fractional Laplace operator. Preprint, 2015.
J. Endal
Local and nonlocal diffusion
General nonlocal diffusion operator
Consider, for ψ ∈ Cc∞ (RN ),
ˆ
ψ(x + z) − ψ(x) − z · Dψ(x)1|z|≤1 dµ(z)
Lµ [ψ](x) =
|z|>0
where µ ≥ 0 is a Radon measure (“regular” Borel measure)
satisfying
ˆ
min{|z|2 , 1} dµ(z) < ∞.
|z|>0
J. Endal
Local and nonlocal diffusion
Why do we want to study such operators?
Symmetric α−stable densities, β = 0, γ = 1, δ = 0
α = 0.5
α = 1.0
α = 1.5
α= 2.0
0.6
0.5
0.4
0.3
0.2
0.1
0
−5
−4
−3
−2
−1
J. Endal
0
1
2
3
4
5
Local and nonlocal diffusion
Why do we want to study such operators?
(Pictures taken from Wikipedia.)
J. Endal
Local and nonlocal diffusion
Why do we want to study such operators?
Assume b ∈ RN is a given vector, a = (aij )i,j is a nonnegative
definite matrix, and µ ≥ 0 is a Radon measure satisfying
ˆ
min{|z|2 , 1} dµ(z) < ∞.
|z|>0
J. Endal
Local and nonlocal diffusion
Why do we want to study such operators?
Assume b ∈ RN is a given vector, a = (aij )i,j is a nonnegative
definite matrix, and µ ≥ 0 is a Radon measure satisfying
ˆ
min{|z|2 , 1} dµ(z) < ∞.
|z|>0
Then any Lévy process has a generator given by
− b · Dψ(x) +
ˆ
+
d
∂2
1 X
aij
ψ(x)
2
∂xi ∂xj
i,j=1
ψ(x + z) − ψ(x) − z · Dψ(x)1|z|≤1 dµ(z)
|z|>0
J. Endal
Local and nonlocal diffusion
Why do we want to study such operators?
Assume b ∈ RN is a given vector, a = (aij )i,j is a nonnegative
definite matrix, and µ ≥ 0 is a Radon measure satisfying
ˆ
min{|z|2 , 1} dµ(z) < ∞.
|z|>0
Then any Lévy process has a generator given by
− b · Dψ(x) +
ˆ
+
d
∂2
1 X
aij
ψ(x)
2
∂xi ∂xj
i,j=1
ψ(x + z) − ψ(x) − z · Dψ(x)1|z|≤1 dµ(z)
|z|>0
and, conversely, for any (b, a, µ), there exists a Lévy process with
the above generator.
D. Applebaum. Lévy Processes and Stochastic Calculus. Cambridge University Press,
Cambridge, UK, 2009.
J. Endal
Local and nonlocal diffusion
Why do we want to study such operators?
Assume X is a Lévy process uniquely defined by (b, a, µ). Then X
decomposes uniquely as X = W + Y , where W is defined by
(b, a, 0) and Y by (0, 0, µ).
J. Endal
Local and nonlocal diffusion
Why do we want to study such operators?
Assume X is a Lévy process uniquely defined by (b, a, µ). Then X
decomposes uniquely as X = W + Y , where W is defined by
(b, a, 0) and Y by (0, 0, µ).
That is, we can decompose the process X into a local and nonlocal
part, or rather, into a Gaussian process and a purely discontinuous
Lévy process.
J. Endal
Local and nonlocal diffusion
Why do we want to study such operators?
As α → 2− , the distributional solution of
α
∂t u + (−4) 2 u m = 0
converges in C ([0, T ]; L1loc (RN )) to the distributional solution of
∂t u − 4u m = 0.
F. del Teso, JE, and E. R. Jakobsen. Uniqueness and properties of distributional solutions
of nonlocal equations of porous medium type. Adv. Math., 305:78–143, 2017.
J. Endal
Local and nonlocal diffusion
Some surprising nonlocal operators
Recall that
ˆ
Lµ [ψ](x) =
ψ(x + z) − ψ(x) − z · Dψ(x)1|z|≤1 dµ(z).
|z|>0
J. Endal
Local and nonlocal diffusion
Some surprising nonlocal operators
Recall that
ˆ
Lµ [ψ](x) =
ψ(x + z) − ψ(x) − z · Dψ(x)1|z|≤1 dµ(z).
|z|>0
For the choice
µh (z) :=
1
(δh (z) + δ−h (z)) ,
h2
J. Endal
Local and nonlocal diffusion
Some surprising nonlocal operators
Recall that
ˆ
Lµ [ψ](x) =
ψ(x + z) − ψ(x) − z · Dψ(x)1|z|≤1 dµ(z).
|z|>0
For the choice
µh (z) :=
1
(δh (z) + δ−h (z)) ,
h2
we have
Lµh [ψ](x) =
1
(ψ(x + h) + ψ(x − h) − 2ψ(x)) .
h2
J. Endal
Local and nonlocal diffusion
Some surprising nonlocal operators
Recall that
ˆ
µ
L [ψ](x) =
ψ(x + z) − ψ(x) − z · Dψ(x)1|z|≤1 dµ(z).
|z|>0
J. Endal
Local and nonlocal diffusion
Some surprising nonlocal operators
Recall that
ˆ
µ
L [ψ](x) =
ψ(x + z) − ψ(x) − z · Dψ(x)1|z|≤1 dµ(z).
|z|>0
For the choice
µh (z) :=
X
µ(zd + Rh )δzd (z),
α6=0
J. Endal
Local and nonlocal diffusion
Some surprising nonlocal operators
Recall that
ˆ
µ
L [ψ](x) =
ψ(x + z) − ψ(x) − z · Dψ(x)1|z|≤1 dµ(z).
|z|>0
For the choice
µh (z) :=
X
µ(zd + Rh )δzd (z),
α6=0
we have
Lµh [ψ](x) =
X
(ψ(x + zd ) − ψ(x)) µ(zd + Rh ).
α6=0
J. Endal
Local and nonlocal diffusion
Thank you for your attention!
J. Endal
Local and nonlocal diffusion