Fractional equations and time-changed processes

Mirko D’Ovidio

Dept. Basic and Applied Sciences for EngineeringSapienza University of Rome

Fifth FCPNLO WorkshopFractional Calculus, Probability and Non-Local Operators

BilbaoNovember 8-10 2017

Plan of the talk - keywords

• subordinators and inverse processes (hitting times)

• fractional/higher-order equations and time-changed processes

• general time-fractional operators

• general results for time-changed processes (on “general” domains)

• examples: regular domains (balls), irregular domains (pre-fractals and fractals)

we use the probabilistic approach in order to solve differential equations

time-changed/subordinated processes can be considered to solve fractional differential equations

some time changes are considered to solve boudary value problems (additive functionals)

1

Subordinator Ht and Inverse Lt, symbol Φ

We recall that E0[exp−λHt] = e−Φ(λ)t where

Φ(λ) =

∫ ∞0

(1− e−λy)Π(dy),Φ(λ)

λ=

∫ ∞0

e−λy

Π((y,∞))dy (1)

(Levy measure Π(dy) and tail of the Levy measure Π((y,∞))). Moreover,

Lt = infs ≥ 0 : Hs /∈ (0, t), t ≥ 0,

P0(Lt ≤ s) = P0(Hs ≥ t)

Notice the relation between λ-potentials

E0

[∫ ∞0

e−λtf(Lt)dt

]=

Φ(λ)

λE0

[∫ ∞0

e−λHtf(t)dt

](2)

We focus only on strictly increasing subordinators with infinite measures Π((0,∞))

Then, L turns out to be a continuous process

2

Subordinated/Time-changed processes

Let Xt, t ≥ 0 be a Markov process with generator A, X ⊥ H ⇔ X ⊥ L

XHt is driven by∂u

∂t= −Φ(−A)u, u0 = f, u(t, x) = Ex[f(XHt)], semig.

XLt is driven by DΦt u = Au, u0 = f, u(t, x) = Ex[f(XLt)], no semig.

−Φ(−A)f(x) =

∫ ∞0

(esAf(x)− f(x)

)Π(ds)

where Π(·) is the Levy measure of a subordinator with symbol Φ and esA is the semigroup

associated with the base process X. Moreover, by following the same spirit, we have that

Φ(Dt)f(t) =

∫ ∞0

(e−sDtf(t)− f(t)

)Π(ds)

with Laplace transform

(∫ ∞0

(e−λs − 1)Π(ds)

)f(λ) = Φ(λ)f(λ)

where e−sDtf(t) = f(t−s) is the translation semigroup. Notice that Φ(Dt) is the representation

of a Riemann-Liouville type fractional operator. (Dt = ∂∂t with symbol λ)

3

Time-fractional operator

Explicit representations of DΦ (in their notation, Toaldo: Caputo-type, Z.-Q. Chen: (R-L)-type)

fDtu(t) = b

d

dtu(t) +

∫ t

0

∂

∂tu(t− s)v(s)ds

∂wt f(t) =

d

dt

∫ t

0

w(t− s)(f(s)− f(0)

)ds

where v(·) and w(·) are the fractional kernels (both generalized Caputo derivatives)

Let M > 0 and w ≥ 0. Let Mw be the set of (piecewise) continuous function on [0,∞) of

exponential order w such that |u(t)| ≤ Mewt. Denote by u the Laplace transform of u. Then,

we define the operator DΦt :Mw 7→ Mw such that∫ ∞

0

e−λt

DΦt u dt = Φ(λ)u−

Φ(λ)

λu0 = (λu− u0)

Φ(λ)

λ, λ > w∫ ∞

0

|DΦt u|

pdt ≤

(∫ ∞0

|u′|dt)(

limλ↓0

Φ(λ)

λ

)p, p ∈ [1,∞)

4

References

[1] Z.-Q. Chen, Time fractional equations and probabilistic representation, Chaos, Solitons &

Fractals, 102, (2017), 168 -174.

[2] Z.-Q. Chen, P. Kim, T. Kumagai, J. Wang, Heat kernel estimates for time fractional equations,

ArXiv:1708.05863v1, (2017).

[3] R. Capitanelli, M. D’Ovidio Fractional equations via convergence of forms, ArXiv:1710.01147,

(2017).

[4] M. Magdziarz, R. L. Schilling, Asymptotic properties of Brownian motion delayed by inverse

subordinators, Journal: Proc. Amer. Math. Soc. 143 (2015), 4485 - 4501.

[5] M. Meerschaert, H. P. Scheffler, Triangular array limits for continuous time random walks,

Stochastic Processes and their Applications, 118 (2008) 1606 - 1633.

[6] B. Toaldo, Convolution-type derivatives, hitting-times of subordinators and time-changed

C∞-semigroups, Potential Analysis, 42 (2015), 115 - 140.

5

Example: time-fractional operators

D=is the ordinary derivative

Dνt u :=D

[1

Γ(1− ν)

∫ t

0

u(s, ·)(t− s)ν

ds

](R-L)-type derivative

Dνt (u− u0) =

1

Γ(1− ν)

∫ t

0

Du(s, ·)(t− s)ν

ds =:∂νu

∂tνCaputo-type derivative

Φ(λ) =λν

: DΦt u =

∂νu

∂tν,

∫ ∞0

e−λt

DΦt u dt = λ

νu−

λν

λu0 (3)

The density u(t, x) of Lt solves DΦt u = −∂u

∂x, x > 0, t > 0 with u0 = δ. Moreover,∫ ∞0

e−γx

u(t, x)dx = Eν(−γtν) =∑k≥0

(−γtν)k

Γ(νk + 1), u(1, x) = Mν(x)

where Eν is the Mittag-Leffler function. Note that

Dνt =

(∂

∂t

)ν6=∂ν

∂tν

6

Example: time-fractional operators

Let us consider stable subordinators, 1Hα ⊥ 2H

α, α ∈ (0, 1) and

Lt = infs ≥ 0 :1H

2ν

s +2Hν

s ≥ t, t ≥ 0, ν ∈ (0, 1/2)

Φ(λ) = λ2ν

+ λν

: DΦt u =

∂2νu

∂t2ν+∂νu

∂tνCaputo derivatives

∫ ∞0

e−λt

DΦt u dt = (λ

2ν+ λ

ν)u−

λ2ν + λν

λu0

The density u(t, x) of Lt solves DΦt u = −∂u

∂x, x > 0, t > 0 with u0 = δ.

Moreover, let r1 = −1 +√

1− γ and r2 = −1−√

1− γ, (Eν is the M-L function)∫ ∞0

e−γx

u(t, x)dx =1

2

[(1 +

1√

1− γ

)Eν(r1t

ν) +

(1−

1√

1− γ

)Eν(r2t

ν)

]

M. D’Ovidio, E. Orsingher, B. toaldo, Time-Changed processes governed by space-time fractional telegraph equations, Stochastic

Analysis and Applications, 32 (2014) 1009-1045.

7

Φ(λ) = λν, ν ∈ (0, 1]

Let hν = hν(t, x) be the density law of the stable subordinator Ht

∂

∂thν = −

(∂

∂x

)νhν, hν(0, x) = δ(x), “hν(t, 0) = 0” (R-L derivative)

Theorem 1. hνn

is the unique solution to

− (−1)n ∂

n

∂tnhνn

= −(∂

∂x

)νhνn, x > 0, t > 0, n ∈ N

(−1)k ∂

k

∂tkhνn

∣∣∣∣t=0+

=x−

νkn −1

Γ(−νkn ), 0 < k < n

hνn(0, x) = δ(x)

M. D’Ovidio. On the fractional counterpart of the higher-order equations. Statistics & Probability Letters, 81, (2011), 1929 - 1939.

8

Φ(λ) = λν, ν ∈ (0, 1]

Let lν = lν(t, x) be the density law of the inverse process Lt

∂ν

∂tνlν = −

∂

∂xlν, lν(0, x) = δ(x), “lν(t, 0) =

t−ν

Γ(1− ν)” Caputo derivative

Theorem 2. lνn

is the unique solution to

∂ν

∂tνlνn

= (−1)n ∂

n

∂xnlνn, x > 0, t > 0, n ∈ N

(−1)k ∂

k

∂xklνn

∣∣∣∣x=0

=t−

ν(k+1)n

Γ(1− ν(k+1)n )

, 0 < k < n

lνn(0, x) = δ(x)

M. D’Ovidio. On the fractional counterpart of the higher-order equations. Statistics & Probability Letters, 81, (2011), 1929 - 1939.

9

Let κn = −(−1)p for n = 2p or κn = ±1 for n = 2p. Let us consider the equation

∂u

∂t= κn

∂nu

∂xn, x ∈ R, t > 0, n ≥ 2, “u ∈ S(R)” (4)

Theorem 3. Let v, w be two solutions to (4). Then

v(t, x) =xm

tmw(t, x), m ∈ N

Notice that (Bertoin)hν(x, t)

lν(t, x)= ν

x

tNotice that

n = 2, l1/2(t, x) =e−

x2

4t√

4πt= u(t, x), x > 0, (y

′ − xy = 0)

n = 3, l1/3(t, x) =1

3√3tAi

(x

3√3t

)= u(t, x), x > 0, (y

′′ − xy = 0)

n > 3, l1/n(t, x) 6=u(t, x)

10

higher-order and fractional equations

Let X be the Feller process (Markov process) with generator (A,D(A))

Let L be the inverse to a stable subordinator with symbol Φ(λ) = λ1/m

Theorem 4.

u(t, x) = Ex[f(XLt)] =

∫ ∞0

Ex[f(Xs)]l 1m

(s, t)ds

solves

∂1/mu

∂t1/m=Au, u0 = f ∈ D(A

m)

and

∂u

∂t=A

mu+

m−1∑k=1

tkm−1

Γ( km)Akf, u0 = f ∈ D(A

m)

11

References

[1] S. Bonaccorsi, M. D’Ovidio, S. Mazzucchi, Probabilistic representation formula for the solution

of fractional high order heat-type equations, arXiv:1611.03364, (2017)

[2] M. D’Ovidio. On the fractional counterpart of the higher-order equations. Statistics &

Probability Letters, 81 (12), 1929–1939, 2011.

[3] V. Keyantuo, C. Lizama. On a connection between powers of operators and fractional Cauchy

problems. J. Evol. Equ., 12, 245–265, 2012.

[4] H. Komatsu. Fractional powers of operators. Pacific J. Math., 19:285 – 346, 1966.

[5] M. Meerschaert, E. Nane, and P. Vellaisamy. Fractional Cauchy problems on bounded domains.

Ann. Probab., 37 (3), 979 – 1007, 2009.

[6] E. Nane. Higher order PDE’s and iterated processes. Trans. Amer. Math. Soc. 360, 2681–2692,

2008.

[7] E. Orsingher, L. Beghin. Fractional diffusion equations and processes with randomly varying

time. Ann. Probab., 37, (1), 206 - 249, 2009.

12

Time-changed (Delayed) Brownian motion on R

XLt, t > 0 (5)

Figure 1: ”from the web page of Prof. Mark M. Meerschaert ”

some bounded domain: trap?, fractal?

13

a/fr: Reflected Brownian Motions and trap domains

(Burdzy, Chen, Marshall 2006)

Consider the reflecting Brownian motion B+ on Ω with Green function G+ and an open connected

set Ω ⊂ Rd, d ≥ 2 with finite volume. Let B ⊂ Ω be an open ball with non-zero radius and

denote by τ∂B = infs ≥ 0 : B+s ∈ ∂B the hitting time of the reflecting BM B+ ∈ Ω \ B.

Definition 1. The set Ω is a trap domain if

supx∈Ω\B

Ex τ∂B =∞ or supx∈Ω\B

∫Ω\B

G+(x, y)dy =∞. (6)

Otherwise, Ω is a non-trap domain.

14

XΦt = XLt, X Feller, (A,D(A))

Let En be locally compact, separable metric spaces (En∂ = En ∪ ∂ one-point compact. of En)

Consider the sequence XΦ,nt = Xn

Lton En, the base process Xn with generator (An, D(An))

associated with the sequence of Dirichlet forms on L2(En,m) with En ⊂ F , E ⊂ F ,

En(u, v) = (√−Anu,

√−Anv), D(En) = D(

√−An) .

Theorem 5. The function u(t, x) = Ex[f(XΦ,nt )], is the unique strong solution in L2(En,m)

to DΦt u = Anu, u0 = f ∈ D(An) in the sense that:

1. ϕ : t 7→ u(t, ·) is such that ϕ ∈ C([0,∞),R+) and ϕ′ continuous and bounded,

2. ϑ : x 7→ u(·, x) is such that ϑ,Anϑ ∈ D(An),

3. ∀ t > 0, DΦt u(t, x) = Anu(t, x) holds m-a.e in En,

4. ∀ x ∈ En, u(t, x)→ f(x) as t ↓ 0.

Let D be the set of continuous functions from [0,∞) to E∂ which are right continuous on [0,∞)

with left limits on (0,∞).

Theorem 6. A sequence of forms En M-converges to a form E in L2(F,m) if and only if

XΦ,n → XΦ in distribution as n→∞ in D.

R. Capitanelli, M. D’Ovidio, Fractional equations via convergence of forms, Submitted, arXiv:1710.01147, (2017).

15

We recall the notion of M -convergence of forms on the Hilbert space L2(F,m)

Definition 2. A sequence of forms En M-converges to a form E in L2(F,m) if

(a) For every vn converging weakly to u in L2(F,m)

lim inf En(vn, vn) ≥ E(u, u), as n→ +∞. (7)

(b) For every u ∈ L2(F,m) there exists un converging strongly to u in L2(F,m) such that

lim sup En(un, un) ≤ E(u, u), as n→ +∞. (8)

We point out that the forms E , En can be defined in the whole of L2(F,m) by setting

E(u, u) = +∞ ∀u ∈ L2(F,m) \D(E).

En(u, u) = +∞ ∀u ∈ L2(F,m) \D(En).

16

The process X on (E,B(E)) with generator (A,D(A)) associated with the Dirichlet form

(E, D(E)) on L2(E,m), the time changes of X and the λ-potentials

Rλf(x) := Ex

[∫ ∞0

e−λtf(Xt)dt

], λ > 0 (9)

RΦλf(x) := Ex

[∫ ∞0

e−λtf(X

Φt )dt

]=

Φ(λ)

λEx

[∫ ∞0

e−λHtf(Xt)dt

], λ > 0 (10)

The sequence of processes Xn on (En,B(En)) with generators (An, D(An)) associated with

the Dirichlet forms (En, D(En)) on L2(En,m), the time changes of Xn and the λ-potentials

Rnλf(x) := Ex

[∫ ∞0

e−λtf(X

nt )dt

], λ > 0

RΦ,nλ f(x) := Ex

[∫ ∞0

e−λtf(X

Φ,nt )dt

]=

Φ(λ)

λEx

[∫ ∞0

e−λHtf(X

nt )dt

], λ > 0

Observe that λRΦλ = ΦRΦ and λRΦ,n

λ = ΦRnΦ

17

application/further result (a/fr)Xnt is a planar Brownian motion on En associated with the Dirichlet form En, D(En) =

H1(En,m) and µn is the Revuz measure of the local time γnt of Xnt on ∂En

u(t, x) =Ex[f(Xnt )] = Ex[f(

∗Xnt )e−cnγnt ]

En(u, v) =

∫En∇u∇v dm+ cn

∫∂En

uv dµn

∂u

∂t= ∆Nu− cnµn u, u0 = f, H

1(R2

) ∩ L2(R2

, µn)

(µn the measure on ∂En, ∆N the Neumann Laplacian, cn ≥ 0 the elastic coefficients)

u(t, x) = Ex[f(XΦ,nt )] = Ex[f(

∗X

Φ,nt )M

Φ,nt ], M

Φ,nt = exp(−cnγnLt) “(= M

nLt

)”

solves DΦt u = Anu, u0 = f where An = ∆N . Let us write MΦ,n

t = MΦ,nt (R),

XRt = (

∗Xt,Mt(R)) X

Nt = (

∗Xt,Mt(N)) X

Dt = (

∗Xt,Mt(D))

the parts process of ∗X on E

we apply Theorem 6

18

a/fr: regular domain (balls Ωl ⊂ Ω` ⊂ Ωr, r = `+ ε and ε > 0)

on ∂Ω` (inside) we have skew reflection - on ∂Ωl and ∂Ω`+ε the process is killed

the gray part indicates the collapsing annulus Ω`+ε \ Ω`

R. Capitanelli, M. D’Ovidio, Asymptotics for time-changed diffusions, Theory of Probability and Mathematical Statistics, special volume

95 in honor of Prof. N.Leonenko, (2017) 37-54

19

a/fr: regular domain (balls Ωl ⊂ Ω` ⊂ Ωr, r = `+ ε and ε = ε(n) > 0)

Let B1, B2 be two independent Brownian motions and define the process

Xnt = B

(α,η)t :=

B

1t on Ω` \ Ωl

B2ηt on Σε = Ωr \ Ω`

on En

= Ωr \ Ωl

with Dirichlet condition on ∂Ωl,∂Ωr and skew condition on ∂Ω`:

∀ x ∈ ∂Ω`, Px(B(α,η)t ∈ Ω` \ Ωl) = 1− α and Px(B

(α,η)t ∈ Σε) = α.

Anu =

σ2(x)

a(x)∇(a(x)∇u), dm =

a(x)

σ2(x)dx (11)

where

a(x) = (1− α)1Ω`\Ωl(x) + α1Ω`+ε\Ω`

(x), α ∈ (0, 1) (12)

and

σ2(x) = 1Ω`\Ωl

(x) + η1Ω`+ε\Ω`(x), η > 0. (13)

We consider α = α(n), η = η(n) under the assumption αε/η → 0 as n→∞ picture

20

a/fr: regular domain (balls Ωl ⊂ Ω` ⊂ Ωr, r = `+ ε and ε > 0)

The generators (An, D(An) associated with the sequence of energy forms in L2(F,m)∫Ω`\Ωl

(1− α)|∇u|2dx+ α∫

Ω`+ε\Ω`|∇u|2dx ifu|Ωr\Ωl ∈ H

10(Ωr \ Ωl)

+∞ otherwise in L2(F )(14)

The perturbed forms are written in terms of cn = α(n)/ε(n).

Theorem 7. The solution to DΦt u = Anu, u0 = f ∈ D(An) converges strongly in L2(Ω` \

Ωl, dx) as n→∞ to the solution to DΦt u = Au, u0 = f ∈ D(A) given by

u(t, x) = Ex[f(XΦt )] = Ex[f(

∗X

Φt )1(Lt<ζ)

]

where A = AD if cn → ∞, A = AN if cn → 0, A = AR if cn → c ∈ (0,∞). Moreover,

we have that

Ex

[∫ ∞0

f(XΦ,nt )dt

]→ Ex

[∫ ∞0

f(∗X

Φt ) exp

(−(

limn→∞

cn

)γ∂Ω`Lt

(∗X)

)1(Lt<τl)

dt

]and ∗X is a reflecting BM on Ω`.

21

lifetime ζΦ of XΦt

limλ↓0

RΦλ1(x) =

(limλ→0

Φ(λ)

λ

)Ex

[∫ ∞0

e−(

limn→∞ cn

)γ∂Ω`t 1(t<τl)

dt

]= Ex[ζ

Φ].

In the case of Dirichlet condition on ∂Ω` (that is cn →∞)

limλ↓0

RΦλ1(x) = lim

λ↓0

1

λEx

[1− e−Φ(λ)(τl∧τ`)

]= Φ

′(0)Ex[(τl ∧ τ`)]

In the case of Neumann condition on ∂Ω` (cn → 0)

limλ↓0

RΦλ1(x) = lim

λ↓0

1

λEx

[1− e−Φ(λ)τl

]= Φ

′(0)Ex[τl]

Finally, Robin condition on ∂Ω` (cn → c ∈ (0,∞))

limλ↓0

RΦλ1(x) = lim

λ↓0

Φ(λ)

λEx

[∫ τl

0

e−sΦ(λ)−cγ

∂Ω`s ds

]= Φ

′(0)Ex

[∫ τl

0

e−cγ

∂Ω`s ds

]

22

liftime ζ of Xt

Xnt is a skew BM on Ω`+ε \ Ωl with generator (An, D(An)), ε = ε(n)→ 0 as n→∞

uε ∈ D(An) s.t. A

nuε = −1

with D(An) = g,Ang ∈ Cb : g|∂Ωl= g|∂Ω`+ε

= 0, g satisfies (15), (16)

u|∂Ω−`

= u|∂Ω+`

continuity on the boundary, (15)

(1− α)∂nu|∂Ω−`

= α∂nu|∂Ω+`

partial reflection, (16)

where ∂nu is the normal derivative of u. α = α(n)→ 0 as n→∞.

We pass to the radial part Rnt , the skew Bessel process on (l, `+ ε) with generator Ln, D(Ln) =

g, Lng ∈ Cb([l, `+ε]) : g(l) = g(`+ε) = 0, g(`−) = g(`+), (1−α)g′(`−) = αg′(`+)

vε ∈ D(Ln) s.t. L

nvε = −1

vε →v(x) = C ln x/l−x2 − l2

2pointwise

CRobin :`c + `2−l2

2

1/2c+ ln `/l, CNeumann : `

2, CDirichlet :

`2 − l2

2 ln `/l

23

liftime ζΦ of XΦt

Ex[ζΦ] = Φ

′(0)Ex[ζ] = Φ

′(0)v(x)

where

Φ′(0) = lim

λ↓0

Φ(λ)

λ

is finite only in some cases, for example:

• inverse Gaussian subordinator with Φ(λ) = σ−2(√

2λσ2 + µ2 − µ)

with σ 6= 0;

• gamma subordinator with Φ(λ) = a ln(1 + λ/b) with ab > 0;

• generalized stable subordinator with Φ(λ) = (λ+ γ)α − γα with γ > 0 and α ∈ (0, 1).

24

a/fr: irregular “ideal” domain : the Koch snowflake

Skew BM across fractal interfaces

R. Capitanelli, M. D’Ovidio, Skew Brownian Diffusions Across Koch Interfaces, Potential Analysis, 46, (2017), 431 - 461.

25

a/fr: irregular domain (snowflake)

contraction factors `−1a with `a : 2.1, 2.7, 3.3, 3.9

Figure 2: only 1 iteration

26

a/fr: irregular domain (snowflake)

Let `a ∈ (2, 4), a ∈ I ⊂ N be the reciprocal of the contraction factor for the family Ψ(a) of

contractive similitudes Ψai : C→ C given by

Ψ(a)1 (z) =

z

`a, Ψ

(a)2 (z) =

z

`aeıθ(`a)

+1

`a,

Ψ(a)3 (z) =

z

`aeıθ(`a)

+1

2+ ı

√1

`a−

1

4, Ψ

(a)4 (z) =

z − 1

`a+ 1

where θ(`a) = arcsin(√`a(4− `a)/2).

Let Ξ = IN with I ⊂ N, |I| = N , and let ξ = (ξ1, ξ2, . . .) ∈ Ξ. Set `ξi = 3 ∀ i. The n-th

pre-fractal curve K(ξ|n) is given by

Φ(ξi) = ∪4

j=1Ψ(ξi)

j , K(ξ|n)

= Φ(ξ1) Φ

(ξ2) · · · Φ(ξn)

(K), n ∈ N

(K is the line segment of unite length and endpoint A = (0, 0), B = (1, 0))

27

a/fr: skew motion on Ω(ξ|n)ε = Ω(ξ|n) ∪ Σ(ξ|n)

An = −div(an∇u) with an = cnσnwn1

Σ(ξ|n) + 1Ω(ξ|n)

∇u · n|y− = cnσn∇u · n|y+, ∀ y ∈ ∂Ω(ξ|n)

(17)

D(An) =

u ∈ L2

(Ω(ξ|n)ε , dx) : u|

Ω(ξ|n) ∈ H2(Ω

(ξ|n)), u|

Σ(ξ|n) ∈ H2(Σ

(ξ|n)), u|

∂Ω(ξ|n)ε

= 0,

and u satisfies the transmission condition (17)

where cn are the elastic coefficients and σn = 4−n∏n

i=1 `ξi = (3/4)n

and wn is the tickness of Σ(ξ|n) (n is the outer normal to Ω(ξ|n))

Proposition 1. Let n ∈ N. The domain Ω(ξ|n) is non-trap for the skew BM Xnt .

28

a/fr: skew motion on Ω(ξ|n)ε = Ω(ξ|n) ∪ Σ(ξ|n)

Xnt = B

α(n)t is a skew Brownian motion on Ω(ξ|n)

ε reflecting on the pre-fractal interfaces ∂Ω(ξ|n)

In particular (s is the measure on the pre-fractal Koch curve ∂Ω(ξ|n))

σn

∫∂Ω(ξ|n)

Px(Xnt ∈ Σ

(ξ|n))s(dx) = α(n)

We have that α(n)/(cnwn)→ 1 that is cn ∼ α(n)/wn where wn is the tickness of Σ(ξ|n)

Throughout, we consider the PCAF A(ξ|n)t = cn γ

(ξ|n)t where γ

(ξ|n)t is the LT of Xn

t on ∂Ω(ξ|n)

Proposition 2. A(ξ|n)t is the unique PCAF with Revuz measure

µn(dx) := cn1∂Ω(ξ|n)(x)ds (18)

where s is the arc-length measure on ∂Ω(ξ|n). (the multiplicative functional Mnt = e−A

(ξ|n)t )

29

a/fr: Φ = Id, Dtu = Au, E = fractal koch domain

Theorem 8. As n→∞,

Xnt

d→ XNt ⇔M

nt (R)

w→Mt(N)⇔ cn → 0

Xnt

d→ XRt ⇔M

nt (R)

w→Mt(R)⇔ cn → c ∈ (0,∞)

Xnt

d→ XDt ⇔M

nt (R)

v→Mt(D)⇔ cn →∞

Proof:

M-convergence of forms⇒ conv. of semigroups

conv. of semigroups⇒ conv. of finite dim. distristibutions (Markov processes)

+ tightness⇒ (weak) convergence of measures

Moreover, the multiplicative functional characterizes uniquely the process (Blu, Get 1964), we

obtain the same results by only considering the convergence of Mnt = Mn

t (µn)

perturbation of the form⇔ Revuz measure µn of the PCAF

30

a/fr: fractal domain E = Ω(ξ), ξ = (ξ1, ξ2, . . .) with `ξi = 3 ∀ i

Theorem 9. As n→∞,

XΦ,nt

d→ XΦ,Nt as cn → 0

XΦ,nt

d→ XΦ,Rt as cn → c ∈ (0,∞)

XΦ,nt

d→ XΦ,Dt as cn →∞

(boundary conditions on E: Neumann, Robin, Dirichlet).

Proof:From the previous theorem we have convergence of Xn

t , by Theorem 6 we have the result

31

a/fr: random Koch Domains

Let `a ∈ (2, 4), a ∈ I ⊂ N be the reciprocal of the contraction factor for the family Ψ(a) of

contractive similitudes Ψai : C→ C given by

Ψ(a)1 (z) =

z

`a, Ψ

(a)2 (z) =

z

`aeıθ(`a)

+1

`a,

Ψ(a)3 (z) =

z

`aeıθ(`a)

+1

2+ ı

√1

`a−

1

4, Ψ

(a)4 (z) =

z − 1

`a+ 1

where θ(`a) = arcsin(√`a(4− `a)/2).

Let Ξ = IN with I ⊂ N, |I| = N , and let ξ = (ξ1, ξ2, . . .) ∈ Ξ. We call ξ an environment

sequence where ξn says which construction we are using at level n. We consider the sequence

(Barlow Hambly 1997)

`ξi, i ∈ N sequence 1/`ξi of contraction factors (19)

32

a/fr: Random Koch Domains

Consider the random vector ξ = (ξ1, ξ2, . . .) with ξi ∈ I with probability mass function

P : I → [0, 1]. The n-th pre-fractal curve K(ξ|n) is given by

Φ(ξi) = ∪4

j=1Ψ(ξi)

j , K(ξ|n)

= Φ(ξ1) Φ

(ξ2) · · · Φ(ξn)

(K), n ∈ N

(K is the line segment of unite length and endpoint A = (0, 0), B = (1, 0)) and depends on the

realization of ξ with probability P (ξi = ξi) for its i-th component. For ξ|n = (ξ1, . . . , ξn) and

ξ|n = (ξ1, . . . , ξn) we obtain the curve K(ξ|n) with probability

P(ξ|n = ξ|n) =

n∏i=1

P(ξi = ξi), ξilaw= ξ1, ∀ i, ξi ⊥ ξj, i 6= j

the fractal (d(ξ)

-set, µ = Hd(ξ)) K

(ξ)= ∪∞n=1K

(ξ|n) with d(ξ)

=ln 4

E[ln ξ1]

Proposition 3. Let n ∈ N. For a given realization (ξ|n) of the r.v. (ξ|n), the domain Ω(ξ|n)

is non-trap for the skew BM Xnt .

33

a/fr: Random Koch Domains

For a given realization (ξ|n) of (ξ|n) we have the domain Ω(ξ|n), for example:

Figure 3: 4 iteration in each picture

4 possible realization of the vector ξ|4 = (ξ1, ξ2, ξ3, ξ4) per fugure

34

a/fr: Brownian Motions and Functionals

Fix ξ ∈ Ξ, n ∈ N, (ξ|n), an environment sequence. Set

γ(ξ|n)t :=

∫ t

0

1∂Ω(ξ|n)(

∗Xns )ds Local time

Robin Laplacian An = ∆R (elastic Brownian motion on Ω(ξ|n))

D(∆R) = u ∈ H1(Ω

(ξ|n)) : ∆u ∈ L2

(Ω(ξ|n)

), ∂nu+ cnu|∂Ω(ξ|n) = 0, a.e.

u(t, x) = Ex[f(Xnt )] = Ex

[f(∗Xnt )e−cnγ

(ξ|n)t

]From pointwise convergence for (ξ|n) we get a.s. convergence for (ξ|n) and

XΦ,nt converges in distribution a.s. in Ω(ξ), the random Koch domain

35

extension

Let us consider the process Zt, t ≥ 0, governed by the fractional problem

DΦt u = −Ψ(−A)u, u0 ∈ D(−Ψ(−A)) ⊂ D(A)

where Ψ is a symbol for some Levy measure.

Let L be the inverse of H with symbol Φ as in the previous sections. Let L∗ be the inverse to H∗

with symbol Ψ. Then Z can be represented through subordination (by H∗) and time-change (by

L∗) of X with generator A. We note that

Ex

[∫ ∞0

e−λtf(Zt)dt

]:= R

Ψ(Φ)λ f(x)

can be written as

RΨ(Φ)λ f(x) =

Φ(λ)

λ

(Ψ(Φ(λ))

Φ(λ)

)2

RΨ(Φ(λ))f(x)

Notice that, as above, λRΨ(Φ)λ can be considered in studying convergence of potentials

Notice also that, we can consider Φn, Ψn uniformly converging

36

References

[1] R. Capitanelli, M. D’Ovidio. Skew Brownian diffusions across Koch interfaces. Potential Analysis 46 (2017), pp 431 - 461.

[2] R. Capitanelli, M. D’Ovidio. Asymptotics for time-changed diffusions. Theor. Probability and Math. Statist. 95 (2016), 37 - 54.

[3] R. Capitanelli, M. D’Ovidio. Fractional equations via convergence of forms. arXiv:1710.01147, (2017).

Other References

[4] M.T. Barlow, B.M. Hambly. Transition density estimates for brownian motion on scale irregular Sierpinski gaskets Annales de l’I.H.P.Probabilits et statistiques 33:5 (1997) 531-557.

[5] R.F. Bass, P. Hsu. Some Potential Theory for Reflecting Brownian Motion in Holder and Lipschitz Domains The Annals of Probability,19:2 (1991), 486–508.

[6] K. Burdzy, Z-Q. Chen, D.E. Marshall. Traps for reflected Brownian motion. Math. Z. 252 (2006), no. 1, 103-132.

[7] R. Capitanelli. Robin boundary condition on scale irregular fractals. Commun. Pure Appl. Anal. 9 (2010),1221 - 1234.

[8] R. Capitanelli, M.A. Vivaldi. Insulating layers and Robin problems on Koch mixtures. J. of Diff. Eq. 251 (2011),1332-1353.

[9] V. Keyantuo, C. Lizama, M. Warma. Existence, regularity and representation of solutions of time fractional diffusion equations, Adv.Differential Equations, 21, (2016), 837 - 886.

[10] A. Kolesnikov. Convergence of Dirichlet forms with changing measure on Rd. Forum Math. 17 (2005), 225-259.

[11] R. M. Blumenthal, R. K. Getoor. Additive functionals of Markov processes in duality, Trans. Amer. Math. Soc. 112 (1964), 131–163.

Books

[12] Billingsley Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc.

[13] Zhen-Qing Chen, Masatoshi Fukushima (2012). Symmetric Markov Processes, Time Change, and Boundary Theory. Princeton universityPress

[14] Fukushima Masatoshi, Oshima Yoichi and Takeda Masayoshi (1994). Dirichlet forms and symmetric Markov processes. Walter de Gruyter& Co

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