probability and analysis at wrocław university of...

Probability and Analysisat Wrocław University of Technology 1978-2015

(personal perspective)

Tomasz Żak

Wrocław University of Technology, Poland

May 6, 2015

Tomasz Żak Probability and Analysis at Wrocław University of Technology 1978-2015

Stochastic processes and Functional Analysis

Every stochastic process (Xt)t0 with real values defines a measurein a space of its trajectories, namely the distribution of the process.One of possible ways to investigate the properties of the process isto examine this distribution.

If the process has continuous trajectories, its distribution isconcentrated on C ([0,∞)), the Banach space of continuousfunctions. Cadlag processes (with jumps) have distributions in theSkorohod space D([0,∞)).

This way of looking at stochastic processes was popular in Poland,because Functional Analysis had a strong influence on Polishmathematics (recall only one name: Stefan Banach).


Gaussian measures in Banach spaces

In the sixties (of XX century) there were very interesting problemsconcerning distributions in separable Banach spaces, let me recallonly one of them:

A distribution of a random vector X in a Banach space E is calledGaussian if for all functionals x∗ ∈ E ∗ the real random variablex∗(X ) is Gaussian.

Question: Is it true that Eeα‖X‖2<∞ for some α > 0 ?


Gaussian measures in Banach spaces

Almost at the same time (1969-1970) three papers appeared, withthe answer YES.

Their authors were: Skorohod, Landau and Shepp, Fernique.

As you can see, mathematicians tried to answer in more generalsettings questions that have well-known answers in Rn.


The beginning

In October 1978 Tomasz Byczkowski organized a seminar inWrocław University of Technology (Politechnika Wrocławska).

During the first two meetings he gave a talk that described resultsof A. de Acosta, published in two papers in Ann. Prob.Stable measures and seminorms (1975) andAsymptotic behavior of stable measures (1977).

de Acosta investigated supports of stable measures andasymptotics of a distribution of seminorms of stable vectors (hencealso moments of ‖X‖).


The beginning

What is seminorm in a Banach space? It is:

a subadditive function q : E → R,


The beginning



p-homogeneous, 0 < p ¬ 1, which means q(ax) = |a|pq(x)for a ∈ R


The beginning




monotone, that is |a| < |b| implies for all x ∈ E inequalityf (ax) < f (bx).


The beginning





Examples: Lr spaces, r 1, q = ‖ · ‖ and then p = 1,


The beginning





Examples: Lr spaces, r 1, q = ‖ · ‖ and then p = 1,

but also Lp for 0 < p < 1.


The beginning

Professor Byczkowski finished his presentation with severalquestions, here is one of them:

Let µ be a stable measure and q a monotone seminorm. Is it truethat

limt→∞tpµ(x : q(t−1x) > ε) = Cε ?


The beginning

Next year (1979) we had a positive answer and our joint paperAsymptotic properties of semigroups of measures on vector spaces

was published in the Annals of Probability (1981).


Distribution of a seminorm

The next important problem was the absolute continuity of thedistribution of a seminorm of a stable process.

This subject started with a seminar discussion of the resultscontained in a paper written by J. Hoffmann-Jorgensen,L. A. Shepp and R. M. Dudley

On the Lower Tail of Gaussian Seminorms, Ann. Prob.(1979).

The authors proved that the distribution function of q(X ) isabsolutely continuous except for one possible jump at thebeginning.


Suprema of stable processes

It turned out that for stable measures the answer is very similar. Itwas given by

T. Byczkowski and K. Samotij in a paper

Absolute continuity of stable seminorm, Ann. Prob. (1986).

The next problem was obvious: if the distribution has a density –give an exact estimate of it.


Khintchine inequality (1928)

Let rk(t) = sin(2kπt), 0 ¬ t ¬ 1, k = 1, 2, 3, ... be theRademacher functions. There exist universal constants Ap, Bpsuch that for all real ak , k = 1, ..., n and all p 1

Ap

(

n∑

k=1

a2k

)1/2

¬

(

∫ 1

0

∣

∣

∣

∣

∣

n∑

k=1

ak rk(t)

∣

∣

∣

∣

∣

p)1/p

¬ Bp

(

n∑

k=1

a2k

)1/2

.

In the language of Functional Analysis: all the p-norms areequivalent on the space of functions generated by Rademachersequence.

In 1964 J.P. Kahane asked a question: what happens if, instead ofnumbers ak , we put vectors xk from a given Banach space E?


Type and cotype of E

It turned out that the answer to Kahane question depends heavilyon the geometry of a Banach space E . Namely, if there exists auniversal constant Bp such that for all (xk)nk=1 ⊂ (E , || · ||E )

(

∫ 1

0||n∑

k=1

xk rk(t)||2E dt

)1/2

¬ Bp

(

n∑

k=1

||xk ||pE

)1/p

,

then (E , || · ||E ) is a space of type p.Examples:

Every normed space is of type 1 (because any norm issubadditve).



It turned out that the answer to Kahane question depends heavilyon the geometry of a Banach space E . Namely, if there exists auniversal constant Bp such that for all (xk)nk=1 ⊂ (E , || · ||E )

(

∫ 1

0||n∑

k=1

xk rk(t)||2E dt

)1/2

¬ Bp

(

n∑

k=1

||xk ||pE

)1/p

,

then (E , || · ||E ) is a space of type p.Examples:

Every normed space is of type 1 (because any norm issubadditve).

Khintchine inequality implies that the type of any spacecannot be greater than 2.



If there exists Aq such that

Aq

(

n∑

k=1

||xk ||qE

)1/q

¬

(

∫ 1

0||n∑

k=1

xk rk(t)||2E dt

)1/2

then (E , || · ||E ) is a space of cotype q.

Cotype of every normed space is greater or equal to 2.



If there exists Aq such that

Aq

(

n∑

k=1

||xk ||qE

)1/q

¬

(

∫ 1

0||n∑

k=1

xk rk(t)||2E dt

)1/2

then (E , || · ||E ) is a space of cotype q.

Cotype of every normed space is greater or equal to 2.

If a Banach space is of type and cotype 2 then it is isomorphicto a Hilbert space (S. Kwapień).


Stable type and cotype

What happens if we put real α-stable random variables in thedefinition of type and cotype? We get stable type or cotype.

We discussed this problem, reading a paper by Araujo and GineType, cotype and Levy measures in Banach spaces

Ann. Prob. 1978.

It turned out that the stable type of a Banach space depends onthe behaviour of the distribution of the norm of stable randomvector. The result is contained in a paper by M. Ryznar

Asymptotic Behaviour of Stable Measures Near the Origin, Ann.Prob.(1986).


Not only Banach spaces ...

To broaden field of our mathematical investigations, professorByczkowski was always looking for interesting questions in differentareas.

For instance, Piotr Graczyk in his Ph.D. dissertation investigatedprocesses with values in groups:

Malliavin calculus for stable processes on homogeneous groups,

Studia Mathematica (1991).


Not only Banach spaces ...

Also, under the influence of prof. Byczkowski, we have takenseveral attempts to start investigations in potential theory ofstochastic processes.

Unfortunately, it turned out that J. Doob’s monograph ClassicalPotential Theory and Its Probabilistic Counterpart (1984) was toodifficult as a text for the first reading in the subject.

The situation changed, when we got the monograph

K.L. Chung, Z. Zhao From Brownian Motion to Schrodinger’sEquation.


Different point of view on Levy processes

At least in two places in his books, David Williams writes that hehas much sympathy with K.L. Chung’s philosophy:

The only proper way for probabilists to proceed is to suppose given

a Levy process and to derive any analysis associated with the

transition function of X from sample path properties.



Properties of trajectories, they are important!

Looking at the distribution of q(X ) we can hardly see thebehaviour of a single trajectory. And natural questions concerninga Levy process X are:

Will X hit a given set (a point)?






If YES, then how long does it take to hit this set?







If X exits from a set D then what is the distribution of exitposition?








How much time does X spend in D before the first exit?









Suppose we kill X on exiting D. Describe the transitionprobability of such process.









Suppose we kill X on exiting D. Describe the transitionprobability of such process.

And much more...


Killed processes

Suppose we have a Levy process with values in Rd and a Borel setD ⊂ R

d . Define the first exit time of X from D:

τD = inf{t > 0 : Xt /∈ D},

and process killed off D:

XDt =

{

Xt , t < τD ,∂, t τD .

We are interested in the behaviour of the process before it is killed.

We can also investigate the distribution of XτD ; this is the so-calledharmonic measure. Its density PD(x , y) (for the process startingfrom x) is called the Poisson kernel of D.


Killed processes

If X is a Levy process with generator L and the transition densityfunction p(t, x , y), then XD also has transition density pD(t, x , y),given by Hunt’s formula:

pD(t, x , y) = p(t, x , y)− E x [p(t,XτD , y); t > τD ].

We call pD(t, x , y) the Dirichlet heat kernel for the process X andthe set D, because it is a solution of the equation

12∂u(x , t)

∂t= Lu(x , t)

with the Dirichlet condition on Dc .


Killed processes

The Green function for X and D

GD(x , y) =

∫ ∞

0pD(t, x , y) dt

measures the amount of time that X , starting from x ∈ D, spendsat y ∈ D, before it exits D.


Harmonic functions

Let X be a Levy process with generator L. Function h isL-harmonic in D if Lh = 0 on D.

Examples of important harmonic functions:

h(x) = GD(x , y) for y ∈ D is harmonic on D \ {y},


Harmonic functions




h(x) = Ex f (XτD ) is harmonic for x ∈ D,


Harmonic functions




h(x) = Ex f (XτD ) is harmonic for x ∈ D,

sD(x) = Ex(τD) is superharmonic, LsD(x) = −1 on D.


Behaviour of harmonic functions

The above examples show that the behaviour of harmonicfunctions is crucial for understanding the behaviour of the samplepaths of X .

In the classical case, that is for Brownian motion in Rd , whereL = 1

2∆, harmonic functions were investigated for a long time, not

only from the probabilistic point of view.

A natural question: what is the behaviour of (−∆)α/2-harmonicfunctions? Here

(−∆)α/2u(x) = Ad ,α

∫

Rd

u(x)− u(x + y)

|y |d+αdy

is the generator of the α-stable, rotation invariant Levy processwith values in Rd .


Breakthrough: BHP

A breakthrough came in 1995 and a dream of professor Byczkowski— to develop the boundary potential theory for α-stable processes— began to fulfill: Krzysztof Bogdan proved the so calledBoundary Harnack Principle for α-harmonic functions:

The boundary Harnack principle for fractional Laplacian, StudiaMathematica (1997).


Breakthrough: BHP

If D is a Lipschitz domain in Rd ,

u, v are two positive functionsvanishing on Dc near a part of ∂D,α-harmonic in D, K is compactand for some x ∈ K u(x) = v(x),then there exist constants 0 < c <C <∞ such that for all y ∈ K ∩D

cu(y) ¬ v(y) ¬ Cu(y).


Schrodinger equation

Later, K. Bogdan and T. Byczkowski developed the

Potential theory for α-stable Schrodinger operator on Lipschitzdomains, Studia Mathematica (1999).

This is the potential theory of the following operator:

L = −(−∆)α/2 + q(x),

where q is a Borel function from the so-called Kato class.If q ¬ 0, this means killing of trajectories, if q 0 — exponentialgrowth of the ”mass” of trajectories in time, with intensity q(x) atpoint x .


Seminar as a branching process

Starting from the year 2000 a number of participants taking part inthe seminar has been growing and there was a need to explore newareas:

Bessel processes (T. Byczkowski, J. Małecki, M. Ryznar)





Brownian motion in hyperbolic spaces (T. Byczkowski,P. Graczyk and A. Stós)






relativistic stable processes on Rd (M. Ryznar)






relativistic stable processes on Rd (M. Ryznar)

spectral theory of α-stable processes (T. Kulczycki)


Main subject: potential theory for Schrodinger operator

Nevertheless, one of the main subject remains the potential theoryfor Schrodinger operator, based on α-stable process (and itsgeneralizations) and different perturbations.

Examples:

gradient perturbations L = −(−∆)α/2 + b(x)∇




Examples:


perturbations with potentials q(t, x) depending on time andspace




Examples:



stable processes with non-symmetric Levy measures




Examples:




processes non-homogeneous in time, that is p(s, x , t, y)




Examples:




processes non-homogeneous in time, that is p(s, x , t, y)

purely analytic approach to Schrodinger and non-localSchrodinger perturbations of arbitrary transition densities



Problems: find formulas (if possible) or estimates for

p(t, x , y) and pD(t, x , y) — K.Bogdan, T. Grzywny,T. Jakubowski, K. Szczypkowski, P. Sztonyk





the Green function, Poisson kernel — K. Bogdan,T. Grzywny, M. Kwaśnicki






gradient perturbations — K. Bogdan, T. Jakubowski






gradient perturbations — K. Bogdan, T. Jakubowski

the eigenvalues and eigenfunctions — K. Kaleta,T. Kulczycki, M. Kwaśnicki


Relativistic processes

A process Xt in Rd is called relativistic α-stable if

E 0e iξXt = e−t((|ξ|2+m2/α)α/2−m).

X behaves like a stable process for small t but like the Brownianmotion if t is large.

M. Ryznar gave formulas and estimates for Poisson kernel andGreen function in Estimates of Green functions for relativisticα-stable process, Potential Analysis (2002),M. Ryznar and T. Grzywny Two-sided optimal bounds for Greenfunctions of half-spaces for relativistic α-stable process, PotentialAnalysis (2008).

T. Kulczycki and B. Siudeja developed the spectral theory forrelativistic Hamiltonian in a paperIntrinsic ultracontractivity of Feynman-Kac semigroup for

relativistic stable processes, Trans. Amer. Math. Soc. (2006)


Processes on fractals

Also processes on the so-called d-sets were investigated:

K. Bogdan, A. Stós, P. Sztonyk Harnack inequality for stableprocesses on d-sets, Studia. Math. (2003)

Class of d-sets contains for example Sierpiński gasket and carpet.


Bessel processes

In order to understand hyperbolic Brownian motion one shouldknow some delicate properties of real Bessel processes:T. Byczkowski, J. Małecki, M. Ryznar:1. Bessel potentials, hitting distributions and Green functions,TAMS (2009)2. Hitting half-spaces by Bessel–Brownian diffussions, PotentialAnal. (2010)3. Hitting times of Bessel processes, Potential Anal. (2013)

J. Małecki and G. Serafin estimated Fourier-Bessel heat kernel

Bessel operator: compare analysis and probabilityA. Nowak, L. Roncal On sharp heat and subordinated kernelsestimates in the Bessel – Fourier settings, Rocky Mountains J.Math. (2014)


Boundary Harnack Principle

Boundary Harnack Principle was generalized in several directions:for different classes of processes, for global or local constants.

Let me mention in this context papers by K. Bogdan,T. Grzywny and M. Kwaśnicki.


More harmonic analysis: relative Fatou theorem

The existence of non-tangential limit limz→Q∈∂B(0,1) f (z) forharmonic function f is known as Fatou theorem.

A generalization of the result for ratios of singular α-harmonicfunctions in Lipschitz domains was given by K. Michalik and M.Ryznar (Illinois J. Math. 2004)


More harmonic analysis: fractional calculus, Hardyinequality

Recall that (−∆)α/2u(x) = Ad ,α∫

Rdu(x)−u(x+y)|y |d+α

dy .

In Fractional calculus for power functions and eigenvalues of thefractional Laplacian, Fract. Calc. Appl. Anal. (2012)B. Dyda gave explicit formulas (in terms of hypergeometricfunction) for

(−∆)α/2(1− |x |2)p+ and (−∆)α/2(1− |x |2)p+xd

for the unit ball and p > −1. For p = n + α/2 these functions arepolynomials. This gives estimates for eigenvalues of (−∆)α/2 forthe unit ball with Dirichlet condition.

K. Bogdan, B. Dyda and T. Luks examined Hardy spaces:On Hardy spaces of local and nonlocal operators, Hiroshima Math.J. (2014)


Levy processes with weak scaling

Consider now a Levy process X such that

Ee iξXt = e−tψ(ξ),

where ψ has some scaling (here — upper scaling) for positive λψ(λξ) ¬ cλαψ(ξ).

For such class of processes:

K. Bogdan, T. Grzywny and M. Ryznar gave estimates of the heatkernel and Green function in Dirichlet heat kernels for unimodalLevy processes, SPA (2014),

maxima were examined by M. Kwaśnicki, J. Małecki and M.Ryznar in Suprema of Levy processes, Ann. Prob. (2013)

and by K. Bogdan, T. Grzywny and M. Ryznar in Barriers, exit timeand survival probability for unimodal Levy processes, TPRF (2015)


Spectral Theory

If we kill a process off an open bounded set D, then its transitionsemigroup (Pt) is compact, so that we get a sequence ofeigenvalues λ1 < λ2 ¬ .... and eigenfunctions (ϕn) such that

Ptϕn(x) = e−tλnϕn(x).

Classical (i.e. for the generator ∆) questions are the following(R. Courant, D. Hilbert Methods of Mathematical Physics):

Estimate the spectral gap, that is λ2 − λ1.


Spectral Theory





Describe the shape of ”nodal domains”: it is known that ϕ1can be choosen positive, what about ϕ2?


Spectral Theory





Describe the shape of ”nodal domains”: it is known that ϕ1can be choosen positive, what about ϕ2?

Examine regularity of ϕ1, for instance if α = 2 and D isconvex, function ϕ1 is log-concave.


Spectral Theory

Here are some results:

connection between spectral theory for Cauchy process andmixed Stieklov problem — R. Banuelos andT. Kulczycki


Spectral Theory



spectral theory for Schrodinger operator (spectral gap,estimates of eigenfunctions) — K. Kaleta, T. Kulczycki,B. Siudeja


Spectral Theory



spectral theory for Schrodinger operator (spectral gap,estimates of eigenfunctions) — K. Kaleta, T. Kulczycki,B. Siudeja

formulas for eigenfunctions for stable and relativistic processeson (0, ∞) — M. Kwaśnicki


Dirichlet and Neumann conditions

Dirichlet condition specify the value of the function on theboundary of D (or on Dc for non-local operators)

Neumann conditions specify the value of the normal derivative ofthe function on the boundary of D

mixed conditions: the solution is required to satisfy a Dirichlet or aNeumann boundary condition in a mutually exclusive way ondisjoint parts of the boundary.


Surprising application

Consider the mixed Stieklov problem for a body of revolution in R3:

{

∆ϕ = 0 in W , ∂ϕ∂z = λϕ on F ,

∂ϕ∂~n = 0 on B,

∫

F ϕdx dy = 0.

y

z

x

B

F

W

r

z

D



T. Kulczycki and M. Kwaśnicki in the paper

On high spots of the fundamental sloshing eigenfunctions in axially

symmetric domains, Proc. London. Math. Soc. (2012)

proved that for λ1 there exist two eigenfunctions ϕ1, ϕ2 with thesame profile and:

for the above situation — when the angle between B and F isless or equal π

2then high spots are attained on ∂F ,



T. Kulczycki and M. Kwaśnicki in the paper

On high spots of the fundamental sloshing eigenfunctions in axially

symmetric domains, Proc. London. Math. Soc. (2012)

proved that for λ1 there exist two eigenfunctions ϕ1, ϕ2 with thesame profile and:

for the above situation — when the angle between B and F isless or equal π

2then high spots are attained on ∂F ,

if the angle between B and F is greater that π2and less than

π then ϕ1, ϕ2 attain their extrema inside F .



high spots

(a) (b)

In a sense: it is easier to spill a liquid from a mug than from asnifter!


Instead of a proof

So that: there is a mathematical proof but can we see it?

For example: Can one take a photograph of a wave on a surface ofa liquid?


Instead of a proof

How to take a photograph of a wave on a surface of a liquid?

sloshing tank

camera

dotted paper


Instead of a proof

In high spots the dots are not blurred.

high spots


Seminar in numbers

Here are numbers concerning the seminar

37 years (started at 1978)


Seminar in numbers



at least 15 permanent participants (from our department)


Seminar in numbers




180 papers (MathRev data) published and reviewed in period2005 – 2015


Seminar in numbers




180 papers (MathRev data) published and reviewed in period2005 – 2015

Estimated number of all papers: more than 300


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