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L´evy-Type Processes: Construction,
Approximation and Sample Path Properties
Bj¨orn B¨ottcher, Ren´e L. Schilling, and Jian Wang
Abstract L´evy-type processes are the next-of-kin of L´evy processes, their relation
being that of an elliptic diffusion to a Brownian motion. In other words, a L´evytype process resembles at any point x of the state space some L´evy process—but the
characteristic exponent of this L´evy process may depend on the current position x.
For example, one could think of a stable-like process where the index of stability
changes while the process moves along. This indicates that L´evy-type processes are
not homogeneous in space.
In this tract we discuss recent developments for L´evy-type processes and more
general stochastic processes that behave locally like a L´evy process. Although our
exposition has the character of a survey, quite a few results are extensions of known
theorems, and some are new. Our focus is on the symbol of a L´evy-type process:
This is a non-random function which is the L´evy-type counterpart of the characteristic exponent of a L´evy process. We characterize those processes admitting a
symbol, we discuss various possibilities to construct a stochastic process for a given
symbol, and we show how one can use the symbol in order to describe the sample
path properties of the underlying process. Finally, we use the symbol to approximate
and simulate L´evy-type processes.
The more complicated nature of these objects requires many tools and techniques
which appear naturally in this context—e.g. from functional analysis, Feller semiBj¨orn B¨ottcher
TU Dresden, Institute of Mathematical Stochastics, D–01062 Dresden, Germany.
e-mail: [email protected];
homepage: http://www.math.tu-dresden.de/˜boettch/
Ren´e L. Schilling
TU Dresden, Institute of Mathematical Stochastics, D–01062 Dresden, Germany.
e-mail: [email protected];
homepage: http://www.math.tu-dresden.de/sto/schilling/
Jian Wang
School of Mathematics and Computer Science, Fujian Normal University, 350007 Fuzhou, China.
e-mail: [email protected];
homepage: http://math.fjnu.edu.cn/newmath/index.html
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Bj¨orn B¨ottcher, Ren´e L. Schilling, and Jian Wang
groups, pseudo-differential operators, function spaces to name but a few—but which
might not be too familiar to a probabilist. We tried to put equal weight on results
and methods, giving a fairly complete survey of the state of the art of all topics
discussed, sometimes with full proofs and often with ideas of the proofs whenever
feasible or enlightening. Open problems and many references are given for future
research.
AMS Subject Classification 2010:
Primary:
60J25, 60J35, 60G17.
Secondary: 35S05, 47D03, 47D06, 47G20, 47G30, 60E10, 60G51, 60J75,
60H10.
Key words: Blumenthal–Getoor index, coupling, evolution equation, Feller process, Feller semigroup, functional inequalities, function spaces, Hausdorff dimension, integro-differential operator, L´evy–Khintchine formula, L´evy-type process,
Markov process, martingale problem, p-variation, path properties, probability estimates, pseudo-differential operator, random time change, semimartingale characteristics, stochastic differential equation, subordination.
Contents
L´evy-Type Processes: Construction, Approximation and Sample Path
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Bj¨orn B¨ottcher, Ren´e L. Schilling, and Jian Wang
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Summary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1
A Primer on Feller Semigroups and Feller Processes . . . . . . . . . . . . 11
1.1
Feller Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2
From Feller Processes to Feller Semigroups—and Back. . 22
1.3
Resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4
Generators of Feller Semigroups and Processes . . . . . . . . 26
1.5
Feller Semigroups and L p -Spaces . . . . . . . . . . . . . . . . . . . . 35
2
Feller Generators and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1
Orientation: Convolution Semigroups and L´evy Processes 38
2.2
Positive and Negative Definite Functions . . . . . . . . . . . . . . 46
2.3
Generator and Symbol of a Feller Semigroup on Rd . . . . 50
2.4
The symbol of a Stochastic Process . . . . . . . . . . . . . . . . . . 61
2.5
The Semimartingale Connection . . . . . . . . . . . . . . . . . . . . . 66
3
Construction of Feller Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.1
The Hille-Yosida Construction . . . . . . . . . . . . . . . . . . . . . . 71
3.2
Stochastic Differential Equations (SDEs) . . . . . . . . . . . . . . 75
3.3
Dirichlet Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.4
Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5
The Martingale Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6
Unbounded Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4
Transformations of Feller Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.1
Random Time Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2
Subordination in the Sense of Bochner . . . . . . . . . . . . . . . . 100
4.3
Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4
Feynman–Kac Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5
Sample Path Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.1
Probability Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
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5.2
Hausdorff Dimension and Indices . . . . . . . . . . . . . . . . . . . . 121
5.3
Asymptotic Behaviour of the Sample Paths . . . . . . . . . . . . 125
5.4
The Strong Variation of the Sample Paths . . . . . . . . . . . . . 128
5.5
Besov Regularity of Feller Processes . . . . . . . . . . . . . . . . . 131
6
Global Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.1
Functional Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.2
Coupling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.3
Transience and Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.1
Constructive Approximation . . . . . . . . . . . . . . . . . . . . . . . . 160
7.2
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8
Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
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Preface
Behind every decent Markov process there is a family of L´evy processes. Indeed, let
(Xt )t>0 be a Markov process with state space Rd and assume, for the moment, that
the limit
1 − Ex eiξ ·(Xt −x)
= q(x, ξ ) ∀x, ξ ∈ Rd
t→0
t
lim
(F)
exists such that the function ξ 7→ q(x, ξ ) is continuous. We will see below that this
is enough to guarantee that q(x, ·) is, for each x ∈ Rd , the characteristic exponent of
a L´evy process; as such, it enjoys a L´evy–Khintchine representation
1
q(x, ξ ) = −il(x) · ξ + ξ · Q(x)ξ +
2
Z
Rd \{0}
1 − eiy·ξ + iξ · y1[−1,1] (|y|) N(x, dy)
where (l(x), Q(x), N(x, dy)) is for every fixed x ∈ Rd a L´evy triplet. The function
q : Rd × Rd → C is called the symbol of the process. The processes which admit a symbol behave locally like a L´evy process, and their infinitesimal generators
resemble the generators of L´evy processes with variable, i.e. x-dependent, coefficients—this justifies the name L´evy-type processes. This guides us to the main
topics of the present tract:
Characterization: For which Markov processes does the limit (F) exist?
Construction:
Is there a L´evy-type process with a given symbol q(x, ξ )?
Is there a 1-to-1 correspondence between symbols and processes?
Sample paths:
Can we use the symbol q(x, ξ ) in order to describe the sample path behaviour of the process?
Approximation: Is it possible to use q(x, ξ ) to approximate and to simulate
the process?
Let us put this point of view into perspective by considering first of all some
d-dimensional L´evy process (Xt )t>0 . Being a (strong) Markov process, (Xt )t>0 can
be described by the transition function pt (x, dy) = Px (Xt ∈ dy) = P(Xt + x ∈ dy)
which, in turn, is uniquely characterized by the characteristic function
Ex eiξ ·Xt =
Z
Rd
eiξ ·y pt (x, dy) = eiξ ·x e−tψ(ξ )
(1)
and the characteristic exponent ψ. Thus,
1−
Z
Rd
eiξ ·(y−x) pt (x, dy) = tψ(ξ ) + o(t) as t → 0
(2)
and, with some effort, we can derive from this the L´evy–Khintchine representation
of the exponent ψ
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Survey B¨ottcher – Schilling – Wang
1
ψ(ξ ) = −il · ξ + ξ · Qξ +
2
Z
Rd \{0}
1 − eiy·ξ + iξ · y1[−1,1] (|y|) ν(dy)
(3)
where (l, Q, ν) is the L´evy triplet. The key observation is that the family of measures
t −1 pt (x, B + x) = t −1 P(Xt ∈ B) converges1 to the L´evy measure ν(B) as t → 0 for
¯
all Borel sets B ⊂ Rd \ {0} satisfying ν(B¯ \ B◦ ) = 0 and 0 ∈
/ B.
In our calculation there is only one place where we used L´evy processes: The
second equality sign in (1) which is the consequence of the translation invariance
(spatial homogeneity) and infinite divisibility of a L´evy process. If we do away with
it, and if we only assume that (Xt )t>0 is strong Markov with transition function
(pt (x, dy))t>0, x∈Rd we still have that
λt (x, ξ ) := Ex eiξ ·(Xt −x) =
Z
Rd
eiξ ·(y−x) pt (x, dy).
(10 )
Assume we knew that t −1 pt (x, B + x) has, as t → 0, for every x ∈ Rd and suitable
Borel sets B ⊂ Rd \ {0}, a limit N(x, B) which is a kernel on Rd × B(Rd \ {0}).
Then we would get, as in (2),
1 − λt (x, ξ ) = −tq(x, ξ ) + o(t) as t → 0.
(20 )
But, what can be said about q(x, ξ )?
With some elementary harmonic analysis this can be worked out. Since ξ 7→
λt (x, ξ ) is a characteristic function, it is continuous and positive definite (see page
46 for the definition); and since 1 > λt (x, 0) > |λt (x, ξ )|, we get from this that
n
∑
λt (x, ξ j − ξk ) − 1 µ j µ¯ k > 0
(4)
j,k=1
for all n > 0, ξ1 , . . . , ξn ∈ Rd and µ1 , . . . , µn ∈ C with ∑nj=1 µ j = 0. This means
that ξ 7→ λt (x, ξ ) − 1 is continuous and conditionally positive definite2 (because
of the condition ∑ j µ j = 0, cf. page 47). The important point is now that every
continuous and conditionally positive definite function enjoys a L´evy–Khintchine
representation.
Obviously, inequality (4) remains valid if we divide by t and let t → 0, so
1 − Ex eiξ ·(Xt −x)
= q(x, ξ ) ∀x, ξ ∈ Rd
t→0
t
lim
(F)
defines a conditionally positive definite function ξ 7→ q(x, ξ ). If it is also continuous,
then it has for every fixed x ∈ Rd a L´evy–Khintchine representation, and each q(x, ·)
is the characteristic exponent of a L´evy process. We will call the function q(x, ξ )
the symbol of the process (Xt )t>0 . In this sense it is correct to say that behind every
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2
The classical proofs use here L´evy’s continuity theorem or the Helly–Bray theorem.
also known as negative definite, and we will prefer this notion in the sequel, cf. Section 2.2
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(x)
decent Markov process (Xt )t>0 there is a family of L´evy processes Lt )t>0,x∈Rd
whose characteristic exponents are given by (F), and we are back at the point where
we started our discussion.
Sufficient conditions for the limit (F) to exist are best described by a list of
L´evy-type processes: L´evy processes, of course, (cf. Section 2.1), any Feller process whose infinitesimal generator has a sufficiently rich domain (Sections 2.3, 2.4),
many L´evy-driven stochastic differential equations (Section 3.2), or temporally homogeneous Markovian jump-diffusion semimartingales (Section 2.5) provided that
their extended generator contains sufficiently many functions. As it turns out, the
symbol q(x, ξ ) encodes, via its L´evy–Khintchine representation and the (necessarily) x-dependent L´evy triplet, the semimartingale characteristics of the stochastic
process; moreover, it yields a simple representation of the infinitesimal generator as
a pseudo-differential operator.
We are not aware of necessary conditions such that (F) defines a negative definite
symbol, although the class of temporally homogeneous Markovian jump-diffusion
semimartingales looks pretty much to be the largest class of decent strong Markov
processes admitting a symbol.
Most of our results hold for any decent strong Markov process admitting a symbol q(x, ξ ), but we restrict our attention to Feller processes where decency comes
from the natural assumption that the compactly supported smooth functions Cc∞ (Rd )
are contained in the domain of the infinitesimal generator. The key results in this direction are our short proof of the Courr`ege–von Waldenfels theorem (Theorem 2.21)
and the probabilistic formula for the symbol, Theorems 2.36 and 2.44.
Let us briefly explain how the material is organized. The Primer on Feller Semigroups, Chapter 1, is included in order to make the material accessible for the
novice, but also to serve as a reference. For the more experienced reader, the ideal
point of departure should be Section 2.1 on L´evy processes which leads directly to
the characterization of Feller processes. Among the central results of Chapter 2 is
the characterization of the generators as pseudo-differential operators and the fact
that Feller processes are semimartingales: In both cases the symbol q(x, ξ ) and its
x-dependent L´evy triplet are instrumental. Chapter 3 is devoted to various construction methods for Feller processes. This is probably the most technical part of our
treatise since techniques from different areas of mathematics come to bearing; it is
already difficult to describe the results, to present complete proofs in this essay is
near impossible. Nevertheless we tried to describe the ideas how things fit together,
and we hope that the interested reader follows up on the references provided. Perturbations and time-changes for Feller processes are briefly discussed in Chapter 4.
In particular, we obtain conditions such that the Feller property is preserved under
these transformations. From Chapter 5 onwards, things become more probabilistic:
Now we show how to use the symbol q(x, ξ ) in order to describe the behaviour of the
sample paths of a Feller process. For L´evy processes this approach has a long tradition starting with the papers by Blumenthal–Getoor [31, 32] in the early sixties;
a survey is given in Fristedt [111]. The principal tool for these investigations are
probability estimates for the running maximum of the process in terms of the symbol (Section 5.1). Using these estimates we can define Blumenthal–Getoor–Pruitt
8
Survey B¨ottcher – Schilling – Wang
indices for Feller processes which, in turn, allow us to find bounds for the Hausdorff dimension of the sample paths, describe the (polynomial) short- and long-time
asymptotics of the paths, their p-variation, their Besov regularity etc. Returning to
the level of transition semigroups we then investigate global properties (in the sense
of Fukushima et al.) in Chapter 6. We focus on functional inequalities and their
stability under subordination and on coupling methods; the latter are explained in
detail for L´evy- and linear Ornstein–Uhlenbeck processes. The classical topics of
transience and recurrence are discussed from the perspective of Meyn and Tweedie,
with an emphasis on stable-like processes. In the final Chapter 7 we show how the
viewpoint of a Feller process being locally L´evy can be used to approximate the
sample paths of Feller processes. This allows us, for the first time, to simulate Feller
processes with unbounded coefficients. We close this treatise with a list of open
problems which we think are important for the further development of the subject.
We cannot cover all aspects of L´evy-type processes in this survey. Notable omissions are probabilistic potential theory, the general theory of Dirichlet forms, heat
kernel estimates and processes on domains. Our choice of material was, of course,
influenced by personal liking, by our own research interests and by the desire to
have a clear focus. Some topics, e.g. probabilistic potential theory and Dirichlet
forms, are more naturally set in the wider framework of general Markov processes
and there are, indeed, monographs which we think are hard to match: In potential
theory there are Chung’s books [69, 70] (for Feller processes), Sharpe [289] (for
general Markov processes), Port–Stone [230] and Bertoin [27] (for Brownian motion and L´evy processes) and for Dirichlet forms there is Fukushima et al. [113]
(for the symmetric case), and Ma–R¨ockner [205] (for the non-symmetric case).
Heat kernel estimates are usually discussed in an L2 -framework, cf. Chen [65] for
an excellent survey, or for various perturbations of stable L´evy processes (also on
domains), e.g. as in Chen–Kim–Song [66, 67]. An interesting geometric approach
has recently been proposed by [158]. Finally, processes on domains with general
Wentzell boundary conditions [339] for the generator are still a problem: While
some progress has been made in the one-dimensional case (cf. Mandl [209], Langer
and co-workers [194, 195]), the multidimensional case is wide open, and the best
treatment is Taira [302].
A few words on the style of this treatise are in order. Some time ago, we have
been invited to contribute a survey paper to the L´evy Matters subseries of the
Springer Lecture Notes in Mathematics, updating the earlier paper L´evy-type processes and pseudo-differential operators by N. Jacob and one of the present authors.
Soon, however, it became clear that the developments in the past decade have been
quite substantial while, on the other hand, much of the material is scattered throughout the literature and that a comprehensive treatise on Feller processes is missing.
With this essay we try to fill this gap, providing a reliable source for reference (especially for those elusive folklore results), making a technically demanding area easily
accessible to future generations of researchers and, at the same time, giving a snapshot of the state-of-the-art of the subject. Just as one would expect in a survey, we do
not always (want to) give detailed proofs, but we provide precise references when-
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9
ever we omit proofs or give only a rough outline of the argument (sometimes also
sailing under the nickname ‘proof’). On the other hand, quite a few theorems are
new or contain substantial improvements of known results, and in all those cases we
do include full proofs or describe the necessary changes to the literature. We hope
that the exposition is useful for and accessible to anyone with a working knowledge
of L´evy- or continuous-time Markov processes and some basic functional analysis.
Finally, it is a pleasure to acknowledge the support of quite a few people. Niels
Jacob has our best thanks, his ideas run through the whole text, and we shall think
it a success if it pleases him.
Without the named (and, as we fear, often unnamed) contributions of our coauthors and fellow scientists such a survey would not have been possible; we are
grateful that we can present and build on their results. Xiaoping Chen, Katharina
Fischer, Julian Hollender, Victorya Knopova, Franziska K¨uhn, Huaiqian Li, Felix
Lindner, Michael Schwarzenberger, and Nenghui Zhu read substantial portions of
various β -versions of this survey, pointed out many mistakes and inconsistencies,
and helped us to improve the text; the examples involving affine processes were
drafted by Michael Schwarzenberger.
Special thanks go to Claudia Kl¨uppelberg and the editors of the L´evy Matters
series for the invitation to write and their constant encouragement to finish this piece.
Finally, we thank our friends and families who—we are pretty sure of it—are
more than happy that this work has come to an end.
Dresden and Fuzhou
April 2013
Bj¨orn B¨ottcher
Ren´e Schilling
Jian Wang
Index
affine process, 14, 18, 23, 28
anisotropic function space, 74, 105
Aronson’s estimates, 14
Bernstein function, 40, 50, 101
Besov space, 132
Beurling–Deny representation, 81
Blumenthal–Getoor–Pruitt index, 122, 127
bounded coefficients, 59
bp-convergence, 11
Brownian motion, 13, 23, 26, 40
coupling, 146
not strongly continuous on Cb , 17
c`adl`ag modification, 24
Cb -Feller property, 17
Chapman–Kolmogorov equations, 16
characteristic exponent, 39, 47
characteristic function, 37
characteristic operator, 34
extends generator, 34
closed operator, 26
complete class, 69
completely monotone, 50
compound Poisson process, 40, 143
conservative, 41, 59, 158
criteria for —, 60
constant coefficients, 55
convolution semigroup, 13, 20, 23, 27, 38, 40
core, 31
— of L´evy generator, 44
— of O-U generator, 28
sufficient conditions, 32
coupling, 142
— of L´evy process, 142, 144
— of subordinate process, 146
— property, 142
— time, 142
— time estimate, 148
successful —, 142
Courr`ege–v. Waldenfels theorem, 51, 65
covariance matrix, 39
differential characteristics, 68
diffusion, 14, 71, 75–76
— with jumps, 67
Dirichlet form, 79
gives Feller process, 84
Dirichlet operator, 29, 80
dissipative, 29
distribution (generalized function), 50
drift coefficient, 39
Dynkin’s formula, 34
Dynkin–Kinney criterion, 35
Dynkin–Reuter lemma, 30
example
not strongly Feller, 20
of discontinuous symbol, 58
of generators, 26–28, 158
of processes, 22–23, 39–40, 66–67, 96–97,
130
of semigroups, 13–15, 17–18, 39–40
of symbols, 39–40, 55–56, 77, 130
transience/recurrence, 157–159
excessive, 100
extended generator, 33
fast positiv, 52
Feller generator, 26
domain, 31
extension, 63
integro-differential form, 51
pseudo-differential form, 54
183
184
Feller process, 22
— — generated by SDE, 76
— — gives Dirichlet form, 83, 84
— — is Itˆo process, 68
— — solves SDE, 78
asymptotic behaviour, 126, 127
c`adl`ag modification, 24
conservative, 60
exponential moment, 119
generated by SDE, 76
solution of martingale problem, 90
Feller property, 12, 25
Cb - — —, 17
strong — —, 19, 107, 108, 145
Feller semigroup, 12, 16
Cb - — —, 17
extension to L p , 35–37
strong — —, 19
vs. Cb -Feller semigroup, 18–19, 70
vs. Feller resolvent, 25
vs. strong Feller semigroup, 20, 21
Feynman–Kac formula, 106
Fourier multiplier, 42
Fourier transform, 37
inverse — —, 37
full class, 69
full generator, 32
fundamental solution, 86
generator, 26
characteristic operator, 34
core, 31
extended —, 33
full —, 32
local operator, 34–35
local part, 51
non-local part, 51
of a Feller process, 51, 54
of a L´evy process, 42, 44
of a Newtonian system, 159
of an affine process, 28
of an O-U process, 27, 28
pointwise —, 31
subordinate —, 101
weak —, 31
h-transform, 99
Hartman–Wintner condition, 149
Has’minskii’s lemma, 107
Hausdorff dimension, 121
Hille–Yosida–Ray theorem, 30, 71
Hunt process, 82
infinitely divisible, 38, 47
Index
integro-differential operator, 44
irreducible, 154
Itˆo process, 68
jump measure, 39, 68
Kato condition, 107
Kolmogorov equation, 86
L´evy
— density, 14
— measure, 39, 68
— process (with killing), 40
— system, 68
— triplet, 39, 48
L´evy process, 13, 23, 40, 81
— — with killing, 40
coupling, 142, 144
operator core for — —, 44
subordinate — —, 102
L´evy–Itˆo decomposition, 45
L´evy–Khintchine formula, 39, 47
for a subordinator, 101
proof, 54
life-time, 22
Liouville property, 151
local operator, 34, 51, 81
log-Sobolev inequality, 138
Lyapunov function, 156, 157
martingale problem, 89
localization, 93
well posedness, 89
maximal inequality, 111, 113
moment, 119
Nash inequality, 137
negative definite function, 47
negative definite symbol, 55
Orlicz space, 20
Ornstein–Uhlenbeck process, 13, 23, 27, 40,
55, 150
p-variation, 128
petite set, 155
Plancherel’s identity, 37
Poincar´e inequality, 139
Poisson process, 13, 17, 20, 22, 26, 40
positive definite, 46
conditionally — —, 47
positive maximum principle (PMP), 29, 50
as limit of Dirichlet operators, 29
limit of Dirichlet operators, 30
Index
potential, 107
potential operator, 25
pr`esque positif, 52
pseudo-differential operator, 55
recurrent, 154
resolvent, 25, 29
— equation, 25
— operator, 25
Riesz representation theorem, 16
sector condition, 80, 82, 113
semigroup, 12
adjoint —, 36
conservative —, 12, 21
contraction —, 12, 159
convolution —, 38
exponential formula, 25
Feller —, 12
inversion formula, 25
Lipschitz —, 162
Markov —, 12
norm continuous —, 17
positive —, 12
strongly continuous —, 12, 15–16, 159
sub-Markov —, 12
subordinate —, 101
symmetric —, 35
uniformly equi-bounded —, 159
vaguely continuous —, 13, 38
semimartingale, 67
— characteristics, 67
differential characteristics, 68
shift semigroup, 13, 17, 20, 22, 26, 39
Sobolev space, 71, 103, 131
spatially homogeneous, 23
stable semigroup, 13, 23, 27, 40
stochastic continuity, 24
vs. strong continuity, 24
stochastic differential equation
— generates Feller process, 76
185
strong continuity, 15–16
vs. stochastic continuity, 24
subordinate process, 101
subordination, 100, 145
— generator, 101
— preserves functional inequality, 137
functional calculus, 102
subordinator, 14, 23, 40
supermedian, 99
symbol, 42, 55
— of a Feller process, 55, 61, 65
— of a L´evy process, 61
— of a stochastic process, 66
— of an SDE, 76
bounded coefficients, 59
continuity in x, 58–60
defines differential characteristics, 68
negative definite —, 55
probabilistic formula, 65
smoothness in ξ , 49, 74–75
smoothness in x, 74–75
subordinate —, 102
T -model, 154
time change, 98
transient, 154
transition function, 22
translation invariant, 38, 81
Triebel–Lizorkin space, 132
truncation function, 39
ultracontractivity, 20
vs. strong Feller property, 20
vague convergence, 11
vanishing at infinity, 11
variable coefficients, 55
wavelet basis, 135
weak convergence, 11, 31