Brownian motions on a metric graph are defined. Their generators are characterized as Laplace operators subject to Wentzell boundary at every vertex. Conversely, given a set of Wentzell boundary conditions at the vertices of a metric graph, a Brownian motion is constructed pathwise on this graph so that its generator satisfies the given boundary conditions.

1.
S.
Albeverio
and
M.
Röckner
, “
Classical Dirichlet forms on topological spaces—Construction of an associated diffusion process
,”
Probab. Theory Relat. Fields
83
,
405
434
(
1989
).
2.
L.
Bachelier
, “
Théorie de la spéculation
,”
Ann. Sci. Ec. Normale Super.
17
,
21
88
(
1900
).
3.
M.
Barlow
,
J.
Pitman
, and
M.
Yor
, “
On Walsh's Brownian motion
,” in
Séminaire de Probabilités XXIII
,
Lecture Notes in Mathematics
Vol.
1372
, edited by
J.
Azèma
,
P. A.
Meyer
, and
M.
Yor
(
Springer–Verlag
,
Berlin/Heidelberg/New York
,
1989
), pp.
275
293
.
4.
M.
Barlow
,
J.
Pitman
, and
M.
Yor
, “
Une extension multidimensionnelle de la loi de l' arc sinus
,” in
Séminaire de Probabilités XXIII
,
Lecture Notes in Mathematics
Vol.
1372
, edited by
J.
Azèma
,
P. A.
Meyer
, and
M.
Yor
(
Springer–Verlag
,
Berlin/Heidelberg/New York
,
1989
), pp.
294
313
.
5.
H.
Bauer
,
Wahrscheinlichkeitstheorie
, 4th ed. (
de Gruyter
,
Berlin/ New York
,
2002
).
6.
J. R.
Baxter
and
R. V.
Chacon
, “
The equivalence of diffusions on networks to Brownian motion
,”
Contemp. Math.
26
,
33
48
(
1984
).
7.
O.
Bénichou
and
J.
Desbois
, “
Exit and occupation times for Brownian motion on graphs with general drift and diffusion constant
,”
J. Phys. A
42
,
015004
(
2009
).
8.
J. M.
Bismut
,
Large Deviations and the Malliavin Calculus
(
Birkhäuser
,
Boston
,
1984
).
9.
J. M.
Bismut
, “
The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs
,”
Invent. Math.
83
,
91
151
(
1986
).
10.
R.
Blumenthal
, “
An extended Markov property
,”
Trans. Am. Math. Soc.
85
,
52
72
(
1957
).
11.
R. M.
Blumenthal
and
R. K.
Getoor
,
Markov Processes and Potential Theory
(
Academic
,
New York/London
,
1968
).
12.
A.
Comtet
,
J.
Desbois
, and
S. N.
Majumdar
, “
The local time distribution of a particle diffusing on a graph
,”
J. Phys. A
35
,
L687
L694
(
2002
).
13.
D. S.
Dean
and
K. M.
Jansons
, “
Brownian excursions on combs
,”
J. Stat. Phys.
70
,
1313
1332
(
1993
).
14.
C.
Dellacherie
and
P.-A.
Meyer
,
Probabilities and Potential C
(
North–Holland
,
1988
).
15.
J.
Desbois
, “
Occupation times distribution for Brownian motion on graphs
,”
J. Phys. A
35
,
L673
L678
(
2002
).
16.
M.
Donsker
and
S. R. S.
Varadhan
, “
Asymptotic evaluation of certain Wiener integrals for large time
,”
Functional Integration and Its Applications
, edited by
A. M.
Arthurs
(
Oxford University Press
,
London/New York
,
1975
), pp.
15
33
.
17.
E. B.
Dynkin
,
Die Grundlagen der Theorie der Markoffschen Prozesse
(
Springer–Verlag
,
Berlin/Göttingen/Heidelberg
,
1961
).
18.
E. B.
Dynkin
,
Markov Processes
(
Springer-Verlag
,
Berlin/Heidelberg/New York
,
1965
), Vol.
1
.
19.
E. B.
Dynkin
,
Markov Processes
(
Springer-Verlag
,
Berlin/Heidelberg/New York
,
1965
), Vol.
2
.
20.
E. B.
Dynkin
and
A. A.
Juschkewitsch
,
Sätze und Aufgaben über Markoffsche Prozesse
(
Springer-Verlag
,
Berlin/Heidelberg/New York
,
1969
).
21.
A.
Einstein
, “
Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen
,”
Ann. d. Phys.
17
,
549
560
(
1905
).
22.
A.
Einstein
, “
Zur Theorie der Brownschen Bewegung
,”
Ann. d. Phys.
19
,
371
381
(
1906
).
23.
N.
Eisenbaum
and
H.
Kaspi
, “
On the Markov property of local time for Markov processes on graphs
,”
Stochastic Proc. Appl.
64
,
153
172
(
1996
).
24.
W.
Feller
, “
The parabolic differential equations and the associated semi-groups of transformations
,”
Ann. Math.
3
,
468
519
(
1952
).
25.
W.
Feller
, “
Diffusion processes in one dimension
,”
Trans. Am. Math. Soc.
77
,
1
31
(
1954
).
26.
W.
Feller
, “
The general diffusion operator and positivity preserving semi-groups in one dimension
,”
Ann. Math.
60
,
417
436
(
1954
).
27.
R. P.
Feynman
, “
Space-time approach to non-relativistic quantum mechanics
,”
Rev. Mod. Phys.
20
,
367
387
(
1948
).
28.
J. L.
Folks
and
R. S.
Chhikara
, “
The inverse Gaussian distribution and its statistical application – A review
,”
J. R. Stat. Soc. Ser. B (Methodol.)
40
,
263
289
(
1978
).
29.
A. I.
Freidlin
and
A. D.
Wentzell
, “
Diffusion processes on graphs and the averaging principle
,”
Ann. Probab.
21
,
2215
2245
(
1993
).
30.
M.
Freidlin
and
S.-J.
Sheu
, “
Diffusion processes on graphs: stochastic differential equations, large deviation principle
,”
Probab. Theory Relat. Fields
116
,
181
220
(
2000
).
31.
A.
Friedman
and
C.
Huang
, “
Diffusion in network
,”
J. Math. Anal. Appl.
183
,
352
384
(
1994
).
32.
M.
Fukushima
,
Dirichlet Forms and Markov Processes
(
North-Holland
,
Amsterdam/Tokyo
,
1980
).
33.
J.
Glimm
and
A.
Jaffe
,
Quantum Physics: A Functional Integral Point of View
(
Springer
,
Berlin/Heidelberg/New York
,
1981
).
34.
L.
Gross
, “
Abstract Wiener spaces
,” in
Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability
(
University California Press
,
Berkeley
,
1967
).
35.
N.
Grunewald
, “
Martingales on graphs
,” Ph.D. dissertation (
Mathematics Institute, University of Warwick
,
1999
).
36.
T.
Hida
,
Analysis of Brownian Functionals
,
Carleton Lecture Notes in Mathematics Nr. 13
(
Carleton University
,
1975
).
37.
T.
Hida
,
H.-H.
Kuo
,
J.
Potthoff
, and
L.
Streit
,
White Noise— An Infinite Dimensional Calculus
(
Kluwer/Academic
,
Dordrecht
,
1993
).
38.
G. A.
Hunt
, “
Some theorems concerning Brownian motion
,”
Trans. Am. Math. Soc.
81
,
294
319
(
1956
).
39.
K.
Itô
, “
Stochastic integrals
,”
Proc. Imp. Acad. (Tokyo)
20
,
519
524
(
1944
).
40.
K.
Itô
, “
On a stochastic integral equation
,”
Proc. Imp. Acad. (Tokyo)
22
,
32
35
(
1946
).
41.
K.
Itô
and
H. P.
McKean
 Jr.
, “
Brownian motions on a half line
,”
Illinois J. Math.
7
,
181
231
(
1963
).
42.
K.
Itô
and
H. P.
McKean
 Jr.
,
Diffusion Processes and their Sample Paths
, 2nd ed. (
Springer–Verlag
,
Berlin/Heidelberg/New York
,
1974
).
43.
D.
Jungnickel
,
Graphs, Networks and Algorithms
(
Springer-Verlag
,
Berlin/Heidelberg/New York
,
2007
).
44.
M.
Kac
, “
On distributions of certain Wiener functionals
,”
Trans. Am. Math. Soc.
65
,
1
13
(
1949
).
45.
M.
Kac
, “
On some connections between probability theory and differential and integral equations
,” in
Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability
, edited by
J.
Neyman
(
University California Press
,
Berkeley
,
1951
), pp.
189
215
.
46.
O.
Kallenberg
,
Foundations of Modern Probability
(
Springer
,
Berlin/Heidelberg/New York
,
2002
).
47.
I.
Karatzas
and
S. E.
Shreve
,
Brownian Motion and Stochastic Calculus
, 2nd ed. (
Springer-Verlag
,
Berlin/Heidelberg/New York
,
1991
).
48.
F. B.
Knight
, “
Essentials of Brownian motion and diffusion
,”
Mathematical Surveys and Monographs
(
American Mathematical Society
,
Providence, RI
,
1981
), Vol.
18
.
49.
V.
Kostrykin
,
J.
Potthoff
, and
R.
Schrader
, “
Heat kernels on metric graphs and a trace formula
,” in
Adventures in Mathematical Physics
,
Contemporary Mathematics
Vol.
447
, edited by
F.
Germinet
and
P.
Hislop
(
American Mathematical Society
,
Providence, RI
,
2007
).
50.
V.
Kostrykin
,
J.
Potthoff
, and
R.
Schrader
, “
Brownian motions on metric graphs: Feller Brownian motions on intervals revisited
,” e-print arXiv:1008.3761.
51.
V.
Kostrykin
,
J.
Potthoff
, and
R.
Schrader
, “
Contraction semigroups on metric graphs
,” in
Analysis on Graphs and Its Applications
edited by
P.
Exner
,
J. P.
Keating
,
P.
Kuchment
,
T.
Sunada
, and
A.
Teplyaev
,
Proceedings of the Symposia in Pure Mathematics
Vol.
77
(
American Mathematical Society
,
Providence
,
2010
), pp.
423
458
.
52.
V.
Kostrykin
,
J.
Potthoff
, and
R.
Schrader
, “
Construction of the paths of Brownian motions on star graphs
,”
Commun. Stoch. Anal.
(to appear),
53.
V.
Kostrykin
,
J.
Potthoff
, and
R.
Schrader
, “
Finite propagation speed for solutions of the wave equation on metric graphs
,” e-print arXiv:1106.0817.
54.
V.
Kostrykin
and
R.
Schrader
, “
Kirchhoff's rule for quantum wires
,”
J. Phys. A
32
,
595
630
(
1999
).
55.
V.
Kostrykin
and
R.
Schrader
, “
Kirchhoff's rule for quantum wires II: The inverse problem with possible applications to quantum computers
,”
Fortschr. Phys.
48
,
703
716
(
2000
).
56.
V.
Kostrykin
and
R.
Schrader
, “
Laplacians on metric graphs: Eigenvalues, resolvents and semigroups
,” in
Quantum Graphs and Their Applications
,
Contemporary Mathematics
Vol.
415
, edited by
G.
Berkolaiko
,
R.
Carlson
,
S. A.
Fulling
, and
P.
Kuchment
(
American Mathematical Society
,
2006
).
57.
V.
Kostrykin
and
R.
Schrader
, “
The inverse scattering problem for metric graphs and the traveling salesman problem
,” e-print arXiv:math-ph/0603010.
58.
W. B.
Krebs
, “
Brownian motion on a continuum tree
,”
Probab. Theory Relat. Fields
101
,
421
433
(
1995
).
59.
P.
Kuchment
, “
Quantum graphs I: Some basic structures
,”
Waves Random Complex Media
14
,
S107
S128
(
2004
).
60.
A.
Lejay
, “
On the constructions of the skew Brownian motion
,”
Probab. Surv.
3
,
413
466
(
2006
).
61.
P.
Lévy
,
Théorie de l'Addition des Variables Aléatoires
(
Gauthier-Villars
,
Paris
,
1937
).
62.
P.
Lèvy
,
Processus Stochastiques et Mouvement Brownien
, 2nd ed. (
Gauthier-Villars
,
Paris
,
1965
), 1st ed. (1948).
63.
P.
Malliavin
,
Géometrie Différentielle Stochastique
(
Presse de l' Université de Montréal
,
Montréal
,
1978
).
64.
P.
Malliavin
,
Stochastic Analysis
(
Springer–Verlag
,
Berlin/Heidelberg/New York
,
1997
).
65.
E.
Nelson
, “
Regular probability measures on function space
,”
Ann. Math.
69
,
630
643
(
1959
).
66.
E.
Nelson
, “
Feynman integrals and the Schrödinger equation
,”
J. Math. Phys.
5
,
332
343
(
1964
).
67.
E.
Nelson
,
Dynamical Theories of Brownian Motion
(
Princeton University Press
,
1967
).
68.
E.
Nelson
, “
Probability theory and Euclidean quantum field theory
,” in
Constructive Quantum Field Theory
,
Lecture Notes in Physics
Vol.
25
, edited by
G.
Velo
and
A.
Wightman
(
Springer
,
Berlin/Heidelberg/New York
,
1973
), pp.
94
124
.
69.
K. R.
Parthasarathy
,
Probability Measures on Metric Spaces
(
Academic
,
New York/London
,
1967
).
70.
D.
Ray
, “
Stationary Markov processes with continuous paths
,”
Trans. Am. Math. Soc.
82
,
452
493
(
1956
).
71.
D.
Revuz
and
M.
Yor
,
Continuous Martingales and Brownian Motion
(
Springer–Verlag
,
Berlin/Heidelberg/New York
,
1999
).
72.
L. C. G.
Rogers
, “
Itô excursion theory via resolvents
,”
Probab. Theory Relat. Fields
63
,
237
255
(
1983
).
73.
T. S.
Salisbury
, “
Construction of right processes from excursions
,”
Probab. Theory Relat. Fields
73
,
351
367
(
1986
).
74.
T. S.
Salisbury
, “
On the Itô excursion process
,”
Probab. Thoery Relat. Fields
73
,
319
350
(
1986
).
75.
M.
Schilder
, “
Some asymptotic formulas for Wiener integrals
,”
Trans. Am. Math. Soc.
125
,
63
85
(
1966
).
76.
R.
Schrader
, “
Finite propagation speed and causal free quantum fields on networks
,”
J. Phys. A: Math. Theor.
42
,
495401
(
2009
).
77.
E.
Schrödinger
, “
Zur Theorie der Fall- und Steigversuche an Teilchen mit Brownscher Bewegung
,”
Phys. Z.
16
,
289
295
(
1915
).
78.
J.
Schwinger
, “
On the Euclidean structure of relativistic field theory
,”
Proc. Natl. Acad. Sci. U.S.A.
44
,
956
965
(
1958
).
79.
J.
Schwinger
, “
Euclidean quantum electrodynamics
,”
Phys. Rev.
115
,
721
731
(
1959
).
80.
M. L.
Silverstein
,
Symmetric Markov Processes
,
Lecture Notes in Mathematics 426
(
Springer
,
Berlin/Heidelberg/New York
,
1975
).
81.
B.
Simon
,
The P(ϕ)2 Euclidean (Quantum) Field Theory
(
Princeton University Press
,
Princeton
,
1974
).
82.
B.
Simon
,
Functional Integration and Quantum Physics
(
Academic
,
New York/San Francisco/London
,
1979
).
83.
K.
Symanzik
Euclidean quantum field theory
,” in
Local Quantum Theory
, edited by
R.
Jost
(
Academic
,
New York
,
1969
).
84.
M. C. K.
Tweedie
, “
Inverse statistical variates
,”
Nature (London)
155
,
453
(
1945
).
85.
N. Th.
Varopoulos
, “
Long range estimates for Markov chains
,”
Bull. Sci. Math.
109
,
225
252
(
1985
).
86.
M.
von Smoluchowski
, “
Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen
,”
Ann. d. Phys.
21
,
756
780
(
1906
).
87.
It seems that Schrödinger (Ref. 77) was the first to introduce the notion of a first passage time, (in German Erstpassagezeit), i.e., a special type of stopping time, in the continuous time context of the Brownian motion process. It is striking that this article and the parallel work of Smoluchowski (Ref. 88) has practically gone unnoticed in the physics literature, while being cited by statisticians, e.g., Refs. 28 and 84.
88.
M.
von Smoluchowski
, “
Notiz über die Berechnung der Brownschen Molekularbewegung bei der Ehrenhaft-Millikan'schen Versuchsanordnung
,”
Phys. Z.
XVI
,
318
321
(
1915
).
89.
J. B.
Walsh
, “
A diffusion with a discontinuous local time
,”
Astérisque
52–53
,
37
45
(
1978
).
90.
N.
Wiener
, “
Differential space
,”
J. Math. Phys.
2
,
131
174
(
1923
).
91.
N.
Wiener
, “
The homogeneous chaos
,”
Am. J. Math.
60
,
897
936
(
1938
).
92.
D.
Williams
,
Diffusions, Markov Processes, and Martingales
(
Wiley
,
Chichester/New York/Brisbane/Toronto
,
1979
).
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