ON THE NON-LOCAL BOUNDARY VALUE PROBLEM FROM THE PROBABILISTIC VIEWPOINT MIRKO DOVIDIO

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ON THE NON-LOCAL BOUNDARY VALUE PROBLEM FROM

THE PROBABILISTIC VIEWPOINT

MIRKO D’OVIDIO

Abstract. We provide a short introduction of new and well-known facts relating

non-local operators and irregular domains. Cauchy problems and boundary value

problems are considered in case non-local operators are involved. Such problems

respectively lead to anomalous behavior on the bulk and on the surface of a given

domain. Such a behavior can be considered (in a macroscopic viewpoint) in order

to describe regular motion on irregular domains (in the microscopic viewpoint).

Contents

1. Introduction 1

2. Markov processes, time changes, non-local operators 2

2.1. Markov Processes 2

2.2. Subordinators and Inverses 3

2.3. Non-local (space) operators 6

2.4. Non-local (time) operators 7

2.5. Non-local operators and subordinators 8

3. Non-local initial value problems 10

3.1. Parabolic problems 10

3.2. Elliptic problems 11

3.3. Delayed and rushed processes 13

4. An irregular domain 14

4.1. Random Koch domains (RKD) 15

4.2. Non-local initial value problem on the RKD Ω(ξ)18

5. Non-local boundary value problems 19

5.1. Singular kernels 19

5.2. Non-singular kernels 23

References 26

1. Introduction

We present a collection of results about the connection between non-local oper-

ators and irregular domains. We are concerned with the interplay between macro-

scopic and microscopic analysis of Feller motions on bounded and unbounded do-

mains. Our discussion relies on the fact that anomalous motions on regular domains

can be considered in order to describe motions on irregular domains. Here, by irreg-

ular domains we mean domains with irregular boundaries or interfaces, for example

of fractal type.

arXiv:2210.02327v2 [math.AP] 27 Oct 2022

2 MIRKO D’OVIDIO

Fractional Cauchy problems have been considered as models for motions on non-

homogeneous domains, fractals for instance. In the same spirit we consider anoma-

lous motions for non-local boundary value problems, that is the motions exhibit

anomalous behavior only near the boundary. Non-local boundary conditions there-

fore introduce models for motions on irregular domains, for example domains with

trap boundaries (trap domains, in the sequel).

The presentation will run as follows. First we recall some facts about non-local

operators and the associated processes. Sometimes non-local operators are also

referred as to generalized fractional operators. However, there is no fractional order

to be considered. We move to the next sections by discussing the non-local initial

value problems and then the non-local boundary value problems. For the latter, we

underline some connection with the case of random Koch domains and, in particular,

with the boundary behaviors introduced by the non-local boundary conditions.

2. Markov processes, time changes, non-local operators

2.1. Markov Processes. Let X={Xt}t≥0with X0=x∈Ω be the Markov

process on Ω with generator (G, D(G)) where (G, D(G)) is the generator of the

semigroup Qt, then we have the probabilistic representation

Qtf(x) = Ex[f(Xt)].

We recall that for the Markovian semigroup Qt, the generator is deﬁned by

Gf(x) = lim

t↓0

Qtf(x)−f(x)

t, f ∈D(G) (2.1)

for fin some set of real-valued functions and for which the limit exists in some

sense. Then we say that fbelongs to the domain of the generator D(G). In

case Xis a Feller process for instance, the domain D(G) consists of continuous

functions vanishing at inﬁnity for which (2.1) exists as uniform limit. Examples of

Feller processes are given by the Brownian motion (with Gf =f00), the Poisson

process (with Gf =λ(F−1)fwhere F f(x) = f(x+ 1) is the forward operator)

and in general, L´evy processes (for which we will consider G=−Ψ(−∆) to be

better deﬁned further on). For the Brownian motion the forward and backward

Kolmogorov’s equation have the same reading in terms of heat equations. A Poisson

process (started at zero for instance) is driven by λ(1−B)fwhere Bf(x) = f(x−1)

is the backward operator. We skip here a discussion on L´evy processes.

We say that Xis a Feller process if Xis a strong Markov process, right-continuous

with no discontinuity other than jumps. If Xis a continuous strong Markov process

we say that Xis a Feller diﬀusion (or simply, diﬀusion).

For the process Xstarted at xwith density p(t, x, y) it holds that

Px(Xt∈D) = ZD

p(t, x, y)dy, D ⊆Ω

and

Ex[f(Xt)] = ZΩ

f(y)p(t, x, y)dy, t > 0, x ∈Ω

PROBABILITY AND NON-LOCAL BOUNDARY VALUE PROBLEMS 3

where Exis the expectation with respect to Px. The function u(t, x) = Qtf(x)

solves the Cauchy problem

∂u

∂t =Gu, u0=f∈D(G).

2.2. Subordinators and Inverses. The Bernstein function

Φ(λ) = Z∞

01−e−λzφ(dz), λ ≥0 (2.2)

where φon (0,∞) with R∞

0(1∧z)φ(dz)<∞is now a L´evy measure can be associated

with a L´evy process. Thus, the symbol Φ is the Laplace exponent of a subordinator

Htwith H0= 0, that is

E0[exp(−λHt)] = exp(−tΦ(λ)).

We also recall that

Φ(λ)

λ=Z∞

e−λzφ(z)dz, φ(z) = φ((z, ∞)) (2.3)

and φis the so called tail of the L´evy measure.

Let us introduce the inverse process

Lt:= inf{s≥0 : Hs≥t}= inf{s≥0 : Hs/∈(0, t)}

with L0= 0. We do not consider (except in some well mentioned case) step-processes

with φ((0,∞)) <∞and therefore we focus only on strictly increasing subordinators

with inﬁnite measures. Thus, the inverse process Lturns out to be a continuous

process. Both random times Hand Lare not decreasing. By deﬁnition, we also can

write

P0(Ht< s) = P0(Ls> t), s, t > 0.(2.4)

It is worth noting that Hcan be regarded as an hitting time for a Markov process

whereas, Lcan be regarded as a local time for a Markov process ([8]). We denote

by hand lthe following densities

P0(Ht∈dx) = h(t, x)dx, P0(Lt∈dx) = l(t, x)dx.

As usual we denote by P0the probability of a process started at x= 0.

Remark 2.1. For Φ(λ) = λα(that is, for stable subordinators), it holds the following

relation between densities

h(v, z)

l(z, v)=αv

z, z, v > 0.

This result has been stated in the famous book [8]without proof, thus we refer to [26].

From (2.4), straightforward calculation gives

Z∞

e−λtl(t, x)dt =Φ(λ)

λe−xΦ(λ), λ > 0.(2.5)

It suﬃces to consider that

l(t, x) = −d

dxP0(Lt> x) = −d

dxP0(t > Hx)

4 MIRKO D’OVIDIO

in (2.5). We recall that ([12]), for a good function f, for λ > Φ−1(w),

E0Z∞

e−λtf(Lt)dt=Φ(λ)

λE0Z∞

e−λHtf(t)dt.(2.6)

Denote by e

l(λ, x) the Laplace transform (2.5). Assume that ∀x > 0, for n≥1,

lim

λ→0+λne

l(λ, x) = 0 and lim

λ→∞ λne

l(λ, x)=0.(2.7)

Then, we conclude that ∀x > 0, for n≥1, λne

l(λ, x)∈C0([0,∞)) and in particular,

∃M > 0 : ∀x > 0,|λne

l(λ, x)| ≤ M.

Thus, under (2.7),

∀x > 0l(·, x)∈C∞((0,∞)).

A further check shows that (n= 0 and formula (2.5))

∀x > 0l(·, x)∈L1((0,∞)) iﬀ lim

λ↓0

Φ(λ)

λ<∞.

Concerning e

h(t, ξ) = e−tΦ(ξ)assume that ∀t > 0, for n≥1, ξne

h(t, ξ)∈C0([0,∞))

and therefore, ξne

h(t, ξ) is bounded as well. Then, we have that

∀t > 0h(t, ·)∈C∞((0,∞)).

Here we obviously have that

∀t > 0h(t, ·)∈L1((0,∞)).

Remark 2.2. We notice that the Laplace transform bu(λ)with λ > ω exists for u

in the set of piecewise continuous functions of exponential order w. That is, given

M > 0,∃Tsuch that |u(t)| ≤ Metω for t > T . This fact will be also discussed

further on.

Proposition 2.1. We have that l(z, 0) = φ(z),z > 0.

Proof. Just considering (2.5). 

Proposition 2.2. Let us write

κ(z) = Z∞

h(t, z)dt, and `(z) = l(z, 0), z > 0.

Then, κand `are associated Sonine kernels. Moreover,

κ(z)`(1 −z)dz =Z1

κ(z)`(1 −z)dz = 1.(2.8)

Proof. It is enough to consider the Laplace transforms of κand `.

We notice that, for a stable subordinator, for which Φ(z) = zα,

E0Z∞

e−λHtdt=1

λα=Z∞

e−λx xα−1

Γ(α)dx

from which we obtain the potential density of Ht, that is

Z∞

h(t, x)dt =xα−1

Γ(α)=: κ(x), x > 0.

PROBABILITY AND NON-LOCAL BOUNDARY VALUE PROBLEMS 5

The inverse of λα−1gives

l(t, 0) = t−α

Γ(1 −α)=: `(t), t > 0.

Thus, formula (2.8) is veriﬁed. The convolutions κ∗uand `∗ucan be used in order

to deﬁne respectively the fractional integral and the fractional derivative of a given

function u. Indeed,

Dα

xu(x) := d

dxI1−α

xu(x) = 1

Γ(1 −α)

dx Zx

u(y)(x−y)−αdy

is the well-known Riemann-Liouville derivative written in terms of the associated

fractional integral. Notice that

Dα

xu(x) = d

dx(u∗`)(x),Iα

xu(x) = (u∗κ)(x).

We also recall the Caputo-Dzherbashian derivative

Dα

xu(x) := 1

Γ(1 −α)Zx

u0(s)(x−s)−αds =Dα

xu(x)−u(0)

where u0=du/ds. The Caputo-Dzherbashian derivative (also termed Caputo deriv-

ative) has been introduced by Caputo in [16,17,18] and previously by Dzherbashian

in [34,35]. The Riemann-Liouville derivatives have a long history.

The previous operators are obtained by convolution and represent well-known

objects in fractional calculus where αhas a clear meaning in terms of fractional order.

Some criticisms have been also reported in the literature about the deﬁnition of

derivative and integral for some operators (see for example [43]). In the following, we

consider non-local operators for which we do not have a direct link with a fractional

order but with a symbol Φ and therefore, with a convolution kernel (singular or not,

see the last section). We basically move from the following ”types” of operators:

i) Marchaud (type) operators,

DΦ

x−u(x) = Z∞

0u(x)−u(x−y)φ(dy)

and

DΦ

x+u(x) = Z∞

0u(x)−u(x+y)φ(dy);

ii) Riemann-Liouville (type) operators,

DΦ

x−u(x) = d

dx Zx

−∞

u(x−y)φ(y)dy

and

DΦ

x+u(x) = d

dx Z∞

u(x−y)φ(y)dy;

iii) Caputo-Dzherbashian (type) operators,

DΦ

tu(t) = Zt

u0(t−s)φ(y)dy

where u0=du/dt is the derivative of u(t).

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ONTHENON-LOCALBOUNDARYVALUEPROBLEMFROMTHEPROBABILISTICVIEWPOINTMIRKOD'OVIDIOAbstract.Weprovideashortintroductionofnewandwell-knownfactsrelatingnon-localoperatorsandirregulardomains.Cauchyproblemsandboundaryvalueproblemsareconsideredincasenon-localoperatorsareinvolved.Suchproblemsrespectivelyleadtoan...

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