Dynkin E.B. Superdiffusions and Positive Solutions of Nonlinear PDEs (AMS,2004)(126s)_MCde_.pdf

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Superdiffusions and positive solutions of
nonlinear partial differential equations
E. B. Dynkin
Department of Mathematics, Cornell University, Malott Hall,
Ithaca, New York, 14853
Contents
Preface
Chapter 1. Introduction
1. Trace theory
2. Organizing the book
3. Notation
4. Assumptions
5. Notes
Chapter 2. Analytic approach
1. Operators
G
D
and
K
D
2. Operator
V
D
and equation
Lu
=
ψ(u)
3. Algebraic approach to the equation
Lu
=
ψ(u)
4. Choquet capacities
5. Notes
Chapter 3. Probabilistic approach
1. Diffusion
2. Superprocesses
3. Superdiffusions
4. Notes
Chapter 4.
N-measures
1. Main result
2. Construction of measures
N
x
3. Applications
4. Notes
Chapter 5. Moments and absolute continuity properties of
superdiffusions
1. Recursive moment formulae
2. Diagram description of moments
3. Absolute continuity results
4. Notes
Chapter 6. Poisson capacities
1. Capacities associated with a pair (k,
m)
iii
v
1
1
3
4
4
6
9
10
11
15
16
17
19
20
24
28
33
35
36
36
40
47
49
49
54
56
59
61
61
iv
CONTENTS
2.
3.
4.
5.
Poisson capacities
Upper bound for Cap(Γ)
Lower bound for Cap
x
Notes
62
63
67
69
71
71
71
72
74
76
77
79
79
80
82
83
84
85
86
89
89
90
93
101
103
107
111
113
117
119
Chapter 7. Basic inequality
1. Main result
2. Two propositions
3. Relations between superdiffusions and conditional diffusions in
two open sets
4. Equations connecting
P
x
and
N
x
with Π
�½
x
5. Proof of Theorem 1.1
6. Notes
Chapter 8. Solutions
w
Γ
are
σ-moderate
1. Plan of the chapter
2. Three lemmas on the conditional Brownian motion
3. Proof of Theorem 1.2
4. Proof of Theorem 1.3
5. Proof of Theorem 1.5
6. Proof of Theorems 1.6 and 1.7
7. Notes
Chapter 9. All solutions are
σ-moderate
1. Plan
2. Proof of Localization theorem
3. Star domains
4. Notes
Appendix A. An elementary property of the Brownian motion
by J.-F. Le Gall
Appendix B. Relations between Poisson and Bessel capacities
I. E. Verbitsky
Notes
References
Subject Index
Notation Index
Preface
This book is devoted to the applications of the probability theory to the
theory of nonlinear partial differential equations. More precisely, we inves-
tigate the class
U
of all positive solutions of the equation
Lu
=
ψ(u)
in
E
where
L
is an elliptic differential operator of the second order,
E
is a bounded
smooth domain in
R
d
and
ψ
is a continuously differentiable positive function.
The progress in solving this problem till the beginning of 2002 was de-
scribed in the monograph [D]. [We use an abbreviation [D] for [Dy02].]
Under mild conditions on
ψ,
a trace on the boundary
∂E
was associated
with every
u
∈ U.
This is a pair (Γ,
�½)
where Γ is a subset of
∂E
and
�½
is a
σ-finite
measure on
∂E
\
Γ. [A point
y
belongs to Γ if
ψ
(u) tends
sufficiently fast to infinity as
x
→
y.]
All possible values of the trace were
described and a 1-1 correspondence was established between these values
and a class of solutions called
σ-moderate.
We say that
u
is
σ-moderate
if
it is the limit of an increasing sequence of moderate solutions. [A moderate
solution is a solution
u
such that
u
≤
h
where
Lh
= 0 in
E.]
In the Epilogue
to [D], a crucial outstanding question was formulated:
Are all the solutions
σ-moderate?
In the case of the equation ∆u =
u
2
in a domain of class
C
4
, a
positive answer to this question was given in the thesis of Mselati [Ms02a] - a
student of J.-F. Le Gall.
1
However his principal tool - the Brownian snake
- is not applicable to more general equations. In a series of publications by
Dynkin and Kuznetsov [Dy04b], [Dy04c], [Dy04d], [Dy04e],[DK03], [DK04],
[Ku04], Mselati’s result was extended, by using a superdiffusion instead of
the snake, to the equation ∆u =
u
α
with 1
< α
≤
2. This required an
enhancement of the superdiffusion theory which can be of interest for any-
body who works on application of probabilistic methods to mathematical
analysis.
The goal of this book is to give a self-contained presentation of these
new developments. The book may be considered as a continuation of the
monograph [D]. In the first three chapters we give an overview of the theory
presented in
[D]
without duplicating the proofs which can be found in [D].
The book can be read independently of [D]. [It might be even useful to read
the first three chapters before reading
[D].]
In a series of papers (including [MV98a], [MV98b] and [MV04]) M. Mar-
cus and L. V´ron investigated positive solutions of the equation ∆u =
u
α
e
1
The dissertation of Mselati was published in 2004 (see [Ms04]).
v

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