Continuity estimates for functions
harmonic with respect to jump processes
joint work with Moritz Kaßmann
Ryad Husseini1
ryad@uni-bonn.de
Institut für Angewandte Mathematik
Rheinische Friedrich-Wilhelms-Universität Bonn
Workshop on stochastic and harmonic analysis of processes with
jumps, Angers 2006
1
Research financed by DFG (German Science Foundation)
through SFB 611, project A9.
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
1 / 16
Outline
1
Basic setting
2
Main result
3
The assumptions in detail
4
Examples
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
2 / 16
Basic setting
Functions harmonic w.r.t. Markov processes
Let X = (Xt , Px ) be a continuous time conservative strong Markov
process with values in Rd and càdlàg (r.c.l.l.) paths.
Definition
A measurable u : Rd → R is called harmonic w.r.t. X in a domain
Ω ⊂ Rd iff
u(x) = Ex u(XτΩ0 )1{τΩ0 <∞}
for all open Ω0 ⊂⊂ Ω, x ∈ Ω0 , where τΩ0 = inf {t : Xt ∈
/ Ω0 }.
X Brownian motion 99K mean value property of harmonic
functions
In general (i.e. if X has jumps) harmonicity is a non-local notion.
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
3 / 16
Basic setting
Regularity of harmonic functions
Regularity generally does not go without saying!
Example: (Xt ) solution of dXt = σ(Xt )dBt for Bt Brownian motion.
A-priori Hölder estimates:
∃C > 0, γ ∈ (0, 1) : ∀R ∈ (0, 1) :
If u : Rd → R is bounded and harmonic in B(x0 , R), then
|x − y | γ
|u(x) − u(y )| ≤ C · kuk∞
∀x, y ∈ B(x0 , R/2).
R
In applications we often need less: a-priori continuity estimates
∀R ∈ (0, 1) : ∃ϑ : [0, 1) → R+ , lim ϑ(t) = 0 :
t→0
If u : Rd → R is bounded and harmonic in B(x0 , R), then
|u(x) − u(y )| ≤ kuk∞ · ϑ(|x − y |) ∀x, y ∈ B(x0 , R/2).
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
4 / 16
Basic setting
Jump-type operators
Let
ν : Rd × B(Rd ) → [0, ∞] be a family of Lévy measures with
Z
(|h|2 ∧ 1) ν(x, dh) < ∞.
sup
x∈Rd
Rd
L be the associated integro-differential operator on Cb2 (Rd )
Z
(L u)(x) =
(u(x + h) − u(x) − 1{|h|<1} h · ∇u(x)) ν(x, dh) .
Rd
Example (fractional Laplacian):
α ∈ (0, 2), ν(x, dh) = A (d, −α) · |h|−d−α dh, L = −(−∆)α/2 .
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
5 / 16
Basic setting
X and ν
Assume: X unique solution to the martingale problem for
L , Cb2 (Rd ) , i.e. for all u ∈ Cb2 (Rd )
Z
u(Xt ) − u(X0 ) −
t
(L u) (Xs ) ds
is a Px -martingale.
0
The relation between the measure ν(x, ·) and the process X is
illustrated by the Lévy system identity:
E
x
X
1{Xs− ∈A,Xs ∈B}
s≤S
|
{z
=E
x
Z
S
1A (Xs )ν(Xs , B − Xs ) ds
0
}
] jumps from A to B before S
where A, B ⊂ Rd are disjoint Borel sets and S is a bounded
stopping time.
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
6 / 16
Basic setting
Former results on regularity of jump-type operators
Hölder regularity:
probabilistic method goes back to Krylov-Safonov (1979) in the
case of diffusions
Bass-Levin ’02, Song-Vrondraček ’04
ν(x, dh) = n(x, h)dh with n(x, h) |h|−d−α , α ∈ (0, 2)
Bass-Kaßmann ’05 (not necessarily absolute continuous ν)
Schilling-Uemura ’06
Other results:
Komatsu ’85
Counter-example:
Barlow-Bass-Chen-Kaßmann ’06
construction of ν(x, dh) such that there are bounded harmonic
functions which are not continuous everywhere
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
7 / 16
Main result
Main theorem
Theorem (H., Kaßmann ’06)
Assume:
the existence of an strong Markov process in the above sense
ν(x, ·) has the properties (A1) and (A2) (see below)
Then:
∃ρ > 0 : ∀R ∈ (0, 1/2) : ∃c > 0 :
If u : Rd → R is harmonic w.r.t. X on B(x0 , R) then
|u(x) − u(y )| ≤ c · kuk∞ · |ln |x − y ||−ρ
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
∀x, y ∈ B(x0 , R/2).
Angers 2006
8 / 16
Main result
Why are we interested in such a result?
Such a theorem (+ one additional assumption) can be used to
prove that the resolvent of X
Z ∞
x
Rλ f (x) := E
e−λt f (Xt )dt, λ > 0,
0
maps bounded functions to uniformly continuous functions.
Our method also works if X is the Hunt process associated to a
certain class of regular Dirichlet forms. We get qualified uniform
continuity for harmonic functions and the resolvent outside an
exceptional set.
continuous ”modification” of Rλ !
In particular, we can construct a Feller process very much like the
counterexample of Barlow-Bass-Chen-Kaßmann.
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
9 / 16
The assumptions in detail
Our assumptions in detail
Define for r ∈ (0, 1)
Z
S(x, r ) =
ν(x, dh)
|h|≥r
1 L(x, r ) = S(x, r ) + r
Z
Z
1
h ν(x, dh) + 2
r
1≥|h|≥r
|h|2 ν(x, dh)
|h|<r
1
N(x, r ) = inf ν(x, A − x) : A ⊂ B(x, 2r ), |A| ≥ |B(x, r )|
3
rot.-inv. α-stable case: S, L and N behave like r −α for r → 0.
lower bounds on N(x, r ) ensure that the process is not
degenerate (enough jumps in ”enough” directions)
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
10 / 16
The assumptions in detail
Assumption (A1)
(A1) ∃κ1 > 0, σ > 0 : ∀x ∈ Rd , r ∈ (0, 1), 1 < λ <
1
:
r
S(x, λr ) ≤ κ1 λ−σ S(x, r ).
Condition (A1) ensures a ”minimal” activity of small jumps:
lim S(x, r )r σ/2 = ∞.
r →0
Example:
ν(x, dh) = n(x, h)dh with n(x, h) ≥ |h|−d−α , α > 0, where |h| ≤ δ
and h belongs to a cone with end 0.
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
11 / 16
The assumptions in detail
(A2) – The degeneration of hitting time estimates
(A2) ∃κ2 > 0 : ∀x, y ∈ Rd , r ∈ (0, 1/2), |x − y | < 2r :
r
N(x, r ) ≥ κ2 · |ln r |−1 · L(y , ).
2
Assumption (A2) implies the following degenerate lower bounds
on probability of hitting ”big” subsets before exiting, which are a
key ingredient of the proof
∃c > 0 : ∀x ∈ Rd , r ∈ (0, 1/2), A ⊂ B(x, r ), |A| ≥
Py TA < τB(x,r ) ≥ c · |ln r |−1
1
3
|B(x, r )|:
r
∀y ∈ B(x, ).
2
Compare with the estimate used by Krylov-Safonov
r
Py TA < τB(x,r ) ≥ c ∀y ∈ B(x, ),
2
i.e. there the lower bound is independent of r !
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
12 / 16
Examples
Examples
Assumptions (A1)-(A2) are fulfilled in the following cases where
ν(x, dh) = n(x, h) dh:
For α ∈ (0, 2):
0 < c1 · |h|−d−α ≤ n(x, h) ≤ c2 · |h|−d−α
For α ∈ (0, 2):
0 < c1 · |h|−d−α ≤ n(x, h) ≤ c2 · |ln(|h| /3)| · |h|−d−α
n(x, h) = 0
for |h| < 2
for |h| ≥ 2
α : Rd → (0, 2), 0 < inf α(x) ≤ sup α(x) < 2, n(x, h) = n(x, −h)
0 < c1 · |h|−d−α(x) ≤ n(x, h) ≤ c2 · |h|−d−α(x)
|α(x) − α(y )| ≤ c3 · |ln |ln |x − y |||−1
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
for |x − y | ≤
1
2
Angers 2006
13 / 16
Examples
Degeneration of hitting time estimates – Example
Let 0 < α < β < 1, a = (1 + β − α)−1 and
F = {(h1 , h2 ) ∈ R2 ; |h2 | ≥ |h1 |a , (h1 )2 + (h2 )2 < 1} .
Let X be the symmetric Lévy process in R2 with characteristic
(0, 0, ν) where ν(dh) = n(h) dh and
n(h) = |h|−2−α + 1F (h) · |ln |h|| · |h|−2−β .
−2−
∣h∣
log∣h∣
∣h∣−2−
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
14 / 16
Examples
There exists a function σ : (0, 1) → R+ with lim σ(r ) = 0 and
r →0
lim P0
r →0
lim
r →0
P0
τB(0,r ) ≤ σ(r ) = 1
r sup |Xs1 | >
=0
16
s≤σ(r )





B(x,r)
Ryad Husseini (Uni Bonn)
⇒
A-priori continuity estimates
degeneration of
hitting times
A
Angers 2006
15 / 16
Examples
Merci beaucoup!
Ryad Husseini (Uni Bonn)
A-priori continuity estimates
Angers 2006
16 / 16