Essential Uses of Probability in Analysis

Krzysztof BurdzyUniversity of Washington

Part I. Brownian Couplings and Neumann Eigenfunctions

Hot Spots Conjecture

Rauch (1974) : In Euclidean domains, the second Neumann Laplacianeigenfunction attains its maximum at the boundary.

0 π

Multidimensional domains

CounterexamplesB and Werner (1999)Bass and B (2000)B (2005)

Positive directionKawohl (1985)Bañuelos and B (1999)Pascu (2002) : Conjecture holds forplanar convex domains with a line of symmetry.Atar and B (2004) : Conjecture holdsfor all lip domains.Jerison and Nadirashvili (2000)Atar (2001), …

Symmetric domains

Lip domains

We say that a set is a “lip domain”if it lies between the graphs of twoLipschitz functions with the Lipschitzconstant 1.

Open problems

(i) Hot spots conjecture for convex domains.

(ii) Hot spots conjecture for simply connected planar domains.

(iii) Hot spots conjecture for triangles.

A partial order

yyx ≤x

temperature at at time (Neumannboundary conditions)

Heat equation and reflected Brownian motion

DA

)()(),( AXPXfEtxu tx

tx ∈==

=),( txu

tX)(1)()0,( xxfxu A==

x t

- reflected Brownian motion in D

x

Neumann eigenfunctionsK++= − texctxu λϕ )(),(

)),(),(()()( tyutxueyx t −≈− λϕϕ

))()(()()( AXPAXPeyx ty

txt ∈−∈≈− λϕϕ

))()(()()( AYPAXPeyx ty

txt ∈−∈≈− λϕϕ

Synchronous couplings

xX =0

yY =0

tYXyx tt ∀≤⇒≤

B and Kendall (2000)

tX tY

(monotonicity)

Effect of reflection

tX

tY

tt YX ≤

Eigenfunction monotonicity

x AtY

tX

y

)()( AYPAXP ty

tx ∈≤∈

)()( yx ϕϕ ≤

tYXyx tt ∀≤⇒≤

Mirror couplings (free BM)

x

y

tX

TtYX tt ≥=

tY

TT YX =

T - coupling time

Mirror couplings for reflected BM

x ytX

Wang (1994)B and Kendall (2000) : reflection on a single lineAtar and B (2004) : piecewise smooth domainsB (2005) : simultaneous reflections

tY

hinge

D∂

Symmetric domains

x

Neumann

Dirichlet

Bañuelos and B (1999)Jerison and Nadirashvili (2000)Pascu (2002)

tX

Scale couplings – Pascu (2002)

D

D

- semidisc

Eigenvalue multiplicity

Maximum eigenvalue multiplicity

Nadirashvili (1986, 1988) : The maximum multiplicity of the second Neumann eigenvalue for a simply connected planar domain is 2.

Long convex domainsBañuelos and B (1999) : If is convexand diameter/width is greater than 3.07then the second eigenvalue is simple.

D

D

diameter

width

Convex domains - conjectureBañuelos and B (1999) : (Conjecture)If is convex and diameter/width is greater than 1.41 then the second eigenvalue is simple.

D

width = 1

diameter = 2

Lip domainsAtar and B (2004) : Second Neumanneigenvalue is simple in lip domains(except squares).

Eigenvalue multiplicity- lip domains

Parabolic boundary Harnackprinciple

mirror

Points outside nodal lines

Bañuelos and B (1999) : If for some point the nodal (zero) line for any second eigenfunction does not pass through this point, then the second eigenvalue is simple.

Bottleneck domains

Nodal line

This is not a nodal line

Bottleneck domains (idea of proof)

Neumann

Dirichlet

Nodal line locationOld results

(i) Rectangles, ellipses, etc.(ii) Domains with symmetry

(Melas (1992), Alessandrini (1994) : Dirichlet nodal lines)

(iii) Jerison (2000) Long and thin domains

Nodal lines and couplingsAtar and B (2002) : Consider a coupling of reflected Brownian motions. If the particles cannot couple in a subset of the domain then this subset is too small to contain a nodal domain.Atar and B (in preparation) : If the particles can couple only in a subset of the domain then this subset must contain the nodal set.

Nodal lines and couplings(idea of proof)

Suppose that the coupling point must lie between the green lines.

This might be the nodal line

Obtuse triangles

a a

Possible nodal line

0 10.63

Synchronous couplings viaunique strong solutionsto Skorohod equation

Xt

t

stt dLXNBxX )(0∫++=

Yt

t

stt dLYNByY )(0∫++=

Lions and Sznitman (1984) : - domains 2C

Skorohod equation:unique strong solutions in

Bass, B and Chen (2004) : Unique strong solutions exist in planar Lipschitz domains with Lipschitzconstant less than 1.Bass and B (2005)

Xt

t

stt dLXNBxX )(0∫++=

2R

Higher dimensional domains

We say that is a -domain if its boundary is represented by a function

and

γ,1CD

Φγ|||)()(| yxyx −≤Φ∇−Φ∇

Skorohod equation:unique strong solutions in

Bass and B (in preparation) : (i) Unique strong solutions exist in

-domains if (ii) Counterexample (?) for some

γ,1C 2/1>

Xt

t

stt dLXNBxX )(0∫++=

3, ≥nRn

γ0>γ