Analysis, Probability, ' }u Çv o] }v NONLOCAL OPERATORSNonlocalOperators Analysis, Probability, Geometry and Applications CenterforInterdisciplinaryResearch(ZiF),Bielefeld July9–14,2012

LOCAL ORGANIZER

Moritz Kaßmann

SCIENTIFIC COMMITTEE

Rodrigo BañuelosCyril Imbert

Grzegorz KarchTakashi Kumagai

József LörincziRené Schilling

NONLOCALAnalysis, Probability,

Geometry and Applicati ons

OPERATORSJuly, 9th - 14th, 2012

Nonlocal OperatorsAnalysis, Probability, Geometry and Applications

Center for Interdisciplinary Research (ZiF), Bielefeld

July 9–14, 2012

Supported by the Collaborative Research Centre SFB 701

Local Organizer:

Moritz Kaßmann

Scientific Committee:

Rodrigo BañuelosCyril ImbertGrzegorz KarchTakashi KumagaiJózsef LörincziRené Schilling

Cover design: Stefan Adamick, ZiFLATEX typesetting: Matthieu Felsinger, SFB 701

Contents

1 Summer School on Nonlocal Operators 7

2 Program 9

3 Abstracts of the talks 19

4 Poster Presentations 59

5 Practical Information 61

6 List of Participants 67

Conference Poster 71

Summer Schoolon Nonlocal OperatorsJuly 4–6 2012, BielefeldPreceding this conference the Department of Mathematics and the CollaborativeResearch Centre SFB 701 hosted a summer school on nonlocal operators.

Zhen-Qing Chen

University of Washingtonon

Potential theory for jumpprocesses

Luis Silvestre

University of Chicagoon

Regularity theory for nonlocaloperators

The purpose of this intense 3-day school on nonlocal operators was to teach recentresults and techniques in the area of jump processes and integro-differential operatorsof fractional order.

Six lectures given by Prof. Z.-Q. Chen addressed methods from probability and stochas-tic analysis in the study of nonlocal operators. The focus was on the potential theory ofLévy-type processes and more general jump processes generated by nonlocal Dirichletforms, including fine estimates on heat kernels.

Six lectures given by Prof. L. Silvestre concentrated on methods from analysis andpartial differential operators. Here the focus was on regularity estimates for solutionsto fully nonlinear equations driven by nonlocal operators.

Further information can be found here:http://www.math.uni-bielefeld.de/nloc-school

Bielefeld, 2012 7

http://www.math.uni-bielefeld.de/nloc-school

Program

Monday, July 9

08:30 – 09:00 Registration

09:00 – 10:00 Opening

10:00 – 10:45 Zhen-Qing ChenPerturbation by non-local Operators

10:45 – 11:20 Ante MimicaRegularity estimates of harmonic functions for jumpprocesses

11:20 – 11:50 Coffee Break

11:50 – 12:25 Christine GeorgelinOn Neumann and oblique derivatives boundary conditionsfor non-local equations

12:25 – 13:00 Mateusz KwaśnickiEstimates of harmonic functions for non-local operators

13:00 – 14:30 Lunch

14:30 – 15:15 Peter ImkellerConstruction of stochastic processes with singular jumpcharacteristics as solutions of martingale problems

15:15 – 15:50 Ilya PavlyukevichSmall noise asymptotics of integrated Ornstein-UhlenbeckProcesses driven by α-stable Lévy processes


16:20 – 16:55 Rupert FrankUniqueness and nondegeneracy of ground states fornon-local equations in 1D

16:55 – 17:40 Yuri KondratievNon-local evolutions as kinetic equations for Markovdynamics

17:45 – 18:45 Poster Session

19:00 – Dinner at ZiF

Bielefeld, 2012 9

Tuesday, July 10

09:00 – 09:45 Mark MeerschaertThe Inverse Stable Subordinator

09:45 – 10:20 Hans-Peter SchefflerFractional governing equations for coupled continuous timerandom walks


10:50 – 11:35 Renming SongStability of Dirichlet heat kernel estimates of non-localoperators under perturbations

11:35 – 12:10 Bartłomiej DydaComparability and regularity estimates for symmetricnon-local Dirichlet forms

12:10 – 12:45 Matthieu FelsingerLocal regularity for parabolic nonlocal operators

12:45 – 14:15 Lunch

14:15 – 15:00 María del Mar GonzálezFractional order operators in conformal geometry

15:00 – 15:35 Giampiero PalatucciAsymptotics of the s-perimeter as s 0


16:05 – 16:50 Francesca Da LioAnalysis of fractional harmonic maps

16:50 – 17:25 Armin SchikorraKnot-energies and fractional harmonic maps

17:25 – 18:10 Julio RossiA Monge-Kantorovich mass transport problem for a discretedistance

10 Nonlocal Operators

Wednesday, July 11

09:00 – 09:45 David ApplebaumMartingale transforms and Lévy processes on Lie groups

09:45 – 10:20 Victoria KnopovaParametrix construction for the transition probabilitydensity of some Lévy-type processes


10:50 – 11:35 Niels JacobNon-locality, non-isotropy, and geometry

11:35 – 12:20 Yannick SireSmall energy regularity for a fractional Ginzburg-Landausystem

12:30 – 13:45 Lunch

13:45 – Excursion

Bielefeld, 2012 11

Thursday, July 12

09:00 – 09:45 Michael RöcknerSub- and supercritical stochastic quasi-geostrophic equation

09:45 – 10:20 Naotaka KajinoNon-regularly varying and non-periodic oscillation of theon-diagonal heat kernels on self-similar fractals


10:50 – 11:35 Alexander Grigor’yanEstimates of heat kernels of Dirichlet forms

11:35 – 12:10 Tomasz GrzywnyEstimates of the Poisson kernel of a half-line forsubordinate Brownian motions

12:10 – 12:45 Nathaël AlibaudContinuous dependence estimates for fractal/fractionaldegenerate parabolic equations

12:45 – 14:15 Lunch

14:15 – 15:00 Richard LehoucqPeridynamic non-local mechanics

15:00 – 15:35 Etienne EmmrichAnalysis of the peridynamic model in non-local elasticity


16:05 – 16:50 Ralf MetzlerAgeing and ergodicity breaking in anomalous diffusion

16:50 – 17:25 Björn BöttcherConstructive approximation of Feller processes withunbounded coefficients

17:25 – 18:10 Alexander BendikovMarkov semigroups on totally disconnected sets


Friday, July 13

09:00 – 09:45 Panki KimAn Lp-theory of stochastic PDEs with random fractionalLaplacian operator

09:45 – 10:20 Mohammud FoondunStochastic heat equation with spatially coloured randomforcing


10:50 – 11:35 Alexis VasseurIntegral variational problems

11:35 – 12:10 Enrico ValdinociNon-local non-linear problems

12:10 – 12:45 Russell SchwabOn Alexandrov-Bakelman-Pucci type estimates forintegro-differential equations

12:45 – 14:15 Lunch

14:15 – 15:00 Krzysztof Bogdan3G, 4G and perturbations

15:00 – 15:35 Yuishi ShiozawaConservation property of symmetric jump-diffusionprocesses


16:05 – 16:50 Enrico PriolaUniqueness for singular SDEs driven by stable processes

16:50 – 17:25 Suleyman UlusoyNon-local conservation laws and related Keller-Segel typesystems

17:25 – 18:10 Piotr BilerBlowup of solutions to generalized Keller-Segel model

Bielefeld, 2012 13

Saturday, July 14

09:00 – 09:45 Igor SokolovFractional subdiffusion and subdiffusion-reaction equations:Physical Motivation and Properties

09:45 – 10:20 Enrico ScalasCharacterization of the fractional Poisson process

10:20 – 10:55 Piotr GarbaczewskiLévy flights, Lévy semigroups and fractional quantummechanics


11:25 – 12:00 Paweł SztonykUpper estimates of transition densities for stable-dominatedsemigroups

12:00 – 12:45 Zoran VondračekPotential theory of subordinate Brownian motions withGaussian components

12:45 – 14:15 Lunch


Bielefeld, 2012 15

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Abstracts of the talksNathaël Alibaud – Continuous dependence estimates for fractal/fractional

degenerate parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . 21David Applebaum – Martingale transforms and Lévy processes on Lie groups 21Alexander Bendikov – Markov semigroups on totally disconnected sets . . . . 22Piotr Biler – Blowup of solutions to generalized Keller–Segel model . . . . . . 22Krzysztof Bogdan – 3G, 4G and perturbations . . . . . . . . . . . . . . . . . . 23Björn Böttcher – Constructive Approximation of Feller Processes with

unbounded coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Zhen-Qing Chen – Perturbation by Non-Local Operators . . . . . . . . . . . . 25Francesca Da Lio – Analysis of Fractional Harmonic Maps . . . . . . . . . . . 26Bartłomiej Dyda – Comparability and regularity estimates for symmetric

nonlocal Dirichlet forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Etienne Emmrich – Analysis of the peridynamic model in nonlocal elasticity . 28Matthieu Felsinger – Local Regularity for Parabolic Nonlocal Operators . . . . 29Mohammud Foondun – Stochastic heat equation with spatially coloured

random forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Rupert Frank – Uniqueness and nondegeneracy of ground states for non-local

equations in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Piotr Garbaczewski – Lévy flights, Lévy semigroups and fractional quantum

mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Christine Georgelin – On Neumann and oblique derivatives boundary

conditions for non-local equations . . . . . . . . . . . . . . . . . . . . . . 31María del Mar González – Fractional order operators in conformal geometry . 32Alexander Grigor’yan – Estimates of heat kernel of Dirichlet forms . . . . . . 33Tomasz Grzywny – Estimates on the Poisson kernel of a half-line for

subordinate Brownian motions . . . . . . . . . . . . . . . . . . . . . . . . 34Peter Imkeller – Construction of stochastic processes with singular jump

characteristics as solutions of martingale problems . . . . . . . . . . . . 35Niels Jacob – Non-locality, Non-Isotropy, and Geometry . . . . . . . . . . . . 36Naotaka Kajino – Non-regularly varying and non-periodic oscillation of the

on-diagonal heat kernels on self-similar fractals . . . . . . . . . . . . . . 36Panki Kim – An Lp-theory of stochastic PDEs with random fractional

Laplacian operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Victoria Knopova – Parametrix construction for the transition probability

density of some Lévy-type processes . . . . . . . . . . . . . . . . . . . . . 38Yuri Kondratiev – Non-local evolutions as kinetic equations for Markov

dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Mateusz Kwaśnicki – Estimates of harmonic functions for non-local operators 39Richard Lehoucq – Peridynamic nonlocal mechanics . . . . . . . . . . . . . . 40

Bielefeld, 2012 19

Mark Meerschaert – Solutions to a nonlocal fractional wave equation . . . . . 41Ralf Metzler – Ageing and Ergodicity Breaking in Anomalous Diffusion . . . . 42Ante Mimica – Regularity estimates of harmonic functions for jump processes 42Giampiero Palatucci – Asymptotics of the s-perimeter as s 0 . . . . . . . . 44Ilya Pavlyukevich – Small noise asymptotics of integrated Ornstein–Uhlenbeck

processes driven by α-stable Lévy processes . . . . . . . . . . . . . . . . . 46Enrico Priola – Uniqueness for singular SDEs driven by stable processes . . . 46Michael Röckner – Sub- and supercritical stochastic quasi-geostrophic equation 47Julio Rossi – A Monge-Kantorovich mass transport problem for a discrete

distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Enrico Scalas – Characterization of the fractional Poisson process . . . . . . . 48Hans-Peter Scheffler – Fractional governing equations for coupled continuous

time random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Armin Schikorra – Knot-energies and Fractional Harmonic Maps . . . . . . . 49Russell Schwab – On Aleksandrov-Bakelman-Pucci type estimates for integro

differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Yuichi Shiozawa – Conservation property of symmetric jump-diffusion processes 50Yannick Sire – Small energy regularity for a fractional Ginzburg-Landau system 52Igor Sokolov – Fractional Subdiffusion and Subdiffusion-Reaction equations:

Physical Motivation and Properties . . . . . . . . . . . . . . . . . . . . . 52Renming Song – Stability of Dirichlet heat kernel estimates of non-local

operators under perturbations . . . . . . . . . . . . . . . . . . . . . . . . 53Paweł Sztonyk – Upper estimates of transition densities for stable-dominated

semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Suleyman Ulusoy – Non-local Conservation laws and related Keller-Segel type

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Enrico Valdinoci – Nonlocal nonlinear problems . . . . . . . . . . . . . . . . . 56Alexis Vasseur – Integral variational problems . . . . . . . . . . . . . . . . . . 56Zoran Vondraček – Potential theory of subordinate Brownian motions with

Gaussian components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57


Nathaël Alibaud

Continuous dependence estimates for fractal/fractional degenerate parabolicequations

This talk will be concerned with the Cauchy problem

ut + div f(u) + (−4)α2 ϕ(u) = 0, u(0) = u0,

where α ∈ (0, 2) and ϕ is a nondecreasing nonlinearity. It will focus on continuousdependence estimates on the data; i.e. given another solution

vt + div g(v) + (−4)β2ψ(u) = 0, v(0) = v0,

we will see how u−v can be bounded by differences between (f, α, ϕ, u0) and (g, β, ψ, v0).For instance, if α = 2 and if u and v have the same data, excepted ψ 6= ϕ, B. Cockburnand G. Gripenberg have shown in 1999 that

‖u(·, t)− v(·, t)‖L1 = O(‖√ϕ′ −

√ψ′‖∞

). (3.1)

In the fractional case, we shall see that

‖u(·, t)− v(·, t)‖L1 =

O(‖(ϕ′) 1

α − (ψ′)1α ‖∞

), α > 1,

O (‖ϕ′ lnϕ′ − ψ′ lnψ′‖∞) , α = 1,O (‖ϕ′ − ψ′‖∞) , α < 1,

giving in particular a new proof of (3.1) as α→ 2. In the case where ψ = ϕ but β 6= α,we shall see that

‖u(·, t)− v(·, t)‖L1 = O(|α− β|).All these results are optimal.

Joint work with Simone Cifani and Espen R. Jakobsen (Norwegian University of Sci-ence and Technology, Trondheim, Norway).

David Applebaum

Martingale transforms and Lévy processes on Lie groups

We obtain martingale transforms that are built in a natural way from a generic Lévyprocess in a Lie group G. Using sharp inequalities due to Burkholder we then constructa class of linear operators that are bounded on Lp(G,m) (where m is a Haar measure)

Bielefeld, 2012 21

for all 1 < p < ∞. When the group is compact, we use Peter-Weyl theory to exhibitspecific examples of these operators as Fourier multipliers. These include second orderRiesz transforms, imaginary powers of the Laplacian and operators associated withsubordinated Brownian motion.Talk based on joint work with Rodrigo Bañuelos (Purdue).

The paper is available at: http://arxiv.org/abs/1206.1560

Alexander Bendikov

Markov semigroups on totally disconnected sets

This is joint work with A. Grigor’yan and Ch. Pittet. Let (X, d) be a metric space.We assume that it is totally disconnected and locally compact. Then there exists anultra-metric d′ which generates the same topology on X. Given a measure m on Xwe construct a symmetric Markov semigroup (Pt) on X. We give upper- and lower-bounds of its transition function and its Green function, give a criterion of transience,estimate moments of the corresponding Markov process and describe the spectrum ofits Markov generator. In a particular case when X is the field of p-adic numbers, ourconstruction recovers The Vladimirov operator. Even in this well-established settingsome of our results are new.

Piotr Biler

Blowup of solutions to generalized Keller–Segel model

The existence and nonexistence of global in time solutions is studied for a class ofequations generalizing the chemotaxis model of Keller and Segel. These equationsinvolve Lévy diffusion operators and general potential type nonlinear terms.

We will consider the following nonlinear nonlocal evolution equation

∂tu+ (−∆)α/2u+∇ · (uB(u)) = 0,

for (x, t) ∈ Rd × R+, where the anomalous diffusion is modeled by a fractional powerof the Laplacian, α ∈ (1, 2), and the linear (vector) operator B is defined as

B(u) = ∇((−∆)−β/2u)

for β ∈ (1, d], and d ≥ 2.


http://arxiv.org/abs/1206.1560

Krzysztof Bogdan

3G, 4G and perturbations

Schrödinger perturbations of transition densities p by functions q are widely studied,see, e.g., [3, 4]. Local integral smallness of q with respect to p plays a role in theseconsiderations. In a series of recent papers (see, e.g., [1]) we propose the followingassumption,

t∫s

∫X

ps,u(x, z)q(u, z)pu,t(z, y)dzdu ≤ [η +Q(s, t)]ps,t(x, y), (3.2)

where q ≥ 0, 0 ≤ η < 1 and 0 ≤ Q(s, u) +Q(u, t) ≤ Q(s, t) for s < u < t. Under (3.2),the perturbation, p, of p by q enjoys the following upper bound,

ps,t(x, y) ≤ ps,t(x, y)

(1

1− η

)1+Q(s,t)/η

.

The assumption (3.2) is quite friendly if p allows for the so-called 3P inequality:

ps,u(x, z) ∧ pu,t(z, y) ≤ Cps,t(x, y), s < u < t.

The Gaussian transition density, however, fails to satisfy such 3P.

I will report a joint work in progress with Karol Szczypkowski on a new bound inspiredby [2] and called 4P, which applies to the Gaussian kernel.

[1] K. Bogdan, T. Jakubowski & S. Sydor: Estimates of perturbation series for kernels.http://arxiv.org/abs/1201.4538

[2] T. Jakubowski & K. Szczypkowski: Time-dependent gradient perturbations of frac-tional Laplacian. J. Evol. Equ. 10(2):319–339 (2010)

[3] V. Liskevich & Y. Semenov: Two-sided estimates of the heat kernel of the Schrödingeroperator. Bull. London Math. Soc. 30(6):596–602 (1998)

[4] Q. S. Zhang: A sharp comparison result concerning Schrödinger heat kernels. Bull.London Math. Soc. 35(4):461–472 (2003)

Bielefeld, 2012 23


Björn Böttcher

Constructive Approximation of Feller Processes with unbounded coeffi-cients

Feller semigroups and Feller processes are well studied objects. But their constructionis in general difficult. In particular, most of the existing construction methods only al-low the construction of Feller processes with bounded coefficients, i.e., they require thatthe symbol of the generator is bounded uniformly with respect to the space variable.We will show how this boundedness assumption can be relaxed.

Let (Xt)t≥0 be an Rd-valued Feller process. In general its generator A is a nonlocaloperator which has, given that the test functions are in the domain, a representationas a pseudo-differential operator with a symbol q. Moreover, the symbol q has a Lévy-Khinchin representation:

q(x, ξ) = c(x)− il(x)ξ + ξQ(x)ξ +

∫y 6=0

(1− eiyξ +

iyξ

1 + |y|2)N(x, dy), (3.3)

where c(x) ≥ 0, l(x) ∈ Rd, Q(x) = (qjk(x))j,k=1,...,d ∈ Rd×d is a positive semidefinitematrix and N(x, .) is a measure which integrates |y|2

1+|y|2 on Rd\0.Conversely, it is a natural to ask:

When is an operator A, with symbol q given by (3.3), the generator of a Fellerprocess?

Thus one wants to construct a Feller process starting with a given symbol q. A commonassumption in such constructions is: limr→∞ supx sup|ξ|≤ 1

r|q(x, ξ)| = 0. Our main

result shows that this condition can be relaxed.

Theorem. Let q(., .) : Rd × Rd → C be a function such that

limr→∞

sup|y|≤r

sup|ξ|≤ 1

r

|q(y, ξ)| = 0.

For each k ∈ N let (Xkt )t≥0 be a Feller process with semigroup (T kt )t≥0, such that its

generator Ak satisfies C∞c ⊂ D(Ak) and the symbol qk(x, ξ) of Ak∣∣C∞c

satisfies

|qk(x, ξ)| ≤ |q(x, ξ)| for all x, ξ ∈ Rd,qk(x, ξ) = q(x, ξ) for all |x| ≤ k, ξ ∈ Rd

andXk.∧τk

Bk(0)

d= X l

.∧τ lBk(0)

for l ≥ k.


Then the operator (−q(x,D), C∞c ) has an extension which generates a Feller processwith semigroup

Ttu = limk→∞

T kt u

for u ∈ C∞.

[1] B. Böttcher: On the construction of Feller processes with unbounded coefficients. Elec-tron. Commun. Probab. 16:545–555 (2011)

Zhen-Qing Chen

Perturbation by Non-Local Operators

Let d ≥ 1 and 0 < β < α < 2. In this talk, we consider fractional Laplacian ∆α/2 :=−(−∆)α/2 on Rd perturbed by non-local operator

Sbf(x) := A(d,−β)

∫Rd

(f(x+ z)− f(x)− 〈∇f(x), z1|z|≤1〉

) b(x, z)|z|d+β

dz,

where b(x, z) is a bounded measurable function on Rd×Rd with b(x, z) = b(x,−z) forx, z ∈ Rd, and A(d,−β) is a normalizing constant so that Sb = ∆β/2 when b(x, z) ≡ 1.We address the existence and uniqueness of the fundamental solution qb(t, x, y) to thenon-local operator Lb = ∆α/2 + Sb. We show that if b(x, z) is continuous in x, thenqb(t, x, y) ≥ 0 if and only if b(x, z) ≥ 0.

The following is one of the main results we will present during the talk. For a ≥0, denote by pa(t, x, y) the fundamental function of ∆α/2 + a∆β/2 (or equivalently,the transition density function of the Lévy process Yt + a1/βZt, where Y and Z are(rotationally) symmetric α-stable process and symmetric β-stable processes on Rd thatare independent to each other). It is known that

pa(t, x, y) (t−d/α ∧ (at)−d/β

)∧(

t

|x− y|d+α+

at

|x− y|d+β

).

Denote by S(Rd) the space of tempered functions on Rd.

Theorem. Let b(x, z) be a bounded non-negative function on Rd × Rd satisfyingb(x, z) = b(x,−z). For each x ∈ Rd, the martingale problem for (Lb,S(Rd)) withinitial value x is well-posed. These martingale problem solutions Px, x ∈ Rd forma strong Markov process Xb with infinite lifetime, which possesses a jointly continu-ous transition density function qb(t, x, y) with respect to the Lebesgue measure on Rd.

Bielefeld, 2012 25

Moreover, the following holds.

(i) The transition density function qb(t, x, y) can be explicitly constructed as follows.Define qb0(t, x, y) := p0(t, x, y) and

qbn(t, x, y) :=

t∫0

∫Rd

qbn−1(t− s, x, z)Sbzp0(s, z, y)dzds for n ≥ 1.

There is ε > 0 so that∑∞

n=0 qbn(t, x, y) converges absolutely on (0, ε] × Rd × Rd

and qb(t, x, y) =∑∞

n=0 qbn(t, x, y).

(ii) qb(t, x, y) = p0(t, x, y) +

t∫0

∫Rd

qb(t− s, x, z)Sbzp0(s, z, y)dzds.

(iii) For every A > 0, there are positive constants ck = ck(d, α, β,A), k = 1, · · · , 4,such that for any non-negative function b(x.z) on Rd×Rd with b(x, z) = b(x,−z)and ‖b‖∞ ≤ A, we have

c1e−c2t p inf b(t, x, y) ≤ qb(t, x, y) ≤ c3e

c4t p‖b‖∞(t, x, y) on (0,∞)× Rd × Rd.

Our study is partially motivated by the consideration of the following stochastic dif-ferential equation dXt = dYt + c(Xt−)dZt. Its solution has infinitesimal generator Lbwith b(x, z) = |c(x)|β .This talk is based on a joint work with Jieming Wang.

Francesca Da Lio

Analysis of Fractional Harmonic Maps

In this talk we present regularity and compactness results for fractional weak harmonicmaps into manifolds. These maps are critical points of nonlocal functionals of the form

L(u) =

∫Rn

|∆α/2u(x)|pdx , (3.4)

where αp = n, u : Rn → N , N is a smooth k-dimensional sub-manifold of Rm whichis compact and without boundary. This kind of variational problems appears as sim-plified models for renormalized energy in general relativity, (see [1]) . For simplicity weconsider here the case of weak 1/2−harmonic maps from the real line into a sphere


(n = 1, α = 1/2, p = 2) . In this particular case the Lagrangian (3.4) is also in-variant under the trace of conformal maps that keep invariant the half space R2

+ (thewell-known Möbius group) .

The key point in our results is first a formulation of the Euler-Lagrange equationfor (3.4) (the so-called 1/2−harmonic map equation) in the form of a nonlocal linearSchrödinger type equation with a 3-terms commutators in the right-hand-side. Wethen establish a sharp estimate for these commutators by using the Littlewood-Paleydecomposition and the theory of para-products.

[1] S. Alexakis & R. Mazzeo: The Willmore functional on complete minimal surfaces inH3: boundary regularity and bubbling.http://arxiv.org/abs/1204.4955

[2] F. Da Lio: Compactness and Bubbles Analysis for Half-Harmonic Maps into Spheres.Preprint

[3] F. Da Lio: Fractional Harmonic Maps into Manifolds in odd dimension n>1.http://arxiv.org/abs/1012.2741

[4] F. Da Lio & A. Schikorra: n/p-Harmonic maps: regularity for the sphere case.http://arxiv.org/abs/1202.1151

[5] F. Da Lio & T. Rivière: Three-term commutator estimates and the regularity of 12 -

harmonic maps into spheres. Anal. PDE 4(1):149–190 (2011)

[6] F. Da Lio & T. Rivière: Sub-criticality of non-local Schrödinger systems with antisym-metric potentials and applications to half-harmonic maps. Adv. Math. 227(3):1300–1348(2011)

[7] A. Schikorra: Regularity of n/2-harmonic maps into spheres. J. Differential Equations252(2):1862–1911 (2012)

Bartłomiej Dyda

Comparability and regularity estimates for symmetric nonlocal Dirichletforms

We will discuss sufficient conditions on the kernel k such that the following twoquadratic forms

EkB(u, u) =

∫∫BB

(u(y)− u(x)

)2k(x, y) dy dx,

E(α)B (u, u) =

∫∫BB

(u(y)− u(x)

)2|x− y|−d−α dy dx,

Bielefeld, 2012 27




are comparable. Namely, the sufficient conditions will be:

[(K)] for almost every x, y ∈ Rd

L(x− y) ≤ k(x, y) ≤ U(x− y) ,

for some functions L,U : Rd → [0,∞) satisfying L(x) = L(−x), U(x) = U(−x)for almost every x ∈ Rd, L 6= 0 on a set of positive measure

[(U)] There exists C1 > 0 such that for every r ∈ (0, 1]∫Rd

(r2 ∧ |z|2

)U(z) dz ≤ C1r

2−α.

[(L)] There exist a > 1 and C2, C3 such that every annulus Ba−n+1 \ Ba−n(n = 0, 1, . . .) contains a ball Bn with radius C2a

−n, such that

L(z) ≥ C3(2− α)|z|−d−α, z ∈ Bn.

We will discuss the independence of the constant on α when α→ 2−.

As an application, we will provide a weak Harnack inequality and a-priori estimates inHölder spaces for solutions u ∈ L∞(Rd) ∩Hα/2

loc (B) to integrodifferential equations ofthe following form

EkB(u, φ) = 0 for every φ ∈ C∞c (B).

The talk is based on a joint work with Moritz Kassmann (Universität Bielefeld).

[1] B. Dyda: On comparability of integral forms. J. Math. Anal. Appl. 318(2):564–577 (2006)

[2] B. Dyda & M. Kassmann: Comparability and regularity estimates for symmetric nonlocalDirichlet forms.http://arxiv.org/abs/1109.6812

[3] M. Kassmann: A new formulation of Harnack’s inequality for nonlocal operators. C. R.Math. Acad. Sci. Paris 349(11-12):637–640 (2011)

Etienne Emmrich

Analysis of the peridynamic model in nonlocal elasticity

Peridynamics is a rather new nonlocal continuum theory that avoids spatial deriva-tives. It is believed to be suited for the description of fracture and other materialfailure. From the mathematical point of view, the peridynamic model exhibits several



difficulties: nonlocality, nonlinearity, time delay, and multiscale behaviour. The anal-ysis and numerical analysis of the peridynamic model is, therefore, still at the verybeginning.

In this talk, we give a survey of results known so far. In particular, we focus on thequestion of existence of solutions and the question of the limit of vanishing nonlocality.

Matthieu Felsinger

Local Regularity for Parabolic Nonlocal Operators

This talk will be concerned with regularity results for weak solutions to parabolicequations

∂tu− Lu = f,

where L is a nonlocal integro-differential operator of differentiability order α ∈ (α0, 2).L can be considered as a generalization of the fractional Laplacian −(−∆)α/2 in thesense that generalized integral kernels kt(x, y) |x− y|−d−α are allowed.

Local a priori estimates of Hölder norms and a weak Harnack inequality are proved.These results are robust with respect to α 2, i.e. all constants appearing in the proofscan be chosen independently of α. In this sense, the presentation is a generalization ofMoser’s result from 1971.

We employ localization techniques and the Lemma of Bombieri-Giusti.

The talk is based on a joint preprint with M. Kassmann, seehttp://arxiv.org/abs/1203.2126.

Mohammud Foondun

Stochastic heat equation with spatially coloured random forcing

The aim of this talk is to present some results concerning the long term behaviour ofa class of stochastic heat equations. More precisely, we look at the following class ofequations.

∂tu(t, x) = Lu(t, x) + σ(u(t, x))W (t, x),

Bielefeld, 2012 29


under suitable conditions. Here L is the generator of a Lévy process and is hence anintegro-differential operator. W is the random term and is assumed to be Gaussian.After explaining how to make sense of these types of equations, we will give conditionsunder which the second-moment of the solution grows exponentially. If time permits,we will specialise to the case where L is a fractional Laplacian and discuss finite timeblow-up of the second moment of the solution under some suitable conditions.

Rupert Frank

Uniqueness and nondegeneracy of ground states for non-local equations in1D

We prove uniqueness of energy minimizing solutions Q for the nonlinear equation(−∆)sQ + Q − Qα+1 = 0 in 1D, where 0 < s < 1 and 0 < α < 4s

1−2s for s < 1/2and 0 < α < ∞ for s ≥ 1/2. Here (−∆)s is the fractional Laplacian. As a techni-cal key result, we show that the associated linearized operator is nondegenerate, in thesense that its kernel is spanned by Q′. This solves an open problem posed by Weinsteinand by Kenig, Martel and Robbiano.

The talk is based on joint work with E. Lenzmann.

Piotr Garbaczewski

Lévy flights, Lévy semigroups and fractional quantum mechanics

Lévy-Schrödinger semigroups derive from additive perturbations of symmetric stablegenerators, [1]-[6]. Their integral kernels can be Doob-transformed into transition prob-ability density functions of the jump-type process which, under confining conditions,admit non-Gaussian invariant (asymptotic) probability density functions (pdfs). Therelated dynamical pattern of behavior is inequivalent to that obtained via gradientperturbations of fractional (e.g. symmetric stable) noise generators, [2, 3]. Undersuitable restrictions, Lévy semigroups admit an analytic continuation in time to theunitary dynamics, viewed as a signature of the quantum behavior, [1, 4]. The spectralproperties of the involved non-local Hamiltonian-type operators (negative semigroupgenerators) are of utmost importance, [4, 6]. Their ground states φ determine invari-ant pdfs ρ = |Φ|2 of jump-type processes, identical with those Doob-deduced fromthe corresponding semigroup. An issue of the abnormal (heavy tailed) asymptotic of


diffusion-type processes is to be mentioned, [7].The talk is supposed to give a brief outline of the present author’s research on Lévystable and quasi-relativistic processes, motivated by physicist’s intuitions (specificallyso-called Schrödinger boundary data problem and the non-Langevin modeling of Lévyflights) and struggles with the understanding and proper usage of the grand mathe-matical formalism for that subject matter.

[1] P. Garbaczewski, J. R. Klauder & R. Olkiewicz: Schrödinger problem, Lévy pro-cesses, and noise in relativistic quantum mechanics. Phys. Rev. E (3) 51(5, part A):4114–4131 (1995)

[2] P. Garbaczewski & R. Olkiewicz: Cauchy noise and affiliated stochastic processes.J. Math. Phys. 40(2):1057–1073 (1999)

[3] P. Garbaczewski & R. Olkiewicz: Ornstein-Uhlenbeck-Cauchy process. J. Math.Phys. 41(10):6843–6860 (2000)

[4] P. Garbaczewski & V. Stephanovich: Lévy flights in confining potentials. Phys. Rev.E 80:031113 (2009)

[5] P. Garbaczewski & V. Stephanovich: Lévy flights in inhomogeneous environments.Phys. A 389(21):4419–4435 (2010)

[6] P. Garbaczewski & V. Stephanovich: Lévy targeting and the principle of detailedbalance. Phys. Rev. E 84:011142 (2011)

[7] P. Garbaczewski, V. Stephanovich & D. Kedzierski: Heavy-tailed targets and(ab)normal asymptotics in diffusive motion. Physica A: Statistical Mechanics and itsApplications 390(6):990–1008 (2011)

Christine Georgelin

On Neumann and oblique derivatives boundary conditions for non-localequations

On this talk we will present different Neumann type boundary value problems for non-local equations related to Lévy processes. Since these equations are nonlocal, Neumanntype problems can be obtained in many ways, depending on the kind of “reflection”we impose on the outside jumps. We will focus on rather simple linear equations setin half-space domains and consider different models of reflection and rather generalnon-symmetric Lévy measures. We derive the Neumann/reflection problems through atruncation procedure on the Lévy measure, and we will present comparison, existence,and some regularity results using a viscosity solution theory. The reflection modelsthat we consider include cases where the underlying Lévy processes are reflected, pro-jected, and/or censored upon exiting the domain.If there is enough time, we will say few words about the approach of Lions & Sznitman

Bielefeld, 2012 31

in order to give a sense to Neumann and oblique derivatives boundary conditions forpartial integro-differential equations set on more general domains Ω.

These results are joint works with G. Barles, E. Chasseigne and E. Jakobsen.

María del Mar González

Fractional order operators in conformal geometry

In the talk we will have a closer look at the relations between scattering operatorsof asymptotically hyperbolic metrics [8] and Dirichlet-to-Neumann operators for uni-formly degenerate elliptic boundary value problems [2], as described in the joint workwith A. Chang [3]. This allows to define the fractional conformal Laplacian opera-tor, which is a nonlocal operator, but still satisfies the geometric conformal covarianceproperty and it coincides with the classical construction of [7] at the integer powers. Inaddition, one may also define the fractional Q-curvature of a manifold. Although theprecise meaning of this new curvature is not clear yet, it may be understood througha variation formula for weighted volume [4].

There are still many open problems: on one hand, on a compact manifold one mayformulate the fractional Yamabe problem (joint with J. Qing [6]). On the other hand,the noncompact case is much less well understood. Some cases of singular metricson spheres were considered in [5], while some other constructions on hyperbolic-likemanifolds are described in [1].

[1] V. Banica, M. d. M. González & M. Saez: Some constructions for the fractionalLaplacian on noncompact manifolds. In preparation

[2] L. Caffarelli & L. Silvestre: An extension problem related to the fractional Lapla-cian. Comm. Partial Differential Equations 32(7-9):1245–1260 (2007)

[3] S.-Y. A. Chang & M. d. M. González: Fractional Laplacian in conformal geometry.Adv. Math. 226(2):1410–1432 (2011)

[4] M. d. M. González: A weighted notion of renormalized volume related to the fractionalLaplacian. To appear in Pacific. J. Math.

[5] M. d. M. González, R. Mazzeo & Y. Sire: Singular solutions of fractional orderconformal Laplacians. To appear in Journal of Geometric Analysis

[6] M. d. M. González & J. Qing: Fractional conformal Laplacians and fractional Yamabeproblems. Preprint.

[7] C. R. Graham, R. Jenne, L. J. Mason & G. A. J. Sparling: Conformally invariantpowers of the Laplacian. I. Existence. J. London Math. Soc. (2), 46(3):557–565, 1992.

[8] C. R. Graham & M. Zworski: Scattering matrix in conformal geometry. Invent. Math.,152(1):89–118, 2003


Alexander Grigor’yan

Estimates of heat kernel of Dirichlet forms

Let (M,d, µ) be a metric measure space and (E ,F) be a regular Dirichlet form inL2 (M,µ). It is well known, any regular Dirichlet form has the generator L that is anon-positive definite self-adjoint operator in L2 (M,µ), which in turn gives rise to theheat semigroup Pt = etL and an associated Hunt process Xtt≥0 onM. If the operatorPt in L2 (M,µ) has an integral kernel, then it is called the heat kernel and is denotedby pt (x, y).

We discuss in the talk two questions:

Q1. What are “nice” estimates of the heat kernels that one can expect?

Q2. How to prove such estimates?

To Q1 the following dichotomy was proved by Grigor’yan and Kumagai in 2008. As-sume that the heat kernel in the general setting satisfies the following estimate

pt (x, y) C

tα/βΦ

(cd(x, y)

t1/β

), (3.5)

with some monotone decreasing function Φ. Then α = dimHM and Φ can be only oftwo types:

1. Φ (s) exp(−cs

ββ−1

)(sub-Gaussian function)

2. Φ (s) (1 + s)−(α+β) (stable-like function)

To Q2. Denote by B (x, r) metric balls in (M,d). It was proved by Grigor’yan andTelcs 2010, that sub-Gaussian estimate (3.5) is equivalent to the conjunction of thefollowing conditions:

(i) the locality, that is, the Hunt process Xt is a diffusion;

(ii) the volume regularity: µ (B (x, r)) Crα;(iii) the mean exit time regularity: ExτB(x,r) Crβ where τΩ is the first exit time ofXt from the set Ω;

(iv) the uniform elliptic Harnack inequality for positive harmonic functions.

Analogous result for stable-like estimate (that corresponds to a jump process) is stillmissing.

Let us discuss upper bound in (3.5). The necessary and sufficient conditions for theon-diagonal upper bound

pt (x, x) ≤ Ct−α/β (3.6)

(that is independent of Φ) are well-known. We present the equivalent conditions forthe upper bounds in (3.5) in terms of (3.6), volume growth, certain tail estimate of the

Bielefeld, 2012 33

heat kernel, and upper bound of the jump kernel J (x, y) of (E ,F). Namely, if J = 0that is, if (E ,F) is local, then the upper bound holds with sub-Gaussian function, andif J (x, y) ≤ C

d(x,y)α+β, then with the stable-like function. These results were proved

by Grigor’yan and Hu 2008 for the local case and by Grigor’yan, Hu, Lau 2011 fornon-local case.

Tomasz Grzywny

Estimates on the Poisson kernel of a half-line for subordinate Brownianmotions

Let Tt be a subordinator i.e. an increasing Lévy process starting from 0. The Laplacetransform of a subordinator is of the form Ee−λTt = e−tψ(λ), λ ≥ 0, where ψ is called theLaplace exponent of T . ψ is a Bernstein function and has the following representation:

ψ(λ) = a+ bλ+

∫(0,∞)

(1− e−λu)µ(du),

where a, b ≥ 0 and µ is a Lévy measure on (0,∞). Let Bt be a Brownian motion in R.

Definition 1. We say that a function f satisfies USC(α) if there exist constants β < αand σ > 0 such that, for all x, λ ≥ 1:

|f(λx)| ≤ σλβ|f(x)|.

We say that a function f satisfies LSC(α) if there exist constants β > α and σ > 0such that, for all x, λ ≥ 1:

|f(λx)| ≥ σλβ|f(x)|.

The purpose of this talk is to present some recent results about subordinate Brownianmotions BTt on R. We give new forms of estimates for the Lévy and potential densityof the subordinator near zero. For example:

Theorem 1. Let µ(du) = µ(u)du, where µ(u) is a non-increasing function. Then thefollowing conditions are equivalent.(i) There exist constants c, t0 such that µ(t) ≤ cµ(2t), for t ≤ t0.(ii) ψ(n) satisfies USC(0), for some n ∈ N.(iii) µ(t) ≈ |ψ(n)|(t−1)

tn+1 , for t ≤ t1 and some n ∈ N.

These results provide us to find estimates for the Lévy and potential density of the


subordinate Brownian motion BTt near origin. Next we show the asymptotic behaviourof the derivative of the renewal function V ′ of the ascending ladder-height process.Using these results we find estimates for the Poisson kernel of a half-line. In particularwe get the following estimates.

Theorem 2. Let ψ be a complete Bernstein function and ψ(∞) = ∞. If ψ′ satisfiesUSC(0) and ψ′

ψ1/2 satisfies LSC(−3/2), then for x, |z| < 1 we have

P(0,∞)(x, z) ≈V (x)

V (|z|)V ′(x− z)V (x− z) ≈

ψ1/2(|z|−2)

ψ1/2(x−2)

ψ′((x− z)−2)

(x− z)3ψ((x− z)−2).

[1] T. Grzywny & M. Ryznar: Potential theory of one-dimensional subordinate Brownianmotions. Preprint (2012)

[2] A. Mimica & P. Kim: Harnack inequalities for subordinate Brownian motions. Electron.J. Probab. 17:no. 37, 1–23 (2012)

[3] P. Kim, R. Song & Z. Vondraček: Potential theory of subordinate Brownian mo-tions revisited. Interdisciplinary Mathematical Sciences - Vol. 13, Stochastic Analysis andApplications to Finance (2012)

Peter Imkeller

Construction of stochastic processes with singular jump characteristics assolutions of martingale problems

We construct Lévy processes with discontinuous jump characteristics in form of weaksolutions of appropriate stochastic differential equations, or related martingale prob-lems with non-local operators. For this purpose we prove a general existence theoremfor martingale problems in which a sequence of operators generating Feller processesapproximates an operator with a range containing discontinuous functions. The ap-proach crucially depends on uniform estimates for the probability densities of the ap-proximating processes derived from properties of the associated symbols. The theoremis applicable to stable like processes with discontinuous stability index.

This talk is based on work with N. Willrich (WIAS Berlin).

Bielefeld, 2012 35

Niels Jacob

Non-locality, Non-Isotropy, and Geometry

Many non-local operators, for example many generators of Lévy or Lévy-type processes,are also non-isotropic, i.e. they have a symbol which grows in the co-variable differentlyin different directions, and in addition, in general no reasonable notion of a principalsymbol exist. This rules out standard geometric considerations when dealing with theseoperators. However, these symbols often define their own natural geometry which issuitable to study these operators and their off-springs such as associated semi-groupsor stochastic processes.

Naotaka Kajino

Non-regularly varying and non-periodic oscillation of the on-diagonal heatkernels on self-similar fractals

The purpose of this talk is to present the author’s result in a recent preprint [2] onon-diagonal oscillatory behavior of the canonical heat kernels on self-similar fractals.Note that this talk is on diffusions on fractals and has nothing to do with non-localoperators.

Let K be either a nested fractal or a generalized Sierpiński carpet, which is a compactsubset of the Euclidean space, and let pt(x, y) be the transition density of the Brownianmotion on K. Then it is well-known that there exist c1, c2 ∈ (0,∞) and ds ∈ [1,∞)such that for any x ∈ K,

c1 ≤ tds/2pt(x, x) ≤ c2, t ∈ (0, 1].

The exponent ds is called the spectral dimension of K. Then it is natural to ask howtds/2pt(x, x) behaves as t ↓ 0 and in particular whether the limit

limt↓0

tds/2pt(x, x) (3.7)

exists. WhenK is a nested fractal, the author has proved in [1] that the limit (3.7) doesnot exist for “generic" (hence almost every) x ∈ K under very weak assumptions on K.The proof of this fact, however, heavily relied on the two important features of nestedfractals — they are finitely ramified (i.e. can be made disconnected by removing finitelymany points) and highly symmetric. In particular, the result of [1] is not applicable togeneralized Sierpiński carpets, which are infinitely ramified.

The main results of [2] have overcome this difficulty by a completely different method,


thereby establishing the non-existence of the limit (3.7) for “generic" x ∈ K when Kis an arbitrary generalized Sierpiński carpet. More strongly, we have the followingassertion under a quite general setting of a self-similar Dirichlet form on a self-similarset K.

[ (NRV)]p(·)(x, x) does not vary regularly at 0 for “generic" x ∈ K, if

lim supt↓0

pt(y, y)

pt(z, z)> 1 for some y, z ∈ K \ V0. (3.8)

[ (NP)]“Generic" x ∈ K does not admit a periodic function G : R → R suchthat

pt(x, x) = t−ds/2G(− log t) + o(t−ds/2) as t ↓ 0, if (3.9)

lim inft↓0

pt(y, y)

pt(z, z)> 1 for some y, z ∈ K \ V0. (3.10)

We will also see that the conditions (3.8) and (3.10) can be easily verified for most(though not all) typical self-similar fractals.

[1] N. Kajino: On-diagonal oscillation of the heat kernels on post-critically finite self-similarfractals. Probability Theory and Related Fields, in press (2012)http://dx.doi.org/10.1007/s00440-012-0420-9

[2] N. Kajino: Non-regularly varying and non-periodic oscillation of the on-diagonal heatkernels on self-similar fractals. Preprint (2012)

Panki Kim

An Lp-theory of stochastic PDEs with random fractional Laplacian operator

In this talk, we introduce an Lp-theory of a class of parabolic stochastic equations withrandom fractional Laplacian operator. The driving noises of the equations are generalLevy processes. Uniqueness and existence results in Sobolev spaces will be introduced.

This is a joint work with Kyeong-Hun Kim.

Bielefeld, 2012 37

http://dx.doi.org/10.1007/s00440-012-0420-9

Victoria Knopova

Parametrix construction for the transition probability density of some Lévy-type processes

The talk is devoted to the parametrix construction of the fundamental solution to theequation

∂

∂tu(t, x) = L(x,D)u(t, x), t > 0, x ∈ R, (3.11)

where for v ∈ C∞0 (R)

L(x,D)v(x) :=

∫R

(u(x+ y)− u(x))m(x, y)µ(dy),

µ is a Lévy measure, and m(x, y) satisfies some mild regularity assumptions. Weshow that the fundamental solution to (3.11) can be constructed by Levi’s parametrixmethod, and derive the estimates for this solution.

The talk is based on the on-going research work with Alexei Kulik.

Yuri Kondratiev

Non-local evolutions as kinetic equations for Markov dynamics

Non-linear PDEs are widely used in phenomenological models of complex systems. Asan important example we may mention reaction-diffusion equations (RDE)

∂u

∂t= 4u+ f(u), u = u(t, x)

which we meet in combustion theory, bacterial growth, nerve propagation, epidemiol-ogy, genetics etc. A particular case is the celebrated Fisher equation corresponding tof(s) = s(1− s).We observe growing attention to non-local versions of RDE:

∂u

∂t= J ∗ u− u+ f(u), u = u(t, x).

u(0, x) = φ(x), x ∈ Rd,where

0 ≤ J ∈ L1, ‖J‖1 = 1


is a jump kernel and non-linearity f may be local or non-local as well. Here are fewreferences: Coville, Dupaigne (2008), Ignat, Rossi (2010), Berestycki, Nadin, Ryzhik(2009), Pan, Li, Lin (2009), Zhang, Li, Sun (2010). Especially, I would like to mentionrecent monograph "Nonlocal Diffusion Problems", F. Andreu-Vaillo et. al., AMS ,MSM v. 165 (2010). Actually, such kind of equations was introduced by Kolmogorov,Petrovsky and Piskunov in 1937 as a way to derive Fisher equation.

The aim of the presented talk is to describe how non-linear non-local evolutional equa-tions appear as kinetic equations for interacting particle systems. More precisely, wewill show that several models of stochastic dynamics for continuous systems (micro-scopic level) in a scaling limit (Vlasov, Lebowitz-Penrose, etc.) may be described bymeans of so-called kinetic equations (mesoscopic level) for the particle density. Theseequations contain (as a rule) certain convolutional terms which may enter as non-localdiffusion generators as well as in other quite different forms.

Mateusz Kwaśnicki

Estimates of harmonic functions for non-local operators

I will present the results of my recent work with Krzysztof Bogdan and Takashi Ku-magai, contained in [1]. We consider a non-local operator A, which generates a Fellersemigroup on a metric measure space X . For example, A can be a fractional powerof the Laplace operator (on a Riemannian manifold or on a fractal set), or a pseudo-differential operator with sufficiently smooth coefficients. We prove local supremumestimate for nonnegative subharmonic functions in a ball. Next, we show the boundaryHarnack inequality (BHI).

More precisely, let∫

(f(y)− f(x))ν(x, y)m(dy) be the non-local part of A. We assumethat ν(x, y) = ν(y, x) is bounded above and below by a positive constant on everycompact subset of X × X \ diagX . The same condition is imposed on the kernel ofthe resolvent operator (1 − A)−1. Furthermore, we require that A generates a Fellerand strong-Feller semigroup, and that the C0 domain of A contains bump functions:functions equal to 1 on a given compact set and vanishing outside a given (larger) openset. Under these assumptions,

supB′

f ≤ c∫

(B′)c

f(z)ν(x0, z)m(dz) (3.12)

for all globally nonnegative functions f subharmonic (with respect to A) in the ballB = B(x0, R). Here B′ = B(x0, r), 0 < r < R and c = c(x0, r, R). The supremum

Bielefeld, 2012 39

estimate (3.12) is used to prove BHI: for any open D,

f(x)

g(x)≤ c f(y)

g(y)(x, y ∈ B′ ∩D) (3.13)

holds for all globally nonnegative functions f , g harmonic (with respect to A) in B∩D,vanishing in B\D and continuous on B∩∂D. Here again B = B(x0, R), B′ = B(x0, r),0 < r < R and c = c(x0, r, R).

Our results cover also the non-symmetric case under an appropriate duality assump-tion. They extend previous works on BHI for non-local operators in several directions,and are new even for many pseudo-differential operators with constant coefficients.Probabilistic methods are used in the proof of (3.12).

[1] K. Bogdan, T. Kumagai & M. Kwaśnicki: Boundary Harnack inequality for Markovprocesses with jumps. In preparation

Richard Lehoucq

Peridynamic nonlocal mechanics

The peridynamic balance of linear momentum was postulated by Silling [1] to allow theconsideration of discontinuous motion, or deformation. The nonlocal force and energydensities are given by ∫

Ω

(t(x,x′)− t(x′,x)

)dx′,

∫Ω

(t(x,x′) · v(x, t)− t(x,x′) · v(x′, t)

)dx′ −

∫Ω

(h(x,x′)− h(x′,x)

)dx′,

and are the nonlocal analogues for the classical force and energy densities

∇ · σ, ∇ · (σv)−∇ · q.

The integral operators sum forces and power expenditures among volumes separated bya finite distance and so represent nonlocal interaction. This is in contrast to the classicaldensities in which interaction only occurs between volumes in direct contact—the in-teraction is therefore deemed local. The integral operators obviate special treatment atpoints of discontinuity conventionally employed because spatial derivatives are avoided.The resulting balance laws extend the classical theory of continuum mechanics to allowfor jumps.

My talk first introduces the nonlocal balances of linear momentum and energy [2]. Acrucial aspect is that the integrands are antisymmetric with respect to an interchange


of x and x′. This is a necessary and sufficient condition for the balance laws to be ad-ditive, an action-reaction principle to hold, and that the nonlocal flux is an alternatingform [3]. The nonlocal balance laws may be derived using the principles of statisticalmechanics [4].

Nonlocal constitutive relations naturally arise when the gradient operator is replacedwith its nonlocal analogue. This generalized notion of kinematics is what enables dis-continuous motion to be modeled, leads to well-posed balance laws and thermodynamicrestriction so that the second law of thermodynamics is not violated. My presentationends with a review of an emerging mathematical theory [5] for nonlocal diffusion [6]and the peridynamic Navier equation. Critical to this analysis is the introduction ofvolume constraints, the nonlocal analogue of boundary conditions. These constraintsenable the use of function spaces where a trace operator may not be defined. The math-ematical analysis is also facilitated by the introduction of a nonlocal vector calculus.

[1] S. A. Silling: Reformulation of elasticity theory for discontinuities and long-range forces.J. Mech. Phys. Solids 48(1):175–209 (2000)

[2] S. A. Silling & R. B. Lehoucq: Peridynamic Theory of Solid Mechanics. Advances inApplied Mechanics 44 (2010)

[3] Q. Du, M. Gunzburger, R. B. Lehoucq & K. Zhou: A nonlocal vector calcu-lus, nonlocal volume-constrained problems, and nonlocal balance laws. To appear in inM3AS:Mathematical Models and Methods in Applied Sciences

[4] R. B. Lehoucq & M. P. Sears: Statistical mechanical foundation of the peridynamicnonlocal continuum theory: Energy and momentum conservation laws. Phys. Rev. E84:031112 (2011)

[5] E. Emmrich & R. B. Lehoucq: Peridynamics: a nonlocal continuum theory. To appearin the proceedings of the Sixth International Workshop Meshfree Methods for PartialDifferential Equations

[6] Q. Du, M. Gunzburger, R. B. Lehoucq & K. Zhou: Analysis and approximation ofnonlocal diffusion problems with volume constraints. To appear in the SIAM review

Mark Meerschaert

Solutions to a nonlocal fractional wave equation

A nonlocal fractional wave equation with attenuation has been proposed to modelsound wave conduction in heterogeneous media. This equation has an exact analyticalsolution written in terms of stable densities. The solution is causal only when the stableindex is less than one. However, applications to medical ultrasound require a stableindex between one and two. In this talk, a stochastic model will be presented to explainthe appearance of the stable density in the solution, and an alternative causal solution

Bielefeld, 2012 41

will be developed, based on the hitting time of a positively skewed stable Lévy motionwith drift. Finally, some open problems will be discussed, including alternatives to thestable density, anisotropic media, and model coefficients that vary in space.

[1] M. Caputo: Linear models of dissipation whose Q is almost frequency independent. II.Fract. Calc. Appl. Anal. 11(1):4–14 (2008), reprinted from Geophys. J. R. Astr. Soc. 13(1967), no. 5, 529–539

[2] W. Chen & S. Holm: Fractional Laplacian time-space models for linear and nonlinearlossy media exhibiting arbitrary frequency power-law dependency. J. Acoust. Soc. Am.(2004)

[3] J. F. Kelly, R. J. McGough & M. M. Meerschaert: Analytical time-domain Green’sfunctions for power-law media. J. Acoust. Soc. Am. 124(5):2861–2872 (2008)

[4] M. Meerschaert, P. Straka, Y. Zhou & R. McGough: Stochastic solution to atime-fractional attenuated wave equation. Submitted.http://www.stt.msu.edu/users/mcubed/StochWave.pdf

[5] T. Szabo: Causal theories and data for acoustic attenuation obeying a frequency power-law. J. Acoust. Soc. Am. 97:14–24 (1995)

Ralf Metzler

Ageing and Ergodicity Breaking in Anomalous Diffusion

In 1905 Einstein formulated the laws of diffusion, and in 1908 Perrin published hisNobel-prize winning studies determining Avogadro’s number from diffusion measure-ments. With similar, more refined techniques the diffusion behaviour in complex sys-tems such as the motion of tracer particles in living biological cells or the tracking ofanimals and humans is nowadays measured with high precision. Often the diffusionturns out to deviate from Einstein’s laws. This talk will discuss the basic mechanismsleading to such anomalous diffusion as well as point out its consequences. In partic-ular the unconventional behaviour of non-ergodic, ageing systems will be discussedwithin the framework of continuous time random walks. Indeed, non-ergodic diffusionin the cytoplasm of living cells as well as in membranes has recently been demonstratedexperimentally.

Ante Mimica


http://www.stt.msu.edu/users/mcubed/StochWave.pdf

Regularity estimates of harmonic functions for jump processes

In this talk we discuss probabilistic methods of proving regularity of harmonic functionsfor a class of non-local operators.

Consider a kernel n : Rd × (Rd \ 0) → [0,∞) that is symmetric and such that thereis a cone V ⊂ Rd whose apex is at origin such that for any x ∈ Rd

n(x, h) ≥ |h|−d−α`(|h|), h ∈ V .

Here α ∈ (0, 2) and ` : (0,∞) → (0,∞) is bounded away from zero and varies slowlyat zero, i. e.

limr→0+

`(λr)

`(r)= 1 for all λ > 0 .

Furthermore, we assume that for some κ > 1,

n(x, h) ≤ κ|h|−d−α`(|h|) for all h ∈ Rd, h 6= 0 .

We can associate a non-local operator L to the kernel n:

Lf(x) =

∫Rd\0

(f(x+ h)− f(x)− 〈∇f(x), h〉1|h|≤1

)n(x, h) dh, f ∈ C2

b (Rd).

Assume that there exists a strong Markov process X = Xt,Pxx∈Rd,t≥0 with pathsthat are right-continuous with left limits that is associated to L in the following sense:for any f ∈ C2

b (Rd) and x ∈ Rd, the stochastic processf(Xt)− f(x)−t∫

0

Lf(Xs) ds

t≥0

is a Px-martingale .

A bounded function u : Rd → R is said to be harmonic in an open set D ⊂ Rd if forany open set B ⊂ Rd such that B ⊂ D and any x ∈ B, the process f(Xt)t≥0 is aPx-martingale.

In this case we have the following a-priori regularity estimates of harmonic functions:there exist constants c > 0 and γ ∈ (0, 1) such that for all r ∈ (0, 1), x0 ∈ Rd and allfunctions u that are harmonic in the ball Br(x0), the following holds

|u(x)− u(y)| ≤ c‖u‖∞( |x− y|

r

)γfor all x, y ∈ Br/2(x0).

We use method of Bass and Levin (see [1]) which is based on an estimate of Krylov-Safonov-type. We show how to prove such an estimate in our case.

Bielefeld, 2012 43

The case α = 0 is also discussed. In this case it is not clear whether Hölder continuityestimates holds. Nevertheless, something can be said about the modulus of continuityof harmonic functions for some classes of non-local operators/stochastic processes.

The talk is based on [2, 3, 4].

[1] R. F. Bass & D. A. Levin: Harnack inequalities for jump processes. Potential Anal.17(4):375–388 (2002)

[2] A. Mimica & M. Kassmann: Analysis of jump processes with nondegenerate jumpingkernels.http://arxiv.org/abs/1109.3678

[3] A. Mimica & P. Kim: Harnack inequalities for subordinate Brownian motions. Electron.J. Probab. 17:no. 37, 1–23 (2012)

[4] A. Mimica: On harmonic functions of symmetric Lévy processes. To appear in Ann. Inst.H. Poincaré Probab. Statist. (2012)http://arxiv.org/abs/1109.3676

Giampiero Palatucci

Asymptotics of the s-perimeter as s 0

Given s ∈ (0, 1) and a bounded open set Ω ⊂ Rn, the s-perimeter of a (measurable)set E ⊆ Rn in Ω is defined as

Pers(E; Ω) := L(E ∩ Ω, (CE) ∩ Ω)

+ L(E ∩ Ω, (CE) ∩ (CΩ)) + L(E ∩ (CΩ), (CE) ∩ Ω),(3.14)

where CE = Rn \ E denotes the complement of E, and L(A,B) denotes the followingnonlocal interaction term

L(A,B) :=

∫A

∫B

1

|x− y|n+sdx dy ∀A,B ⊆ Rn.

This notion of s-perimeter and the corresponding minimization problem were intro-duced in [4] (see also [14, 15], where some functionals related to the one in (3.14) havebeen analyzed in connection with fractal dimensions).

Recently, the s-perimeter has inspired a variety of literature in different directions,both in the pure mathematical settings (for instance, as regards the regularity of sur-faces with minimal s-perimeter, see [7, 1, 3, 13]) and in view of concrete applications(such as phase transition problems with long range interactions, see [5, 11, 12]). Ingeneral, the nonlocal behavior of the functional is the source of major difficulties, con-




ceptual differences, and challenging technical complications. We refer to [9, 8] for anintroductory review on this subject.

The limits as s 0 and s 1 are somehow the critical cases for the s-perimeter, sincethe functional in (3.14) diverges as it is. Nevertheless, when appropriately rescaled,these limits seem to give meaningful information on the problem. In particular, it wasshown in [6, 2] that (1− s)Pers approaches the classical perimeter functional as s 1(up to normalizing multiplicative constants), and this implies that surfaces of minimals-perimeter inherit the regularity properties of the classical minimal surfaces for ssufficiently close to 1 (see [7]).

As far as we know, the asymptotic as s 0 of sPers was not studied yet (see how-ever [10] for some results in this direction), and this is the question that we would liketo address in this talk. That is, we are interested in the quantity

µ(E) := lims0

sPers(E; Ω)

whenever the limit exists.

We will prove necessary and sufficient conditions for the existence of such limit, byalso providing an explicit formulation in terms of the Lebesgue measure of E and Ω.Moreover, we will construct examples of sets for which the limit does not exist.

Work in collaboration with S. Dipierro, A. Figalli and E. Valdinoci. Available athttp://arxiv.org/abs/1204.0750

[1] M. C. Caputo & N. Guillen: Regularity for non-local almost minimal boundaries andapplications.http://arxiv.org/abs/1003.2470

[2] L. Ambrosio, G. De Philippis & L. Martinazzi: Gamma-convergence of nonlocalperimeter functionals. Manuscripta Math. 134(3-4):377–403 (2011)

[3] B. Barros Barrera, A. Figalli & E. Valdinoci: Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces.http://arxiv.org/abs/1202.4606v1

[4] L. Caffarelli, J.-M. Roquejoffre & O. Savin: Nonlocal minimal surfaces. Comm.Pure Appl. Math. 63(9):1111–1144 (2010)

[5] L. A. Caffarelli & P. E. Souganidis: Convergence of nonlocal threshold dynamicsapproximations to front propagation. Arch. Ration. Mech. Anal. 195(1):1–23 (2010)

[6] L. Caffarelli & E. Valdinoci: Uniform estimates and limiting arguments for nonlocalminimal surfaces. Calc. Var. Partial Differential Equations 41(1-2):203–240 (2011)

[7] L. Caffarelli & E. Valdinoci: Regularity properties of nonlocal minimal surfaces vialimiting arguments. Preprinthttp://www.ma.utexas.edu/mp_arc-bin/mpa?yn=11-69

[8] E. D. Nezza, G. Palatucci & E. Valdinoci: Hitchhiker’s guide to the fractionalSobolev spaces. Bull. Sci. Math.

Bielefeld, 2012 45



http://arxiv.org/abs/1202.4606v1

http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=11-69

http://dx.doi.org/10.1016/j.bulsci.2011.12.004

[9] G. Franzina & E. Valdinoci: Geometric analysis of fractional phase transition inter-faces. Preprinthttp://cvgmt.sns.it/paper/1782/

[10] V. Maz’ya & T. Shaposhnikova: On the Bourgain, Brezis, and Mironescu theoremconcerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195(2):230–238 (2002)

[11] O. Savin & E. Valdinoci: Density estimates for a variational model driven by theGagliardo norm.http://arxiv.org/abs/1007.2114

[12] O. Savin & E. Valdinoci: Γ-convergence for nonlocal phase transitions.http://arxiv.org/abs/1007.1725

[13] O. Savin & E. Valdinoci: Regularity of nonlocal minimal cones in dimension 2.http://arxiv.org/abs/1202.0973

[14] A. Visintin: Nonconvex functionals related to multiphase systems. SIAM J. Math. Anal.21(5):1281–1304 (1990)

[15] A. Visintin: Generalized coarea formula and fractal sets. Japan J. Indust. Appl. Math.8(2):175–201 (1991)

Ilya Pavlyukevich

Small noise asymptotics of integrated Ornstein–Uhlenbeck processes drivenby α-stable Lévy processes

We study the behaviour of a one-dimensional integrated Ornstein–Uhlenbeck processdriven by an α-stable Lévy process of small amplitude. We show that the continuousintegrated Ornstein–Uhlenbeck process converges to the driving càdlàg α-stable Lévyprocess in the Skorokhod M1-topology. In particular this allows us to determine thelimiting distribution of its first passage times.

This is a joint work with R. Hintze (FSU Jena).

Enrico Priola

Uniqueness for singular SDEs driven by stable processes


http://dx.doi.org/10.1016/j.bulsci.2011.12.004

http://cvgmt.sns.it/paper/1782/




In [2] the authors have established pathwise uniqueness for the following one dimen-sional SDE

dXt = b(Xt)dt+ dLt, X0 = x ∈ R, (3.15)

where b : R → R is continuous and bounded and L = (Lt) is a standard α-stableprocess with α ∈ [1, 2). They have also shown an example of non-uniqueness when0 < α < 1 and b is β-Hölder continuous and bounded with α+ β < 1.

Simultaneously with [2], the limit case α = 2 (i.e., when L is a Wiener process) hasbeen considered in [4] where uniqueness was proved even if b : R → R is only Boreland bounded (see also [3], [1] and the references therein for uniqueness results in moredimensions).

In this talk we show a multidimensional extension of [2] when b : Rn → Rn, n ≥ 1,is β-Hölder continuous and bounded and L is a non-degenerate symmetric α-stableprocess. The proof uses analytic regularity results at the level of the non-local Kol-mogorov equation plus an Itô-Tanaka trick which is related to the Zvonkin method. Wealso prove the stochastic flow property of the solutions and their differentiability withrespect to the initial data. In the final part possible generalizations of the previousresults will be considered as well.

[1] N. V. Krylov & M. Röckner: Strong solutions of stochastic equations with singulartime dependent drift. Probab. Theory Related Fields 131(2):154–196 (2005)

[2] H. Tanaka, M. Tsuchiya & S. Watanabe: Perturbation of drift-type for Lévy pro-cesses. J. Math. Kyoto Univ. 14:73–92 (1974)

[3] A. J. Veretennikov: Strong solutions and explicit formulas for solutions of stochasticintegral equations. Mat. Sb. (N.S.) 111(153)(3):434–452, 480 (1980)

[4] A. K. Zvonkin: A transformation of the phase space of a diffusion process that willremove the drift. Mat. Sb. (N.S.) 93(135):129–149, 152 (1974)

Michael Röckner

Sub- and supercritical stochastic quasi-geostrophic equation

Consider the 2D stochastic quasi-geostrophic equation on the torus T2 for generalparameter α ∈ (0, 1)

dΘ(t) + (−∆)αΘ(t)dt+ u(t) · ∇Θ(t)dt = G(t,Θ(t))dW (t),

u(t) = −∇⊥((−∆)12 Θ(t)),

G(0) = Θ0

Bielefeld, 2012 47

on the Hilbert space

H :=

Θ ∈ L2(T2) :

∫T2

Θ(ξ)dξ = 0.

The talk will give a survey on recent results on this equation. These include existenceof weak solutions for additive noise, existence of martingale solutions and Markovselections for multiplicative noise and under some condition pathwise uniqueness for allα ∈ (0, 1). Further- more, in the subcritical case α > 1

2 , we prove existence and unique-ness of (probabilistically) strong solutions. In addition, we prove ergodicity for α > 2

3 ,provided the noise is non-degenerate. In this case, the convergence to the (unique)invariant measure is exponentially fast. We establish the large deviation principle forthe stochastic quasi-geostrophic equations for α > 1

2 with small multiplicative noise.An analogous result is also obtained for the small time asymptotics. (These resultsare joint with Wei Liu.) The existence of a random at- tractor for the solutions ofthe stochastic quasi-geostrophic equation for α > 1

2 driven by real multiplicative noiseand additive noise is also established. Time permitting, we shall also report on a veryrecent result about ergodicity in the general subcritical case α > 1

2 and for degeneratenoise.

Joint work with Rongchan Zhu and Xiangchan Zhu.

Julio Rossi

A Monge-Kantorovich mass transport problem for a discrete distance

This talk is concerned with a Monge-Kantorovich mass transport problem in which inthe transport cost we replace the Euclidean distance with a discrete distance. We fixthe length of a step and the distance that measures the cost of the transport dependsof the number of steps that is needed to transport the involved mass from its originto its destination. For this problem we construct special Kantorovich potentials, andoptimal transport plans via a nonlocal version of the PDE-formulation given by Evansand Gangbo for the classical case with the Euclidean distance. We also study howthis problems, when rescaling the step distance, approximate the classical problem.In particular we obtain, taking limits in the reescaled nonlocal formulation, the PDE-formulation given by Evans-Gangbo for the classical problem.

Joint work with N. Igbida, J. M. Mazon and J. Toledo.


Enrico Scalas

Characterization of the fractional Poisson process

The fractional Poisson process (FPP) is a counting process with independent andidentically distributed inter-event times following the Mittag-Leffler distribution. Thisprocess is very useful in several fields of applied and theoretical physics including modelsfor anomalous diffusion. Contrary to the well-known Poisson process, the fractionalPoisson process does not have stationary and independent increments. It is not a Levyprocess and it is not a Markov process. I present formulas for its finite-dimensionaldistribution functions, fully characterizing the process. Some recent applications tofinance of these results are briefly discussed.

[1] M. Politi, T. Kaizoji & E. Scalas: Full characterization of the fractional Poissonprocess.http://arxiv.org/abs/1104.4234

[2] E. Scalas & M. Politi: A parsimonious model for intraday European option pric-ing. Economics Discussion Papers, No. 2012-14, Kiel Institute for the World Econ-omy. Discussion paper available at: http://www.economics-ejournal.org/economics/discussionpapers/2012-14

Hans-Peter Scheffler

Fractional governing equations for coupled continuous time random walks

In a continuous time random walk (CTRW), a random waiting time precedes eachrandom jump. The CTRW is coupled if the waiting time and the subsequent jumpare dependent random variables. The CTRW is used in physics to model diffusingparticles. Its scaling limit is governed by an anomalous diffusion equation. Someapplications require an overshoot continuous time random walk (OCTRW), where thewaiting time is coupled to the previous jump. This talk develops stochastic limit theoryand governing equations for CTRW and OCTRW. The governing equations involvecoupled space-time fractional derivatives. In the case of infinite mean waiting times,the solutions to the CTRW and OCTRW governing equations can be quite different.

Armin Schikorra

Bielefeld, 2012 49


http://www.economics-ejournal.org/economics/discussionpapers/2012-14

http://www.economics-ejournal.org/economics/discussionpapers/2012-14

Knot-energies and Fractional Harmonic Maps

We will present a proof that curves (knots) which are stationary for the Moebius energyare smooth in the critical dimension. This energy was introduced by O’Hara (Topology,30(2):241–247, 1991), and it was shown by Freedman, He and Wang (Ann. of Math.(2), 139(1):1–50, 1994), and He (Comm. Pure Appl. Math., 53(4):399–431, 2000) thatminimizers of this energy are smooth, using a geometric argument employing cruciallythe Moebius invariance of this energy.

Our approach, however, which shows regularity even for critical points, does notrely on this geometric invariance, but rather uses analytic methods from HarmonicAnalysis and Potential Theory inspired by the regularity arguments of fractional har-monic maps by Da Lio-Riviere (APDE, 4(1):149–190, 2011; Advances in Mathematics,227:1300–1348, 2011), and S. (J. Differential Equations, 252:1862–1911, 2012; Preprint,arXiv:1103.5203, 2011).

Joint work with S. Blatt and P. Reiter.

Russell Schwab

On Aleksandrov-Bakelman-Pucci type estimates for integro differential equa-tions

Despite much recent (and not so recent) attention to solutions of integro-differentialequations of elliptic type, it is surprising that a result such as a comparison theoremwhich can deal with only measure theoretic norms (e.g. L-n and L-infinity) of the righthand side of the equation has gone unexplored. For the case of second order equationsthis result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back tocirca 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f,the supremum of u is controlled by the L-n norm of f (n being the underlying dimen-sion of the domain). We will discuss this estimate in the context of fully nonlinearintegro-differential equations and present a recent result in this direction.

Joint work with Nestor Guillen, see http://arxiv.org/abs/1101.0279.

Yuichi Shiozawa

Conservation property of symmetric jump-diffusion processes



We say that a Markov process is conservative if the associated particle stays at thestate space forever. This property is one of important global path properties of Markovprocesses. In particular, there are many results on the conservativeness criterion ofsymmetric diffusion processes, in terms of the volume growth of the underlying measureand the growth of the “coefficient”, established by Grigor’yan, Davies, Ichihara, Takeda,Oshima, Sturm,....

Motivated by the recent progress on the analysis of jump processes, there also havebeen results on the conservativeness criterion of symmetric jump(-diffusion) processesgenerated by regular Dirichlet forms ([1, 2, 3, 4]). In [1, 2, 3], the volume of theunderlying measure is allowed to grow exponentially, but the coefficients are assumedto be bounded. In contrast with this, we allow in [4] the coefficients to be unbounded;however, since the explicit form of the L2-generator is needed for the proof, we assumethat the state space is Rd and the underlying measure is the Lebesgue measure on Rd.Furthermore, we also need the assumption on the “drift” parts which may entail thecontinuity on the coefficients.

A purpose in this talk is to establish a conservativeness criterion for symmetric jump-diffusion processes generated by regular Dirichlet forms, in terms of the volume growthof the underlying measure and the growth of the coefficients. Moreover, by using thiscriterion, we remove the conditions in [4] as we mentioned before. We also generalizethe results in [1, 2, 3, 4] so that we allow the volume of the underlying measure to growexponentially and the coefficients to be unbounded at the same time. We do not knowabout the sharpness of our criterion in general; however, we can show the sharpnessfor a class of time changed Dirichlet forms.

We finally give examples related to symmetric stable-like processes and censored stable-like processes.

Example(cf. [4, Example 3.1]) Fix α ∈ (0, 2). Let c(x, y) be a nonnegative Borelfunction on Rd × Rd \ diag such that c(x, y) = c(y, x) for any (x, y) ∈ Rd × Rd \ diag.Let m be a positive Radon measure on Rd such that, for some positive constants β > 0and C1, C2 with C2 > C1 > 0,

C1rβ ≤ m(Bx(r)) ≤ C2r

β for any x ∈ Rd and r > 0. (3.16)

We define

E(u, v) =1

2

∫∫Rd×Rd\diag

(u(x)− u(y))(v(x)− v(y))c(x, y)

|x− y|α+βm(dx)m(dy). (3.17)

We now assume the following on c(x, y).

For 0 < |x− y| < 1,

c(x, y) (1 + |x|2) log(2 + |x|) + (1 + |y|2) log(2 + |y|);

Bielefeld, 2012 51

For |x− y| ≥ 1,c(x, y) (1 + |x|2)p + (1 + |y|2)p

for some p ∈ [0, α/2).

By our result, (E ,F) is conservative.

[1] A. Grigor’yan, X. Huang & J. Masamune: On stochastic completeness of jumpprocesses. To appear in Math. Z.http://dx.doi.org/10.1007/s00209-011-0911-x

[2] J. Masamune & T. Uemura: Conservation property of symmetric jump processes. Ann.Inst. Henri Poincaré Probab. Stat. 47(3):650–662 (2011)

[3] J. Masamune, T. Uemura & J. Wang: On the conservativeness and the recurrence ofsymmetric jump-diffusion. Preprint

[4] YY. Shiozawa & T. Uemura: Explosion of jump-type symmetric dirichlet forms onRd;. To appear in J. Theoret. Probab.http://dx.doi.org/10.1007/s10959-012-0424-5

Yannick Sire

Small energy regularity for a fractional Ginzburg-Landau system

I will describe an epsilon-regularity result for a non local Ginzburg-Landau equationinvolving the fractional laplacian.

Joint work with V. Millot.

Igor Sokolov

Fractional Subdiffusion and Subdiffusion-Reaction equations: Physical Mo-tivation and Properties

The main topic of the talk is: How do kinetic equations with nonlocal operators (e.g.with time-fractional derivatives describing slowly fading memory of the system) emergewhen describing physical models. This understanding is important both in order toget an intuition about possible types of the corresponding equations and in order tosee strengths and limitations of the corresponding approaches. For this we first discussphysical models leading to subdiffusion, and consider the whole chain of reasoning and


http://dx.doi.org/10.1007/s00209-011-0911-x

http://dx.doi.org/10.1007/s10959-012-0424-5

approximations leading from the physical model of a particle (say, a charge carrier)in a random potential field via continuous-time random walks and generalized mas-ter equation to the time-fractional subdiffusion equation. We moreover discuss howour intuition based on the time-local (in this case – Markovian) behavior gets uselesswhen passing to systems with long-time memory. We do this considering an exampleof derivation of the subdiffusive-reaction equations, which are found to contain non-linearity in the memory kernel depending on reaction, a property which is absent inmemoryless variants of the theory. The properties of the corresponding equations andtheir solutions are discussed using the simplest examples of the isomerization (A⇒ B)reaction and of the autocatalytic conversion (A + B ⇒ 2B) reaction (a nonlocal vari-ant of the FKPP-equation), the last one leading to a very peculiar front propagationproperties.

Renming Song

Stability of Dirichlet heat kernel estimates of non-local operators underperturbations

Recently, sharp two-sided Dirichlet heat kernel estimates have been obtained for severalclasses of discontinuous processes (or non-local operators), including symmetric stableprocesses, censored stable processes, relativistic stable processes and mixtures of stableprocesses. In this talk I will present some results on the stablility of Dirichlet heatkernel estimates of non-local operators under gradient perturbations and Feynman-Kac perturbations.

I will first present stability results for the Dirichlet heat kernel estimates of symmetricα-stable processes, α ∈ (1, 2), under gradient perturbations. A Radon measure µ onRd is said to be in the Kato class Kd,α−1 if

limr→0

supx∈Rd

∫B(x,r)

|ν|(dy)

|x− y|d+1−α = 0.

For µ = (µ1, · · · , µd) with µj ∈ Kd,α−1, we define an α-stable process with drift µ as aweak solution of an SDE. We can show that this SDE has a unique solution and, whenD is a C1,1 open set, the Dirichlet heat kernel of the α-stable process with drift µ inD is comparable that of the α-stable process in D.

Then I will present stability results for the Dirichlet heat kernels of (not necessarilysymmetric) Hunt processes under nonlocal Feynman-Kac perturbations. Our assump-tions on the nonlocal Feynman-Kac perturbation are also Kato class type conditions.

The results presented are contained in the 4 references.

[1] Z.-Q. Chen, P. Kim & R. Song: Dirichlet heat kernel estimates for fractional Laplacian

Bielefeld, 2012 53

with gradient perturbation. To appear

[2] Z.-Q. Chen, P. Kim & R. Song: Stability of Dirichlet heat kernel estimates for non-localoperators under Feynman-Kac perturbation. Preprint

[3] P. Kim & R. Song: Stable process with singular drift. Preprint

[4] P. Kim & R. Song: Dirichlet heat kernel estimates for stable processes with singulardrifts in unbounded C1,1 open sets. Preprint


Paweł Sztonyk

Upper estimates of transition densities for stable-dominated semigroups

We consider Feller semigroups with symmetric jump intensity f(x, y) dominated bythat of the rotation invariant stable Lévy process, i.e.,

f(x, y) ≤M φ(|y − x|)|y − x|α+d

, x, y ∈ Rd, y 6= x,

for a positive constant M and a bounded function φ(s) satisfying some additionalassumptions. The assumptions are satisfied for example by φ(s) =

(em(1−sγ) ∧ 1

)(1 ∨

s)β , where γ ∈ (0, 1] and β ∈ (−∞, d/2 + α− 1/2), m > 0.

For the semigroup of operators Pt, t ≥ 0 with the generator

Aϕ(x) = limε↓0

∫|y−x|>ε

(ϕ(y)− ϕ(x)) f(x, y) dy,

we obtain the following estimate.

There exists p : (0,∞)× Rd × Rd → [0,∞) such that

Ptϕ(x) =

∫Rd

ϕ(y)p(t, x, y) dy, x ∈ Rd, t > 0, ϕ ∈ C∞(Rd),

andp(t, x, y) ≤ C1e

C2t min

(t−d/α,

tφ(|y − x|)|y − x|α+d

), x, y ∈ Rd, t > 0.

Suleyman Ulusoy

Non-local Conservation laws and related Keller-Segel type Systems

Non-local conservation laws have gained a lot of attention recently. In this talk, wewill briefly discuss our proposed Levy mixed hyperbolic-parabolic equations. If timepermits, our recent findings on a related Keller-Segel type system will also be intro-duced.

Joint work with Kenneth H. Karlsen.

Bielefeld, 2012 55

Enrico Valdinoci

Nonlocal nonlinear problems

I would like to discuss some questions related to some semilinear equations driven by anonlocal elliptic operator. As a specific example, we consider the fractional Allen-Cahnequation

(−∆)su = u− u3, s ∈ (0, 1),

in which the classical Laplace operator is replaced by a fractional Laplacian.

Particular interest will be devoted to the qualitative properties of the solutions, suchas:

symmetry (e.g., whether or not a global, bounded, monotone solution dependsonly on one Euclidean variable),

density estimates of the level sets (e.g., what measure of space is occupied by|u| < 1/2),asymptotic behaviors (e.g., the limit properties of the blow-up uε(x) = u(x/ε)and the corresponding Γ-convergence issues).

The limit interfaces of these problems are related to both the local and the nonlocalperimeter functionals, depending on whether s ∈ [1/2, 1) or s ∈ (0, 1/2). On this topic,I would like to discuss some rigidity and regularity results (for instance, regularity ofs-minimizers for any s ∈ (0, 1/2) in dimension n = 2, and for any s ∈ ((1/2)− ε0, 1/2)in dimension n ≤ 7). Some open problems will be presented as well.

Alexis Vasseur

Integral variational problems

We will present, in this talk, the proof of the existence of classical solutions for a classof non-linear integral variational problems. Those types of equations are typically usedin nonlocal image and signal processing. They involve nonlinear versions of fractionaldiffusion operators. The method is based on De Giorgi-Nash-Moser techniques. Itextends to fully nonlinear settings a previous work on the regularity of solutions to theSurface Quasi-Geostrophic equations (SQG).

Joint work with L. Caffarelli and Ch.-H. Chan.


Zoran Vondraček

Potential theory of subordinate Brownian motions with Gaussian compo-nents

In this talk I will look at a subordinate Brownian motion with a Gaussian componentand a rather general discontinuous part. The assumption on the subordinator is thatits Laplace exponent is a complete Bernstein function with a Levy density satisfying acertain growth condition near zero. The main result that I will present is a boundaryHarnack principle with explicit boundary decay rate for non-negative harmonic func-tions of the process in C1,1 open sets. I will also discuss an example showing that theboundary Harnack principle fails for processes with finite range jumps. As a conse-quence of the boundary Harnack principle, one can establish sharp two-sided estimateson the Green function of the subordinate Brownian motion in any bounded C1,1 openset D and identify the Martin boundary of D with respect to the subordinate Brownianmotion with the Euclidean boundary.

Joint work with Panki Kim and Renming Song.

Bielefeld, 2012 57

Poster Presentations

Ishak Derrardjia

Fixed point techniques and stability for neutral nonlinear differential equa-tions with unbounded delay

We use the contraction mapping theorem to obtain stability results of the scalar non-linear neutral differential equation with functional delay

x′(t) = −ax(t) + b(t)x2(t− r(t)) + c(t)x(t− r(t))x′(t− r(t)).

[1] T. A. Burton: Liapunov functionals, fixed points, and stability by Krasnoselskii’s theo-rem. Nonlinear Stud. 9(2):181–190 (2002)

[2] T. A. Burton: Stability and periodic solutions of ordinary and functional-differentialequations, volume 178 of Mathematics in Science and Engineering. Academic Press Inc.,Orlando, FL (1985)

[3] T. A. Burton: Stability by fixed point theory or Liapunov theory: a comparison. FixedPoint Theory 4(1):15–32 (2003)

[4] T. A. Burton & T. Furumochi: Fixed points and problems in stability theory forordinary and functional differential equations. Dynam. Systems Appl. 10(1):89–116 (2001)

Luz Roncal

Fractional Laplacian on the torus

We consider the fractional powers of the Laplacian (−∆T)σ/2, 0 < σ < 2, on thetorus T. This operator can be defined in the obvious way by using Fourier series.We are interested in obtaining pointwise formulas for (−∆T)σ/2f(x) as well as Hölderestimates and interior and boundary Harnack’s inequalities.

To deal with these problems we use a novel approach to fractional operators based onthe application of the language of semigroups, as introduced by Stinga and Torrea.Indeed, by using the Poisson semigroup on the torus, we derive integro-differentialpointwise formulas (avoiding the computation of inverse Fourier series) and regularityproperties on Hölder spaces. Clearly, this semigroup method permits us to see thatthe fractional Laplacian on the torus is a nonlocal operator.

Bielefeld, 2012 59

On the other hand, we generalize the Caffarelli-Silvestre extension problem to fractionalpowers of second order differential operators Lγ , for any noninteger positive γ. Againthe semigroup language allows us to write an explicit formula for the solution of theextension which involves the heatdiffusion semigroup e−tL. Our result extends theone obtained by Stinga and Torrea for 0 < γ < 1. With this we prove interior andboundary Harnack’s inequalities for (−∆T)σ/2 .

Joint work with P. R. Stinga.

Pablo Raúl Stinga

Harnack’s inequality for fractional nonlocal equations

We show Harnack’s inequalities for solutions to fractional nonlocal equations of theform

Lσu = 0, u ≥ 0, in Ω ⊆ Rn,

where Lσ, 0 < σ < 1, is the fractional power of a Laplacian L. Our examples includesecond order divergence form elliptic operators with potentials; the radial Laplacianor, more generally, Bessel operators; the Laplacian on bounded domains and also oper-ators arising in classical orthogonal expansions (Hermite, Laguerre and ultrasphericaloperators).

The main idea is to apply a novel point of view of the Caffarelli–Silvestre extensionproblem of [1] for the fractional Laplacian. This new viewpoint was introduced byStinga and Torrea in [3]. It consists in the use of the language of semigroups, theprincipal tool being the heat-diffusion semigroup e−tL. Such a general method appliesto any Laplacian L. In this way we can take advantage of local methods, like Gutiérrez’sHarnack inequality for degenerate Schrödinger equations of [2], to obtain our results.

[1] L. Caffarelli & L. Silvestre: An extension problem related to the fractional Lapla-cian. Comm. Partial Differential Equations 32(7-9):1245–1260 (2007)

[2] C. E. Gutierrez: Harnack’s inequality for degenerate Schrödinger operators. Trans. Am.Math. Soc. 312(1):403–419 (1989)

[3] P. R. Stinga & J. L. Torrea: Extension problem and Harnack’s inequality for somefractional operators. Comm. Partial Differential Equations 35(11):2092–2122 (2010)


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Bielefeld, 2012 65

Here is a list of pubs that we recommended to the participants of the summer school. For specificrecommendations of restaurants please contact your colleagues from Bielefeld or the ZiF-staff.

Accumulation Points

• Corner Arndstr./Große-Kurfürsten-Str.5mins walking distance from the tram station “Siegfriedplatz” and the central station “Haupt-bahnhof”

Wunderbar, Arndtstraße 21Bar & Restaurant

Café Berlin, Große-Kurfürsten-Str. 65Café & Restaurant

Desperado, Arndtstr. 20Bar

Westside Lounge, Arndtstr. 18(Cocktail-)Bar & Restaurant

• Neues BahnhofsviertelWithin spitting distance on the North side of the central station “Hauptbahnhof”. Beside thebars, clubs and restaurants listed below there are also a cinema and a bowling center located inthis area.

Edelweiß, Boulevard 1(Cocktail-)Bar & Restaurant

Mexim’s, Ostwestfalenplatz 2(Cocktail-)Bar & Mexican Restaurant

Stereo, Boulevard 1Club

La Pampa, Boulevard 4Steakhouse

Puccini’s, Boulevard 4Italian Restaurant

Wok and Roll, Boulevard 5Asian Restaurant

• Arndtstr. (Downtown)5mins walking distance from the tram station “Jahnplatz” and the central station “Hauptbahnhof”

Mokkaklatsch, Arndtstr. 11(Cocktail-)Bar

Nichtschwimmer, Arndtstr. 6(Cocktail-)Bar & Restaurant

Las Tapas, Arndtstr. 7Spanish Restaurant

Mellow Gold, Karl-Eilers-Str. 22Bar

• Old Town / Klosterplatz5-10mins walking distance from the tram stations “Jahnplatz” or “Rathaus”

Brauhaus Joh. Albrecht, Hagenbruchstr. 8German Restaurant

Irish Pub, Mauerstr. 38

Rockcafé, Neustädter Str. 25Bar

3eck, Ritterstraße 21Bar & Restaurant

Atoms

Plan B, Friedrichstr. 65Tram stations nearby: Hauptbf., SiegfriedplatzBar

Bernstein, Niederwall 2Tram stations nearby: Jahnplatz(Cocktail-)Bar

List of ParticipantsLuis Guillermo Alcuna ValverdePurdue University, USA

Nathaël AlibaudUniversité de Besançon, France

David ApplebaumUniversity of Sheffield, UK

Jong-Chun BaeSeoul National University, Republic of Ko-rea

Rodrigo BañuelosPurdue University, USA

Ali Ben AmorUniversität Bielefeld, Germany

Mahmoud Ben FredjFaculté des Sciences de Monastir, Tunisia

Alexander BendikovUniversity of Wroclaw, Poland

Piotr BilerUniversity of Wroclaw, Poland

Krzysztof BogdanWroclaw University of Technology, Poland

Björn BöttcherTechnische Universität Dresden, Germany

Toralf BurghoffFriedrich-Schiller-Universität Jena, Ger-many

Zhen-Qing ChenUniversity of Washington, USA

Francesca Da LioUniversita di Padova, Italy

Latifa DebbiUniversität Bielefeld, Germany

Zhang DengUniversität Bielefeld, Germany

Ishak DerrardjiaEl Tarf, Algeria

Bartłomiej DydaUniversität Bielefeld, Germany

Khalifa El MabroukEcole Supérieure des Sciences et Technolo-gie, Tunisia

Etienne EmmrichTechnische Universität Berlin, Germany

Mouhamed Moustapha FallGoethe-Universität Frankfurt, Germany

Matthieu FelsingerUniversität Bielefeld, Germany

Mohammud FoondunLoughborough University, UK

Rupert FrankUniversity of Princeton, USA

Uta FreibergUniversität Siegen, Germany

Piotr GarbaczewskiUniversity of Opole, Poland

Bielefeld, 2012 67

Christine GeorgelinUniversité François Rabelais, France

María del Mar GonzálezUniversitat Politècnica de Catalunya,Spain

Alexander Grigor’yanUniversität Bielefeld, Germany

Tomasz GrzywnyTechnische Universität Dresden, Germany

Walter HohUniversität Bielefeld, Germany

Julian HollenderTechnische Universität Dresden, Germany

Jiaxin HuUniversität Bielefeld, Germany

Cyril ImbertUniversité Paris Est Créteil CNRS, France

Peter ImkellerHumboldt-Universität Berlin, Germany

Niels JacobSwansea University, UK

Sven JarohsGoethe-Universität, Frankfurt am Main,Germany

Naotaka KajinoUniversität Bielefeld, Germany

Diana KämpfeUniversität Bielefeld, Germany

Grzegorz KarchUniversity of Wroclaw, Poland

Moritz KaßmannUniversität Bielefeld, Germany

Panki KimSeoul National University, Republic of Ko-rea

Victoria KnopovaTaras Shevchenko National University ofKyiv, Ukraine

Yuri KondratievUniversität Bielefeld, Germany

Takahashi KumagaiKyoto University, Japan

Mateusz KwaśnickiWroclaw University of Technology, Poland

Richard LehoucqSandia National Laboratories, USA

Xiaohua LiUniversität Bielefeld, Germany

Luis LópezMarseille, France

József LörincziLoughborough University, UK

Antonios ManoussosUniversität Bielefeld, Germany

Mark MeerschaertMichigan State University, USA

Ralf MetzlerUniversität Potsdam, Germany

Ante MimicaUniversität Bielefeld, Germany

Giampiero PalatucciUniversità degli Studi di Parma, Italy


Ilya PavlyukevichFriedrich-Schiller-Universität Jena, Ger-many

Ling PeiUniversität Bielefeld, Germany

Xuhui PengUniversität Bielefeld, Germany

Xue PengUniversität Bielefeld, Germany

Enrico PriolaUniversità degli Studi di Torino, Italy

Diana Camelia PutanUniversität Bielefeld, Germany

Marcus RangUniversität Bielefeld, Germany

Michael RöcknerUniversität Bielefeld, Germany

Luz RoncalUniversidad de La Rioja, Spain

Julio RossiUniverisad de Alicante, Spain

Nikola SandrićUniversity of Zagreb, Croatia

Enrico ScalasUniversità del Piemonte Orientale, Italy

Hans-Peter SchefflerUniversität Siegen, Germany

Armin SchikorraMax-Planck-Institut Leipzig, Germany

René SchillingTechnische Universität Dresden, Germany

Russell SchwabCarnegy Mellon University, USA

Marina SertićUniversität Bielefeld, Germany

Yuichi ShiozawaOkayama University, Japan

Yannick SireUniversité Aix-Marseille III, France

Igor SokolovHumboldt-Universität Berlin, Germany

Renming SongUniversity of Illinois, USA

Pablo Raul StingaUniversidad de La Rioja, Spain

Kohei SuzukiUniversity Kyoto, Japan

Paweł SztonykWroclaw University of Technology, Poland

Felix ThielHumboldt-Universität Berlin, Germany

Erwin ToppUniversidad de Chile, Chile

Suleyman UlusoyZirve University, Turkey

Enrico ValdinociUniversità di Roma Tor Vergata, Italy

Alexis VasseurUniversity of Texas, USA

Paul VoigtUniversität Bielefeld, Germany

Bielefeld, 2012 69

Zoran VondračekUniversity of Zagreb, Croatia

Vanja WagnerUniversity of Zagreb, Croatia

Sven WiesingerUniversität Bielefeld, Germany

Zhuo-Ran XiaoUniversität Bielefeld, Germany


Analysis, Probability, Geometry and Applications

SCIENTIFIC COMMITTEERODRIGO BAÑUELOS CYRIL IMBERT GRZEGORZ KARCH

Nonlocal OperatorsJuly 9th - 14th, 2012 at ZiF Bielefeld

TAKASHI KUMAGAI JÓZSEF LÖRINCZI RENÉ SCHILLING

LOCAL ORGANIZER - MORITZ KASSMANN

Zentrum für interdisziplinäre Forschung (ZiF)Universität BielefeldWellenberg 1D-33615 Bielefeld

Sonderforschungsbereich (SFB) 701Universität BielefeldPostfach 10 01 31D-33501 Bielefeld

Analysis, Probability, ' }u Çv o] }v NONLOCAL OPERATORSNonlocalOperators Analysis, Probability, Geometry and Applications CenterforInterdisciplinaryResearch(ZiF),Bielefeld July9–14,2012

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