Traveling waves for reaction-diffusion equations withbistable nonlinearity and nonlocal diffusion

Franz Achleitner Christian Kuhn

Anacapri, September 2015

evolution equation + traveling wave solutions

for u : R× R+ → R, (x , t) 7→ u(x , t),

∂tu = L[u(·, t)](x) + f (u), x ∈ R, t > 0,

where L : C 2b (R)→ Cb(R) and f : R→ R.

example Allen and Cahn (1978) modeling phase transitionsvia excess free energy of interfaces

E [u] =

∫R

12 (∂xu)2 − 1

2 (1− u2)2 dx

u long-range order parameter, x distance normal to surface

∂tu = ∂2xu + u(1− u2), x ∈ R, t > 0.

phase transitions: TWS u(x , t) = U(x − ct) with limξ→±∞ U(ξ) = ±1

bistable reaction-diffusion equation

Aronson and Weinberger (1975) Fife and McLeod (1977)

∂tu = ∂2xu + f (u), x ∈ R, t > 0,

with bistable f ∈ C 1(R) such that f (u±) = 0 and f ′(u±) < 0:

uu− w u+ f (u)

potential F with F ′ = fF (u+) = F (u−) → balanced FF (u−) 6= F (u+) → unbalanced F

→ existence of TWS u(x , t) = U(x − ct) with limξ→±∞ U(ξ) = u±via phase plane analysis of traveling wave equation −cU ′ = U ′′ + f (U)→ sgn c = sgn

(F (u+)− F (u−)

)→ (c = 0 ⇔ balanced F )

→ asymptotic stability of TWS with exponential rate of decay viavariational structure

local reaction-nonlocal diffusion modelBates, Fife, Ren and Wang (1997)

∂tu = J ∗ u − u + f (u) , x ∈ R , t > 0 , (1)

for even non-negative functions J ∈ C 1(R) with∫R

J(y) dy = 1 ,

∫R|y |J(y) dy <∞ , J ′ ∈ L1(R) ,

and bistable function f ∈ C 2(R) with f (u±) = 0, f ′(u±) < 0.

free energy

E [u] = 14

∫R

∫R

J(x − y)(u(x)− u(y))2 dx dy +

∫R

F (u(x)) dx

→ existence of TWS via homotopy to local reaction-diffusion model→ sgn c = sgn

(F (u+)− F (u−)

)→ regularity of TWS under additional assumptions on f→ balanced case: asymptotic stability of TWS with exponential decay rate

Chen: abstract evolution equation

∂tu(x , t) = A[u(·, t)](x) , x ∈ R , t > 0 ,

nonlinear operator A

� A is independent of t

� A generates Cb semigroup

� A is translational invariant

∀u ∈ domA ∀h, x ∈ R A[u(·+ h)](x) = A[u(·)](x + h)

⇒ ∃f : R→ R : A[α1] = f (α)1 ∀α ∈ R

� function f ∈ C 1(R) is bistable

� A satisfies comparison principle

If ∂tu ≥ A[u], ∂tv ≤ A[v ] and u(·, 0) v(·, 0),then u(·, t) > v(·, t) for all t > 0.

Chen’s class of evolution equations

∂tu = D∂2xu + G (u, J1 ∗ S1(u), . . . , Jn ∗ Sn(u)) (2)

D1 f (u) := G (u, u, . . . , u) is bistable, s.t. f ∈ C 1(R) and w ∈ (0, 1)

uu− w u+ f (u)

f (u)

{> 0 in (−1, 0) ∪ (w , 1) ,

< 0 in (0,w) ∪ (1, 2) ,

f ′(0) < 0 , f ′(1) < 0 , f ′(w) > 0 .

D2 all Ji ∈ C 1 satisfies Ji ≥ 0, ‖Ji‖L1 = 1 and∫R |J

′i (y)| dy <∞

D3 G = G (u, p) and S j(u) are smooth functions with

∂pi G (u, p) ≥ 0 , ∂uS j(u) ≥ 0 ∀u , pi ∈ [−1, 2]

D4 Either D > 0OR ∂uG (u, p) < 0 and ∂p1G (u, p)∂uS1(u) > 0 on [−1, 2]n+1

Chen’s results

→ existence of TWS with monotone increasing C 1 profile U

Take a smoothed Heaviside function as initial datum u0, let itevolve according to (2), and show that the solution u(·+ ξ(t), t)with u(ξ(t), t) = 1/2 approaches the profile U of a travelingwave in the limit t →∞.

→ uniqueness of smooth TWS up to translations→ global exponential stability of TWS

If u0 ∈ Cb(R) such that

0 ≤ u0 ≤ 1, lim infx→∞

u0(x) > w , lim supx→−∞

u0(x) < w ,

then the solution of (2) with u(·, 0) = u0 satisfies

‖u(·, t)− U(· − ct + ξ)‖L∞(R) ≤ Ke−κt ∀t ≥ 0

for some constants ξ and K depending on u0.

normal vs. anomalous diffusion

Brownian motion vs. 32 -stable Levy process

mean square displacementsubdiffusion 〈x2(t)〉 = d tγ with 0 < γ < 1

normal diffusion 〈x2(t)〉 = d tsuperdiffusion 〈x2(t)〉 = d tγ with 1 < γ

for some diffusion coefficient d > 0

references: Bouchaud and Georges, Metzler and Klafter, etc.

continuous time random walks (CTRW)

jump probability distribution function (PDF) ψ(x , t)

length of a given jump ∼ w(x) =∫∞

0 ψ(x , t) dt

waiting time between jumps ∼ φ(t) =∫R ψ(x , t) dx

PDF p(t, x) of particles following CTRW satisfies

p(x , t) = δ(x)κ(t) +

∫ ∞0

∫ ∞−∞

ψ(x − y , t − s) p(y , s) dy ds

with survival PDF κ(t) = 1−∫ t

0 φ(s) ds .

CTRW ↔ diffusion equationConsider CTRW with independent jump length and waiting time PDFs

ψ(x , t) = w(x) φ(t) with σ2 :=∞∫−∞

x2 w(x) dx and T :=∞∫0

t φ(t) dt .

Study large-scale, long-time limit of CTRW

� normal diffusion: If σ2,T <∞, then ∂tp = d∂2xp.

� superdiffusion: If T <∞ and jump length PDF is heavy tailed withw(x) ∼ |x |−1−a for some a ∈ (1, 2) as |x | → ∞, then

∂tp = d Daθ[p] with Riesz-Feller operator Da

θ .

� subdiffusion: If σ2 <∞ and waiting time PDF is heavy tailed withφ(t) ∼ t−1−γ for some 0 < γ < 1 as t →∞, then ∂tp = d D1−γ

t ∂2xp

with Riemann-Liouville fractional derivative

D1−γt = 1

Γ(γ)∂t

∫ t

0

f (s)(t−s)1−γ ds

for 0 < γ < 1. → subdiffusion is non-Markovian. → great care toderive reaction-subdiffusion equations.

fractional diffusion equation

∂tu = Daθu , x ∈ R , t > 0 ,

for some fixed parameters 0 < a ≤ 2 and |θ| ≤ min(a, 2− a).

Riesz-Feller operator1 (FDaθ f )(ξ) = −ψ(−ξ)(F f )(ξ)

Id

−Id

∂x

−∂x

∂2x

∂3x

a

θ

2

1

0

−13210

ψ(ξ) = |ξ|a exp(i sgn(ξ)θ π2

)real-valued parameters

a index of stability0 < a ≤ 2

θ skewness parameter|θ| ≤ min(a, 2− a)

1Mainardi, Luchko and Pagnini (2001)

fractional diffusion equation

∂tu = Daθu , x ∈ R , t > 0 ,

for some fixed parameters 1 < a < 2 and |θ| ≤ min(a, 2− a).

strongly continuous, convolution semigroup

Tt : Lp(R)→ Lp(R) , u0 7→ Ttu0 = u(t, x) = G aθ (t, ·) ∗ u0 ,

with 1 ≤ p <∞ and kernel G aθ (t, x) = F−1[exp(−ψ(·)t)](x).

G aθ (t, x) is a Levy strictly stable distribution on R

LC H N

a

θ

2

1

0

−13210

L Levy-Smirnov PDF x−3/2

2√π

exp(− 1

4x

), x > 0.

C Cauchy(-Lorentz) PDF 1π

11+x2

H Holtsmark

N Normal (Gaussian) PDF 1√2πσ2

exp(− x2

2σ2

)

local reaction-nonlocal diffusion equation + TW problem

∂tu = Daθu + f (u) , x ∈ R , t > 0 , (3)

for 0 < a ≤ 2, |θ| ≤ min{a, 2− a} and f ∈ C (R).

Definition

A traveling wave solution of (6) is a solution of the form u(t, x) = U(ξ),for some wave speed c ∈ R, a traveling wave variable ξ := x − ct, and afunction U connecting different endstates limξ→±∞ U(ξ) = u±.

traveling wave equation (TWE) −cU ′ = DaθU + f (U) for ξ ∈ R.

How to make sense of DaθU? If 1 < a < 2, f ∈ S(R) and x ∈ R, then

Daθ f (x) = c1

∫ ∞0

f (x+ξ)−f (x)−f ′(x) ξξ1+a dξ + c2

∫ ∞0

f (x−ξ)−f (x)+f ′(x) ξξ1+a dξ

for c1, c2 ≥ 0 and c1 + c2 > 0.

fractional Allen-Cahn equation - balanced case

Theorem (Cabre and Sire (2011))

Let a ∈ (0, 2), f ∈ C 1,γ(R) with γ > max(0, 1− a) and F ′ = −f . Then,

0 = Da0u + f (u) in R (4)

has a solution u with u′ > 0 in R and limx→±∞ u(x) = u± if and only if

f (u±) = 0 and F (u) > F (u−) = F (u+) in u ∈ (u−, u+) .

If—in addition—f ′(u±) < 0 then the solution is unique up to translations.

problem equivalent to nonlinear BVP{div(yα∇v) = 0 in Rn+1

+ = { (x , y) ∈ Rn × R | y > 0 } ,(1 + α)∂ναv = f (v) in ∂Rn+1

+ ,

where α = 1− a, v = v(x , y) is real-valued and ∂ναv = − limy→0 yα∂yv .

Theorem (Palatucci, Savin and Valdinoci (2012))

Let a ∈ (0, 2), F ∈ C 1(R) with F ′ = f . If f is bistable and balanced, thena global minimizer u of E [u,R] = ‖u‖H2

a/2(R) +

∫R F (u(x)) dx exists.

Moreover, u is a solution of (4) and is unique up to translations in theclass of monotone solutions of (4).

fractional Allen-Cahn equation - balanced case

Theorem (Cabre and Sire (2011))

Let a ∈ (0, 2), f ∈ C 1,γ(R) with γ > max(0, 1− a) and F ′ = −f . Then,

0 = Da0u + f (u) in R (4)

has a solution u with u′ > 0 in R and limx→±∞ u(x) = u± if and only if

f (u±) = 0 and F (u) > F (u−) = F (u+) in u ∈ (u−, u+) .

If—in addition—f ′(u±) < 0 then the solution is unique up to translations.

Theorem (Palatucci, Savin and Valdinoci (2012))

Let a ∈ (0, 2), F ∈ C 1(R) with F ′ = f . If f is bistable and balanced, thena global minimizer u of E [u,R] = ‖u‖H2

a/2(R) +

∫R F (u(x)) dx exists.

Moreover, u is a solution of (4) and is unique up to translations in theclass of monotone solutions of (4).

fractional Allen-Cahn equation - unbalanced case

∂tu = Da0u + f (u), x ∈ R, t > 0, (5)

for 0 < a ≤ 2, θ = 0 and bistable f ∈ C 1(R).

Theorem (Chmaj (2013))

Let a ∈ (0, 2) and f be bistable. ∃ TWS u(t, x) = U(x − ct) of (5) withsgn c = sgn

∫ u+

u−f (v) dv such that limx→±∞ U(x) = u± and U ′ > 0.

� TWE cU ′ = Da0U + f (U);

� Approximate Jε ∗ U − (∫

Jε)Uε→0−→ Da

0U;� TWE cU ′ = Jε ∗ U − (

∫Jε)U + f (U) ∀ε ∃!(Uε, cε) with U ′ε > 0

� U0 = limε→0 Uε is a TWS of (5).

Result (Gui and Zhao (2014))

(unbalanced) double well potential F ∈ C 2,γ(R) with F ′ = f→ existence of unique TWS u ∈ C 2(R) via homotopy to balanced case;→ asymptotic behavior of front tails.

local reaction-nonlocal diffusion equation - Cauchy problem

∂tu = Daθu + f (u) , x ∈ R , t > 0 , (6)

for 1 < a ≤ 2, |θ| ≤ min{a, 2− a}, and bistable f ∈ C 1(R) such that

f (0) = f (1) = 0 , f ′(0) < 0 , f ′(1) < 0 . (7)

Theorem (FA and Kuhn (2013))

Suppose 1 < a ≤ 2, |θ| ≤ min{a, 2− a} and f ∈ C∞(R) satisfies (7). TheCauchy problem (6) with initial condition u(0, ·) = u0 ∈ Cb(R) and0 ≤ u0 ≤ 1 has a solution u(t, x) in the following sense: for all T > 0

� u ∈ Cb((0,T )× R) and u ∈ C∞b ((t0,T )× R) for all t0 ∈ (0,T );

� u satisfies (6) on (0,T )× R;

� u(t, ·)→ u0 locally uniformly on R as t → 0;

� 0 ≤ u(t, x) ≤ 1 for all (t, x) ∈ (0,∞)× R;

� ∀k ∈ N ∀t0 > 0 ∃C > 0 such that ‖u(t, ·)‖C kb (R) ≤ C ∀0 < t0 < t.

sketch of proof

� comparison principle for classical solutions

� mild formulation of CP

u(t, x) =(

G aθ (t, ·) ∗ u0

)(x) +

∫ t

0

[G aθ (t − τ, ·) ∗ f (u(τ, ·))

](x) dτ

� consider f ∈ C∞b (R) with f (u) = f (u) for all u ∈ [0, 1]⇒ exists a unique mild solution u to Cauchy problem (CP){

∂tu = Daθu + f (u) for (t, x) ∈ (0,T ]× R ,

u(0, ·) = u0 for x ∈ R .(8)

⇒ u is a (unique) mild solution to the original CP

� CP generates a nonlinear semigroup of mild solutions, whereG aθ (t, ·) ∈W∞,1(Rx) for all t > 0.⇒ uniform C k

b estimates follow from bootstrap argument.

⇒ u is a classical solution

local reaction-nonlocal diffusion equation - TW problem

∂tu = Daθu + f (u) , x ∈ R , t > 0 , (9)

for 1 < a ≤ 2, |θ| ≤ min{a, 2− a}, and bistable f ∈ C 1(R).

Theorem (FA and Kuhn (2013))

→ existence and uniqueness of monotone traveling wave solution→ asymptotic stability of TWS u(x , t) = U(x − ct)→ wave speed c is bounded

via Chen’s approach: nonlinear operator A[u] = Daθ [u(t, ·)](x) + f (u)

� A is independent of t

� A generates Cb semigroup

� A is translational invariant

∀u ∈ domA ∀h, x ∈ R A[u(·+ h)](x) = A[u(·)](x + h)

� f : R→ R, u 7→ f (u) = A[u1](0), is bistable;

� A satisfies comparison principle

Chen’s assumptions

C1 f ∈ C 1(R) is bistable with u− = 0 and u+ = 1

C2 ∃ a positive function η(x , t) in C([0,∞)× (0,∞)

)such that if

u, v ∈ C (R× (0,∞)) satisfy −1 ≤ u, v ≤ 2, ∂tu ≥ A[u], ∂tv ≤ A[v ],and u(·, 0) ≥ v(·, 0), then

u(x , t)−v(x , t) ≥ η(|x |, t)

∫ 1

0u(y , 0)− v(y , 0) dy ∀x ∈ R , t > 0 .

C3 ∃ positive constants K1, K2, K3, and a probability measure ν suchthat for any u, v ∈ C 2

b (R) with −1 ≤ u, v ≤ 2∣∣∣∣A[u+v ](x)−A[u](x)

∣∣∣∣ ≤ K1

∫R|v(x−y)|ν(dy)+K2‖∂2

xv‖C0([x−1,x+1])

for x ∈ R and similar estimates on Frechet derivative A′[·](·) of A.

C4 If u0 ∈ C 3b (R) with 0 ≤ u0 ≤ 1, then the solution u of CP (9) with

u(·, 0) = u0(·) satisfies supt∈[0,∞) ‖u(·, t)‖C2b (R) <∞.

Theorem (uniqueness)

Suppose (C1) holds and (U, c) is a TWS of (9) satisfying

U ∈ C 1(R), limξ→±∞

U(ξ) = u±, U ′(ξ) > 0 on R, limξ→±∞

U ′(ξ) = 0. (10)

If (U, c) is a TWS of (9) with U ∈ C (R), limξ→±∞ U(ξ) = u± and

u− ≤ U ≤ u+ on R, then c = c and U(·) = U(·+ ξ0) for some ξ0 ∈ R.

Lemma (construction of sub- and supersolutions)

Suppose (U, c) is a TWS of (9) satisfying (10). Define functions

w±(x , t) := U(x − ct + ξ0 ± σ∗δ[1− exp(−βt)]

)± δ exp(−βt) (11)

where β := 12 min{−f ′(u±)}, ξ0 ∈ R and δ, σ∗ > 0. Then,

∃δ∗ > 0 ∃σ∗ > 0 : ∀δ ∈ (0, δ∗] ∀ξ0 ∈ R

{w + is a supersolution of (9)

w− is a subsolution of (9)

remark: δ∗ is independent of U.

proof of uniqueness

step 1 Since U(ξ) and U(ξ) have the same limits as ξ → ±∞,

∃ξ1 ∈ R ∃h� 1 : U(·+ ξ1)− δ∗ < U(·) < U(·+ ξ1 + h) + δ∗ on R .

using comparison principle and sub- and supersolutions

U(x − ct − σ∗δ∗[1− exp(−βt)]

)− δ∗ exp(−βt)

< U(x − ct) < U(x − ct + h + σ∗δ∗[1− exp(−βt)]

)+ δ∗ exp(−βt)

keeping ξ := x − ct fixed, sending t →∞, yields c = c and

U(ξ − σ∗δ∗) ≤ U(ξ) ≤ U(ξ + h + σ∗δ∗) ∀ξ ∈ R .

step 2 The shifts ξ∗ := inf{ ξ ∈ R | U(·) ≤ U(·+ ξ) } ≥ −σ∗δ∗ andξ∗ := sup{ ξ ∈ R | U(·) ≥ U(·+ ξ) } ≤ h + σ∗δ∗ satisfy ξ∗ ≤ ξ∗.In fact ξ∗ = ξ∗, otherwise a contradiction to strict comparison principle.

stability

Theorem

Suppose (C1)–(C4) hold, (U, c) is a TWS of (9) satisfying (10), and

0 < δ ≤ min{

min{1, 1/σ∗} δ∗2 ,w2 ,

1−w2

},

for some σ∗, δ∗ > 0. Then, ∃κ > 0 such that for all u0 ∈ Cb(R) satisfying

0 ≤ u0 ≤ 1, lim infx→+∞

u0(x) > 1− δ > w , lim supx→−∞

u0(x) < δ < w ,

the solution u(x , t) of CP (9) with u(·, 0) = u0 satisfies

‖u(·, t)− U(· − ct + ξ)‖L∞(R) ≤ Ke−κt ∀t ≥ 0 ,

where ξ and K are constants depending on u0.

proof of stabilitymathematical induction on k ∈ N0 shows

U(x − cτ + ξ)− δ ≤ u(x , τ) ≤ U(x − cτ + ξ + h) + δ ∀x ∈ R , (12)

for some ξ = ξk ∈ R, T k := T1 + kt∗, δk := (1− κ∗)kδ∗, hk := (1− κ∗)k .

⇒ (12) holds ∀k ∈ N0 ∀τ ≥ T k , δ = δk , h = hk + 2σ∗δk , ξ = ξk − σ∗δk .

h = h(t)

δ = δ(t)

δ

δ∗

δ1

δ2

δk

h

hh1

h2

hk

tT 1 T 2 T 3 T k T k+1

proof of stability

� To deduce the (best) estimate for t ≥ T1, define

δ(t) := δk , ξ(t) := ξk − σ∗δk , h(t) := hk + 2σ∗δk

on each interval t ∈ [T k ,T k+1) for all k ∈ N0.

� T k ≤ t < T k+1 is equivalent to k ≤ t−T1t∗ < k + 1.

δ(t) ≤ δ∗ exp{(

t−T1t∗ − 1

)ln(1− κ∗)

},

h(t) ≤ (1 + 2σ∗δ∗) exp{(

t−T1t∗ − 1

)ln(1− κ∗)

}for all t ≥ T1 where ln(1− κ∗) < 0.

� ξ(t) ∈ [ξ(τ)− σ∗δ(τ) , ξ(τ) + h(τ) + σ∗δ(τ)] for any t ≥ τ ≥ T1.

⇒ ξ∞ = limt→∞ ξ(t) exists

� stability result follows with κ = − ln(1−κ∗)t∗ .

existence

Theorem

Suppose (C1)-(C4) hold. There exists a TWS (U, c) of (9) satisfying

U ∈ C 1(R), limξ→±∞

U(ξ) = u±, U ′(ξ) > 0 on R, limξ→±∞

U ′(ξ) = 0.

idea of proof Consider

∂tv = A[v ] in R× (0,∞), v(·, 0) = ζ(·) in R, (13)

for some function ζ ∈ C∞(R) with

ζ ≡ 0 if s ≤ 0, 0 < ζ ′ < 1, |ζ ′′| ≤ 1 if s ∈ (0, 4), ζ ≡ 1 if s ≥ 4,

Show that, for some diverging sequence (tj)j∈N with tj →∞, the sequence(v(·+ ξ(tj), tj))j∈N—where v(ξ(t), t) = w for all t ≥ 0—has a pointwiselimit U(·) which is the profile of a TWS of (9).

proof of U being profile of a TWS of (9)

Consider {∂tU = A[U] in R× (0,∞) ,

U(·, 0) = U(·) in R .

First, limj→∞ v(ξ + z(a, tj), tj + t) = U(ξ, t) for all (ξ, t) ∈ R× (0,∞).

⇒ U(ξ −m0 − σ2δ0, 1) ≤ U(ξ) ≤ U(ξ + m0 + σ2δ0, 1) for all ξ ∈ R .

The shifts

{ξ∗ := sup{ ξ ∈ R | U(·+ ξ, 1) ≤ U(·) }ξ∗ := inf{ ξ ∈ R |U(·) ≤ U(·+ ξ, 1) }

satisfy −m0 − σ2δ0 ≤ ξ∗ ≤ ξ∗ ≤ m0 + σ2δ0. In fact ξ∗ = ξ∗.

Comparing U(·, t) with U(·) for t ∈ (1, 2]⇒ ∃c ∈ C ([1, 2];R) with c(1) = ξ∗ = ξ∗ such that U(·, t) = U(· − c(t)).

∂tU = A[U] implies −c ′(t)U ′(ξ) = A[U](ξ). ⇒ c ′(t) is constant for all t⇒ (U, c) is a TWS of (9).

conclusion

evolution equation for u : R+ × R→ R, (t, x) 7→ u(t, x),

∂tu = L[u(t, ·)](x) + f (u), x ∈ R, t > 0,

where L : C 2b (R)→ Cb(R) and f : R→ R.

Traveling wave problem for bistable f

2� L = Daθ with 1 < a < 2 and |θ| ≤ 2− a

2 L = Daθ with 0 < a ≤ 1 and |θ| ≤ a

references:

� F. Achleitner, C. Kuehn: Analysis and numerics of traveling waves forasymmetric fractional reaction-diffusion equations, Commun. Appl. Ind. Math. 6,No. 2 (2015).

� F. Achleitner, C. Kuehn: Traveling waves for a bistable equation withnonlocal diffusion, Adv. Differential Equations 20, No. 9-10 (2015), 887–936.

Thank you for your attention.

infinitely divisible probability measure

Definition

A probability measure µ on Rd is infinitely divisible if, for any n ∈ N, thereis a probability measure µn on Rd such that µ = µn ∗ . . . ∗ µn︸ ︷︷ ︸

=n−times

.

prob. measure µ is determined by its characteristic function

F [µ](ξ) = exp(ψ(ξ)) ;

F [µa ∗ µb] = F [µa]F [µb] for prob. measures µa, µb;

prob. measure µ is infinitely divisible ⇔

∀n ∈ N ∃ prob. measure µn : F [µ] =(F [µn]

)n.

Theorem

1 If µ is an infinitely divisible distribution on Rd , then

F [µ](ξ) = exp

[− (σξ) · ξ + i b · ξ

+

∫Rd

[exp(i z · ξ)− 1− i(z · ξ) h(z)

]ν( dz)

], ξ ∈ Rd , (14)

where σ is a symmetric positive semi-definite d × d matrix, b ∈ Rd ,ν is a measure on Rd satisfying

ν({0}) = 0 and

∫Rd

min(1 , |x |2) ν( dx) <∞ . (15)

2 The representation of F [µ](ξ) in (1) by σ, ν, and b is unique.

3 Conversely, if σ is a symmetric positive semi-definite d ×d matrix, ν isa measure satisfying (15), and b ∈ Rd , then there exists an infinitelydivisible distribution µ whose characteristic function is given by (14).

Levy operator LLevy operator L defined as Fourier multiplier operator

F [L[u]](ξ) = ψ(ξ)F [u](ξ)

with symbol ψ(ξ); its Levy-Khinchine representation w.r.t. h(x) = 11+|x |2

ψ(ξ) = −(σξ) · ξ + i b · ξ +

∫Rd

[exp(i z · ξ)− 1− i(z · ξ) h(z)

]ν( dz)

σ ∈ Rd×d symmetric positive semi-definite matrixb ∈ Rd

ν is a measure on (Rd ,B(Rd)) satisfying

ν({0}) = 0 and

∫Rd

min(1 , |x |2) ν( dx) <∞

examplesoperator symbol Levy triplet (σ, b, ν)∆x −|ξ|2 (Id×d , 0, 0)

−(−∆x)α/2 −|ξ|α (0d×d , 0, ν) with ν( dx) = c |x |−d−α dx

Levy operators: integration by parts

For a Levy operator L with Fourier symbol ψ and Levy triplet (σ, b, ν) andsuff. smooth and integrable functions u and v∫

Rd

u L[v ] dx =

∫Rd

F [u](ξ)ψ(ξ)F [v ] dξ

=

∫Rd

ψ(ξ)F [u](ξ)F [v ] dξ =

∫Rd

L[u] v dx

where L is a Levy operator with Fourier symbol ψ(ξ) and Levy triplet(σ,−b, ν) with ν( dx) = ν(− dx).

Levy strictly α-stable distributions on R

A random variable is said to be strictly stable (or to have a strictly stabledistribution), if it has the property that linear combinations of twoindependent copies of the variable have the same distribution, up to ascaling.

random variable X with Levy strictly stable distribution

characteristic function E [exp(iξX )] = exp(ψ(ξ))

characteristic exponent ψ(ξ) := −c0|ξ|α exp(− i sgn(ξ)θαπ2

)parameters

α index of stability 0 < α ≤ 2

θ skewness parameter |θ| ≤ min(

2−αα , 1

)c0 scaling parameter c0 > 0

Levy α-stable distributions on RA random variable is said to be stable (or to have a stable distribution), ifit has the property that linear combinations of two independent copies ofthe variable have the same distribution, up to location and scaleparameters.

random variable X with Levy stable distribution

characteristic function E [exp(iξX )] = exp(ψ(ξ))

characteristic exponent

ψ(ξ) =

{−c |ξ|α

(1− iβ(sgn ξ) tan απ

2

)+ iτξ for α 6= 1 ,

−c |ξ|(1− iβ 2

π (sgn ξ) log |ξ|)

+ iτξ for α = 1 .

parameters

α index of stability 0 < α ≤ 2

β skewness parameter −1 ≤ β ≤ 1

c scaling parameter 0 < c

τ location parameter τ ∈ R

Fourier multiplier operator (FDaθ f )(ξ) = −ψ(−ξ)(F f )(ξ)

Id

−H

−Id

H

∂x

−∂xH

−∂x

∂xDαu

∂2x

−H∂2x ∂3

x

1− α

1 + αa

θ

2

1

0

−13210

Riesz-Feller operator Daθ

ψ(ξ) = |ξ|a exp(i sgn(ξ)θ

π

2

)real-valued parameters

a index of stability0 < a ≤ 2

θ skewness parameter|θ| ≤ min(a, 2− a)

Hilbert transform

Hf :=1

πPV

∫ ∞−∞

f (y)

x − ydy

Levy operator

Theorem (Sato (1999) Theorem 31.5)

Suppose {Xt} is a Levy process on Rd with generating triplet (σ, b, ν),whereat σ = (σjk) ∈ Rd×d and b = (bj) ∈ Rd . The associated family ofoperators {Pt | t ≥ 0} is a strongly continuous semigroup on C0(Rd) withnorm ‖Pt‖ = 1. Let L be its infinitesimal generator. Then C∞c (Rd) is acore of L, C 2

0 (Rd) ⊂ D(L), and

Lf (x) = 12

d∑j ,k=1

σjk∂2f

∂xj∂xk(x) +

d∑j=1

bj∂f

∂xj(x)+

+

∫Rd

(f (x + y)− f (x)−

d∑j=1

yj∂f

∂xj(x)1D(y)

)ν( dy) (16)

for f ∈ C 20 (Rd) and D = { x ∈ Rd | |x | ≤ 1 }.

Levy operator: examples for d = 1

1 Any non-trivial α-stable distribution with 0 < α < 2 has absolutelycontinuous Levy measure

ν(dx) =

{c1x−1−α on (0,∞) ,

c2|x |−1−α on (−∞, 0) ,

with c1 ≥ 0, c2 ≥ 0, c1 + c2 > 0.

2 If the Levy measure ν is non-singular, then∫R

(f (x + y)− f (x)− y

∂f

∂x(x)1D(y)

)ν( dy) = (K ∗ f − µf )(x)

for some K ∈ L1(R) and µ ∈ R.