Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics
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Index
3-vector, 306
4-acceleration, 379, 385
4-momentum, 304
4-velocity, 304
A
0
, 260
A, 112
A, 65, 136, 137
ˆ
A, 138
BM,16
D, 187
D
∗
, 188
D
y
x(t), 189
D
y
∗
x(t), 189
D
F
y
∗
x(t), 189
D
F
∗
, 188
D
P
y
, 189
D
P
, 188
D
+
, 376
D
−
, 376
D
s
(M), 105
D
s
μ
(M), 105
D, 61
e, 169
F-mean derivative, 188
GL(n, R), 11
H, 40, 61
H
E
, 40, 41
H,66
M
4
, 300
∂M,4
O(n), 12
OM,64
P-mean derivative, 188
{P
t
}-progressively measurable, 347
p,5
p,60
Q,66
Q, 112
R
I
, 170
R
EE
, 169
(r, s)-tensor, 21
(r, s)-tensor bundle, 23
(r, s)-tensor field, 23
r-vector, 25
r-vector field, 25
SO(n), 12
so(3), 14
S, 87, 177
S
+
(n), 192
S
+
(n), 192
s-form, 26
T
n
,11
t ∧ τ, 119
V, 36, 60
α-lemma, 106
Γ , 43, 51
Γ
s,t
, 49, 177
Γ ,51
Γ
e
(·, ·), 169
ε-approximation, 99
κ(t), 206
ω-lemma, 106
,97
a.s., 115
acceleration, 367
4-acceleration, 379, 385
of a curve, 66
of a stochastic process, 357
action of a group, 12
left, 12
right, 12
action of the real line, 38
admissible
Y.E. Gliklikh, Global and Stochastic Analysis with Applications
to Mathematical Physics, Theoretical and Mathematical Physics,
DOI 10.1007/978-0-85729-163-9, © Springer-Verlag London Limited 2011
427
428 Index
curve, 267
tangent vector, 267
vector field, 405
algebra of a group, 108
almost lower semicontinuous
set-valued force field, 289
set-valued mapping, 98
anticipating integral, 125
antisymmetric mean derivative, 194
relativistic, 384
associated bundle, 16
atlas
continuous, 77
uniform Riemannian, 144
backward admissible stochastic process,
405
backward differential, 126
backward Itˆo bundle, 150
backward Itˆo equation (section of backward
Itˆo bundle), 151
backward Itˆo formula, 128
backward Itˆo integral, 125
backward Itˆo process, 127
backward martingale, 119
backward mean derivative, 188, 198
relative to the future, 188
with respect to H, 225
covariant, 405
relativistic, 376
backward stochastic differential, 228
backward stochastic flow, 138
backward Wiener process, 201
Banach manifold, 4
barotropic fluid, 390
base of a bundle, 14
basic vector field, 63
Bianchi identity, 61
bilinear form
non-degenerate, 19
signature, 300
body coordinates, 258
Bohnenblust-Karlin fixed point theorem,
295
Borel σ-algebra, 119
boundary, 4
bracket
Lie, 7
of vector fileds, 13
Poisson, 96
trivial, 13
Bressan-Colombo Theorem, 99
Brownian motion, 122
bundle, 14
associated, 16
base, 14
cotangent, 9
cross-section, 16
frame, 16
principal, 15
projection, 14
standard fiber, 14
structure group, 14
tangent, 5
tensor, 23
total space, 14
vector, 15
zero-section, 17
Burgers equation, 403
without viscous term, 389
canonical 1-form, 95
canonical 2-form, 95
canonical correspondence with respect to a
connection, 151
Cartan’s development, 87
Cauchy problem
locally well-posed, 71, 75
regular, 75
Chow-Rashevsky Theorem, 270
Christoffel symbols
of the first kind, 57
of the second kind, 45, 51
tetrad, 52
closed form, 28
closed set-valued mapping, 98
co-gradient, 108
codifferential, 31
commuting vector fields, 34
compactness lemma, 89
compatible connection and metric, 155
complete
connection, 54
flow of a vector filed, 67
Riemannian manifold, 19
Riemannian metric, 19
stochastic flow, 135, 158
vector field, 67
completely continuous vector field, 78
components of a tensor, 22
conditional expectation, 117
conditional probability, 117
configuration space, 255
conjugate
operator, 23
point, 54
connection
complete, 54
Index 429
connector, 42
Euclidean, 41, 326
flat, 55
form, 61
Levi-Civit´a, 58
on a manifold, 50
on a principal bundle, 61
on a vector bundle, 40
reduced, 268
Riemannian, 56
connector, 42
local, 43, 326
conservation law, 263
of angular momentum, 266
conservation of energy law, 259
conservative mechanical system, 258
constraint, 267
holonomic, 267
non-holonomic, 267
totally non-holonomic, 267
continuous
atlas, 77
covector field, 10
scalar field, 4
set-valued mapping, 98
vector field, 6
contraction of a tensor, 24
contravariant rank, 22
controllability problem, 284
convective representation, 258
coordinate process, 116
coordinate-momentum phase space, 255
coordinate-velocity phase space, 255
coordinates
curvilinear, 3
local, 3
cotangent
bundle, 9
mapping, 11
space, 8
vector, 8
coupled Itˆo equations, 230
covariant
backward mean derivative, 405
derivative, 46, 48, 51
derivative (reduced), 267
differential, 61
formula, 47
forward mean derivative, 405
mean derivative, 232
rank, 22
covector, 8
physically equivalent, 20
covector field, 10
continuous, 10
smooth, 10
cross-section
of a bundle, 16
parallel, 49
current 4-velocity
relativistic, 384
current velocity, 195
curvature
form, 61
Gaussian, 59
scalar, 59
tensor, 55
Ricci, 59
curve, 4
acceleration, 66
admissible, 267
variation, 260
variation with fixed ends, 260
curvilinear coordinates, 3
cut locus, 54
cylinder set, 116
decomposable set, 99
degree, 25
density of electric charge, 313
derivative
covariant, 46, 48, 51
of a function in the direction of a vector
field, 7
differential
s-form, 25
backward, 126
covariant, 61
exterior, 28
forward, 126
of a function, 10
of a mapping, 8
symmetric, 126
differential equation with mean derivatives
first order, 239
differential form, 26
differential inclusion, 100
forward mean derivatives, 211
with forward P-mean derivatives, 222
diffusion
coefficient, 129
process, 128
type process, 128
type stochastic differential equation, 130
direction
to future, 303
to past, 303
displacement form, 63
430 Index
dissipative set-valued vector force field, 347
distance
internal, 19
Riemannian, 19
distribution, 17
integrable, 17
involutory, 17
of a process, 116
divergence, 32, 34
domain of possible motion, 263
double of a manifold, 4
drag force, 340
drift, 129
dual operator, 23
Eells-Elworthy development, 169
Einstein equation, 314, 315
Einstein tensor, 314
Einstein’s Convention, xxii
Einstein-de Sitter space-time, 301
electric charge
density, 313
three-dimensional density, 313
electric field
potential, 313
strength, 313
electromagnetic field, 312
elementary tensor, 22
EMD, 199
equation in backward differentials, 203
equation with P-mean derivatives, 221
equation with mean derivatives (EMD),
199
kth-order, 199
essential extension of a vector field, 272
Euclidean connection, 41, 326
Euler equation, 258, 392, 394
Eulerian representation, 258
event, 303
exact form, 28
explosion time, 136
exponentialmap,54
exterior
s-form, 25
codifferential, 31
differential, 28
form, 26
product, 26
extremal, 260
values of controlling force, 274
with fixed end-points, 260
extreme point, 274
fiber bundle, 14
field of variation, 260
Filippov’s method, 271
filtration, 117
first fundamental form, 18
first integral, 263
first order differential equation
with current velocities, 219
with forward mean derivatives, 209
flat connection, 55
flow
of a vector field, 7
complete, 67
stochastic, 135
L
1
-complete, 163
backward, 138
complete, 135, 158
continuous at infinity, 160
of diffeomorphisms, 138
of homeomorphisms, 138
strictly complete, 135
strongly complete, 135
force, 9
force field, 255
form
s-form, 26
bilinear, 19
canonical 1-form, 95
canonical 2-form, 95
closed, 28
connection, 61
curvature, 61
differential, 26
differential s-form, 25
displacement, 63
exact, 28
exterior, 26
exterior s-form, 25
first fundamental form, 18
horizontal 1-form, 255
identically zero, 27
tangent, 110
torsion, 63
volume, 30
forward admissible stochastic process, 405
forward differential, 126
forward mean derivative, 187, 198
with respect to H, 225
covariant, 405
relative to the past, 188
relativistic, 376
forward stochastic differential, 152
frame bundle, 16
Friedman universe, 301
Frobenius’ Theorem, 17
Index 431
functional
calculus of variations, 260
of action, 260
of length, 260
variation, 260
fundamental vector field, 60
future
σ-algebra, 116
domain, 306
proper, 306
Gaussian curvature, 59
general linear group, 11
generalized Langevin equation, 332
generator
of a stochastic flow, 137
backward, 138
of a one-parameter diffeomorphism
group, 264
geodesic, 53
spray, 93
Glicksberg-Ky Fan Theorem, 100, 291
global chart, 166
gradient, 20
group action, 12
gyroscopic force field, 263
Hamilton’s principle of least action, 262
Hamiltonian, 95
system, 96
vector field, 95
Hausdorff
continuous set-valued mapping, 98
metric, 98
submetric, 98
Hilbert manifold, 4
holonomic constraint, 267
Hopf equation, 389
Hopf-Rinow Theorem, 95, 262
horizontal
1-form, 255
component, 94
lift, 49, 50, 169–171
subspace, 40, 61
vector, 40
identically zero form, 27
induced Riemannian metric, 168
inertia
operator, 257
tensor, 257
infinite-dimensional manifold, 4
infinitesimal generator, 137
backward, 138
integrable distribution, 17
integral
curve of a vector field, 6
manifold, 17
of motion, 263
interior
of light cone, 303
product, 29
internal
distance, 19
energy, 389
involutory distribution, 17
isometrical embedding, 18
isotropic
vector, 303
world line, 304
Itˆo algebra, 147
Itˆo bundle, 148
backward, 150
principal, 148
Itˆo condition, 131
Itˆo development, 170, 171
Itˆoequation
backward, 151
coupled, 230
in backward differentials, 228
in Baxendale’s form, 153
in Belopolskaya-Daletskii form, 153
section of an Itˆo bundle, 148
Itˆo formula, 127
backward, 128
Itˆo group, 147
Itˆo integral, 123
backward, 125
line Itˆo integral with Riemannian
parallel translation, 178
Itˆo process, 126
backward, 127
on a Lorentz manifold, 383
on a manifold, 176
Itˆo stochastic differential equation, 125,
129
Itˆo
vector field, 140, 157
Jacobi identity, 8
Jacobi metric, 262
Killing form, 14
kinetic energy, 255
Kodaira-Hodge Laplacian, 31
Kolmogorov-Fokker-Planck equation, 382
Lagrangian hydrodynamical system (LHS)
of an ideal barotropic fluid, 391
432 Index
of an ideal incompressible fluid, 393
of diffuse matter, 387
Lagrangian of a natural mechanical system,
260
Lagrangian representation, 258
Langevin equation, 336
Langevin inclusion, 342
Laplace-Beltrami operator, 59
Laplace-de Rham operator, 31
least constrained non-holonomic geodesic,
269
left action of a group, 12
left-invariant vector field, 12
length minimizing non-holonomic geodesic,
269
less than quadratic growth, 288
Levi-Civit´a connection, 58
LHS
of an ideal barotropic fluid with external
force, 391
of an ideal barotropic fluid without
external force, 391
of an ideal incompressible fluid with
external force, 393
of an ideal incompressible fluid without
external forces, 393
Lie algebra, 13
of a Lie group, 13
Lie bracket of vector fields, 7
Lie derivative, 33, 34
Lie group, 11
light cone, 303
interior, 303
light-like
vector, 303
world line, 304
line Itˆo integral with Riemannian parallel
translation, 178
linear constraint, 267
local connection coefficient, 43, 326
local connector, 43, 51, 326
local coordinates, 3
local martingale, 119
local solution, 165
locally well-posed Cauchy problem, 71, 75
Lorentz equation, 312
Lorentz manifold, 300
Lorentz metric, 300
Lorentz transformation, 311
standard form, 311
lower semicontinuous set-valued mapping,
97
magnetic field
strength, 313
vector potential, 313
manifold
Banach, 4
complete Riemannian, 19
connection, 50
double of, 4
Hilbert, 4
infinite-dimensional, 4
integral, 17
Lorentz, 300
non-orientable, 30
orientable, 30
oriented, 30
parallelizable, 17
Poisson, 96
Riemannian, 18
second order differential equations, 92
semi-Riemannian, 19
smooth, 3
stochastically complete Riemannian, 173
symplectic, 96
topological, 3
trivializable, 17
uniformly complete, 178
with boundary, 4
Markov process, 118
simple, 118
Markov time, 118
martingale, 118
backward, 119
with respect to a connection, 149
mass, 304
material coordinates, 258
mathematical expectation, 341
Maupertuis principle of least action, 262
maximal set-valued vector force field, 347
Maxwell’s equations, 312, 314
mean derivative
F-mean derivative, 188
P-mean derivative, 188
antisymmetric, 194
antisymmetric relativistic, 384
backward, 188, 198
with respect to H, 225
backward covariant, 232, 405
backward relative to the future, 188
backward relativistic, 376
covariant, 232
forward, 187, 198
with respect to H, 225
forward covariant, 232
quadratic, 192, 234, 335
quadratic P-mean derivative, 192
Index 433
quadratic relative to the past, 192
relativistic symmetric, 384
symmetric, 194
mechanical system, 255
on a Lie group, 257
metric
complete Riemannian, 19
Hausdorff, 98
induced Riemannian, 168
Lorentz, 300
Riemannian, 18
semi-Riemannian, 19
weak Riemannian, 107
metric tensor, 20, 299
Michael’s Theorem, 98
Minkowski space, 301
model space, 3
momentum
4-momentum, 304
conservation law, 265
of a mechanical system, 257
Nash’s theorem, 18
Navier-Stokes equation, 404
Newton-Nelson equation, 358, 368, 385
Noether’s theorem, 264
non-anticipative process, 117
non-conjugate point, 297
non-degenerate bilinear form, 19
non-holonomic constraint, 267
non-holonomic dynamics, 270
non-orientable manifold, 30
normal chart, 54
normal neighborhood, 54
one-parameter diffeomorphism group, 263
orbit
of a flow, 67
of a one-parameter diffeomorphism
group, 263
of a stochastic flow, 135
orientable manifold, 30
orientation, 30
oriented manifold, 30
Ornstein-Uhlenbeck coordinate process,
341
Ornstein-Uhlenbeck velocity process, 341
orthogonal
matrix, 12
operator, 310
osmotic velocity
of a process, 195
relativistic, 384
parallel cross-section, 49
parallel translation, 49, 52, 62
along a stochastic process, 176
parallel vector field, 52
parallelizable manifold, 17
parameter of velocity, 309
past
σ-algebra, 116
domain, 306
proper, 306
path, 4
phase space
coordinate-momentum, 255
coordinate-velocity, 255
physically equivalent
covector, 20
vector, 20
Poisson bracket, 96
Poisson manifold, 96
polyvector, 26
field, 26
potential
energy, 259
of an electric field, 313
power, 9
present σ-algebra, 116
pressure, 390
principal bundle, 15
principal Itˆo bundle, 148
principal lemma of Riemannian geometry,
57
principle of least action
Hamilton’s form, 262
Maupertuis form, 262
probability measure, 119
projection of a bundle, 14
Prokhorov’s Theorem, 120
proper
function, 68
future, 306
mapping, 68
past, 306
time, 305
pull-back, 11
quadratic P-mean derivative, 192
quadratic bound, 289
quadratic mean derivative, 192, 234, 335
relative to the past, 192
random diffeomorphism, 138
random homeomorphism, 138
random time, 118
434 Index
realization of a Wiener process in tangent
space, 172
redshift, 302
reduced connection, 268
reduced covariant derivative, 267
reference frame, 306
regression, 118, 192
regular Cauchy problem, 75
Regularity Theorem, 397
relatively weakly compact set, 77
relativistic antisymmetric mean derivative,
384
relativistic backward mean derivative, 376
relativistic current 4-velocity, 384
relativistic forward mean derivative, 376
relativistic mass, 308
relativistic Newton-Nelson equation, 380
relativistic osmotic velocity, 384
relativistic symmetric mean derivative, 384
rest mass, 308
Ricci curvature tensor, 59
Riemannian connection, 56
Riemannian distance, 19
Riemannian manifold, 18
complete, 19
stochastically complete, 173
uniformly complete, 178
Riemannian metric, 18
complete, 19
induced, 168
weak, 107
Riemannian volume form, 30
right action of a group, 12
right-invariant vector field, 12
rotation, 32
sample path, 116
scalar, 4
scalar curvature, 59
scalar field, 4
continuous, 4
smooth, 4
Schwarzschild space-time, 302
screw gradient, 95
second order differential equation on a
manifold, 92
second order tangent bundle, 65
second order tangent space, 65
second order tangent vector, 65
second tangent bundle, 38
selector of a set-valued mapping, 98
semi-martingale, 119
semi-Riemannian manifold, 19
semi-Riemannian metric, 19
set-valued force field, 272
almost lower semicontinuous, 289
set-valued mapping, 97
almost lower semicontinuous, 98
closed, 98
continuous, 98
Hausdorff continuous, 98
lower semicontinuous, 97
selector, 98
upper semicontinuous, 97
value, 97
set-valued vector force field, 272
dissipative, 347
maximal, 347
signature of a bilinear form, 300
simple Markov process, 118
skew-symmetric tensor, 25
smooth
change of coordinates, 4
covector field, 10
manifold, 3
mapping of manifolds, 4
scalar field, 4
vector field, 6
space coordinates, 258
space of sample paths, 116
space of trajectories, 116
space-like
vector, 303
world line, 304
space-time, 300
time-oriented, 303
special orthogonal group, 12
special vector field, 92
specific internal energy, 390
spray, 93
standard fiber of a bundle, 14
standard form of a Lorentz transformation,
311
state equation, 390
stochastic differential
backward, 228
forward, 152
stochastic differential equation
in Itˆo form, 129
in Stratonovich form, 129
in Stratonovich form on a manifold, 140
section of an Itˆo bundle, 148
with constraints, 271
stochastic evolution family, 135
stochastic flow, 135
L
1
-complete, 163
complete, 135
continuious at infinity, 160
Index 435
of diffeomorphisms, 138
of homeomorphisms, 138
strictly complete, 135
strongly complete, 135
stochastic process, 115
acceleration, 357
backward admissible, 405
forward admissible, 405
stochastic-mechanical trajectory, 357, 368
stochastically complete Riemannian
manifold, 173
stopping time, 118
Stratonivich process, 128
Stratonovich integral, 125
Stratonovich stochastic differential
equation, 129
stress-energy tensor, 315
strictly complete stochastic flow, 135
strong Riemannian metric, 111
strong solution, 130, 182, 337, 343
strongly complete stochastic flow, 135
strongly unique solution, 131
structure equation, 61
structure group of a bundle, 14
submanifold, 4
symmetric
differential, 126
mean derivative, 194
tensor, 25
tensor product, 25
symplectic
form, 96
manifold, 96
tangent bundle, 5
second, 38
second order, 65
tangent form, 110
tangent mapping, 8
tangent space, 5
second order, 65
tangent vector, 5
admissible, 267
second order, 65
tensor
(r, s)-tensor field, 23
bundle, 23
components, 22
contraction, 24
curvature, 55
elementary, 22
field, 22, 23
force field, 333
inertia, 257
metric, 20
of type (r, s), 21
Ricci curvature, 59
skew-symmetric, 25
stress-energy, 315
symmetric, 25
torsion, 55
trace, 24
type (r, s), 21
tensor product, 22, 24
symmetric, 25
tetrad
Christoffel symbols, 52, 58
connector, 52
three-dimensional current density of
electric charge, 313
time-like
vector, 303
world line, 304
time-oriented space-time, 303
topological manifold, 3
torsion
form, 63
tensor, 55
total energy, 259
total space of a bundle, 14
totally non-holonomic constraint, 267
trace of a tensor, 24
trajectory, 256
of a one-parameter diffeomorphism
group, 263
of a relativistic stochastic-mechanical
system, 385
of a stochastic process, 116
of a system with discontinuous force
field, 273
stochastic-mechanical, 357, 368
trivial bracket, 13
trivializable manifold, 17
trivialization, 6, 15
uniform Riemannian atlas, 144
uniformly complete Riemannian manifold,
178
upper Carath´eodory condition, 98, 289
upper semicontinuous set-valued mapping,
97
vakonomic dynamics, 270
valency, 22
value of a set-valued mapping, 97
variation
of a curve, 260
of a curve with fixed ends, 260