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The Fractional Laplacian for the Fractional PDEsI

Kailai Xu1

AbstractRecent years have witnessed a notable boom in the research interest in the modeling usingnonlocal operators. The fractional Laplacian, which is the generator of a symmetric -stableprocess, has been used for modeling lossy media, option price, turbulence, etc. However,it is still not clear how the fractional Laplacian should be defined for the bounded domainmodeling. This note has two objectives: the first is to derive the fractional PDEs from theprobability view of the fractional Laplacian; the second is to define Dirichlet-type boundaryconditions based on this interpretation. It serves as a clarification of the current confusionover the fractional PDE modeling via the fractional Laplacian.

Keywords: fractional PDE, fractional Laplacian, symmetric -stable process

IUpdated on May 9, 2018.Email address: kailaix@stanford.edu (Kailai Xu)

Contents

1 Introduction 3

2 Stochastic Process and the Fractional Laplacian 32.1 Lvy Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Special Case: Symmetric -stable Process . . . . . . . . . . . . . . . . . . . 52.3 Stochastic Differential Equations Driven by Levy Process . . . . . . . . . . . 72.4 Connection with the fractional Laplacian . . . . . . . . . . . . . . . . . . . . 82.5 Feynman-Kac Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Generation of the Symmetric -stable Distribution . . . . . . . . . . . . . . 102.7 Generation of the Symmetric -stable Process . . . . . . . . . . . . . . . . . 12

3 The Fractional Laplacian in the Bounded Domain 123.1 Definition of the Fractional Laplacian on the Bounded Domain . . . . . . . . 123.2 Integration By Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Fractional PDE in Rd 154.1 Fractional Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Fractional Advection-diffusion Equation . . . . . . . . . . . . . . . . . . . . . 184.3 Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Fractional PDE in the Bounded Domain 205.1 The Fractional Poisson Problem . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Fractional PDEs in the Bounded Domain . . . . . . . . . . . . . . . . . . . . 215.3 Remarks on 1D Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Conclusion 22

7 References 22

1. Introduction

The fractional PDEs have attracted considerable attention recently due to its applicationin soft matter [1], elasticity [2], turbulence [3], anomalous diffusion [4], finance [5], imagedenoising [6], porous media flow [7], etc. Among different fractional operators, the fractionalLaplacian has been intensively studied in the recent literature. For example, [8] uses thefractional Laplacian for linear and nonlinear lossy media. [9] uses the fractional Laplacianfor option pricing in jump diffusion and exponential Lvy models. Recently, [10] providesthe first ever derivation of the fractional Laplacian operator as a means to represent themean friction in the turbulence modeling. For more application examples, see [11].

In this note, we consider the fractional Laplacian

()/2u(x) := c,dP.V.R2

u(x) u(y)|x y|d+

dy

where P.V. stands for the Cauchy principal value integration and

c,d =2

(d+2

)d/2

(2

)Over the years, researchers have investigated the fractional Laplacian from many perspec-

tives such as the probabilistic view, the potential theory or the PDEs. Different views leadto various definitions for the fractional Laplacian in Rd, which, under certain assumptions,are shown to be equivalent [12].

However, the community has not reached an agreement on how the fractional Laplacianshould be defined on a bounded domain, which is crucial for physical modeling. There aremainly three definitions in the literature: the restricted fractional Laplacian, the spectralfractional Laplacian, and the regional Laplacian. The definitions and discussions will becarried out as we proceed through the notes.

A main application of the fractional Laplacian is PDE modeling via the fractional Lapla-cian. The derivations of nonlocal models via the fractional Laplacian are very similar to thatof the classical models via the typical Laplacian such as the diffusion equation, except thatwe replace the underlying Gaussian process by its counterpart of the symmetric -stable (orLvy) process. The derivation in this note assumes no advanced knowledge of probabilitytheory and will be self-contained.

2. Stochastic Process and the Fractional Laplacian

2.1. Lvy ProcessThroughout this note, (,F ,F, P ) is a filtered probability space with filtration F =

{Ft, t 0}. We first define the Levy process L = {Lt, t 0} with value in Rd and definedon (,F ,F, P ).

Lvy process can be defined through some fundamental characterizations of its properties.

Definition 1 (Lvy process [13]). Let Lt, t [0,) be a stochastic process Xt : Rdsatisfying

L0 = 0, a.s.

stationary increments Lt Ls Xts X0 s tindependent increments Lt Ls(Xr, r s) s tcontinuity in probability lim

t0P(|Xt X0| > ) = 0 > 0

where stands for same distribution, stands for stochastic independence, () standsfor the sigma algebra.

Then L = {Lt, t 0} is called Lvy process.We see that Levy process shares many similar properties with the Brownian motion. As

we will see, the Brownian motion is merely a special case of Levy process. The full extentto which we can characterize Lvy process is to describe its characteristic function, knownas Lvy-Khintchine formulaTheorem 1 (Lvy-Khintchine formula). Let Lt, t 0 be a Lvy process, then the charac-teristic function of the probability law at time t is

EeiuLt = exp(t

(i u 1

2 u2 +

Rd

(eiux 1 iu x1|x| 0, then the corresponding Lvy tuple is (0, 0, 1).

Gaussian process. Assume that the process have the linear drift and covarianceT , then the Lvy tuple is (, , 0).

We can reorganize the exponential in eq. (1) into term terms

t

(i u+ 1

2 u2

Rd

(eiux 1 iu x1|x| 1)

|x|>1

(eiux 1

) (dx)(|x| > 1)

|x|1

(eiux 1 iux

)(dx) (3)

Actually, this decomposition leads to a remarkable decomposition of Lvy process

Theorem 2 (Lvy-It decomposition). Assume satisfies eq. (2), then there exists a proba-bility space on which three independent Lvy process exists: X(1) is a Gaussian process; X(2)is a compound Poisson process; X(3) is a square integrable martingale. The characteristicexponents are given by eq. (3) respectively. Then X = X(1) +X(2) +X(3) is a Lvy processwith characteristic exponent given by eq. (3).

We can also decompose the Lvy process into

Xt = dt+ dWt + dJt (4)

where there is a one-to-one correspondence between the terms in eq. (4) and eq. (1). Dueto the uniqueness of the Lvy triple, such a decomposition is also unique.

2.2. Special Case: Symmetric -stable ProcessThe symmetric -stable process is the basic tool to understand the fractional Laplacian

as the generator of a stochastic process. The symmetric -stable process can be definedthrough its Lvy-Khintchine representation

EeiuXt = exp(t|u|), (0, 2] (5)

where the dispersion represents the spread of the distribution. Symmetric -stable processhas the scaling property cXct Xt. Some authors will also include a drift term in eq. (5)

EeiuXt = exp(t( u |u|)), (0, 2]

in this case, the distribution is symmetric around , which is the mean of the distributionfor 1 < 2 and median for 0 < 1.

Proposition 1. Let the levy measure be

(dx) =Cdx

|x|+d(6)

then the Levy process with the triplets (0, 0, ) is a symmetric -stable process with charac-teristic function

exp

( tCc,d

|u|)

Proof. Let Lt be the Lvy process with the triplets (0, 0, ), then we have

EeiuLt = exp(Ct

Rd

(eiux 1 iu x1|x|

then g(u) is well-defined. In the following, we prove

g(u) = 1c,d

|u|

For all orthogonal matrix R we have

g(Ru) =

Rd

(eiRux 1 iRu x1|x|

For the case d = 1, there are three cases where we have explicit formulas for the proba-bility distribution of the symmetric -stable process with drift .

f1/2(, ;x) =

2(x )3/2 exp

(

2

4(x )

)Lvy

f1(, ;x) =1

2 + (x )2Cauchy

f2(, ;x) =142

exp

((x )

2

42

)Gaussian

Here the subscript in the density function f is the fractional index .

2.3. Stochastic Differential Equations Driven by Levy ProcessHere we consider the fractional stochastic differential equation driven by the Lvy

processdXt = b(Xt)dt+ (Xt)dWt + a(Xt)dJt, t [0, T ] (9)

where T > 0 is given and b, , a : [0, T ]Rd Rd are FB(Rd)-progressively measurablefunctions. A distinguishing difference from classical stochastic differential equations is thatthe discontinuity in Levy process is allowed and so is that in Xt.

A strong solution to (9) will be defined as

Xt = X0 +

t0

b(Xs)ds+

t0

(Xs)dWs +

t0

a(Xs)dJt, t [0, T ] (10)

Similar to the classical stochastic differential equation, we have the existence and unique-ness result

Theorem 3. [17] If b, , a are bounded and Lipschitz, there exists a solution to eq. (10)and that solution is pathwise unique. In addition, if Lt is a symmetric stable process withexponent (1, 2) and |G(