Appendix Basic Concepts of Probability
Theory and Stochastic Processes
Probability Theory
Probability theory, stochastic processes and related properties can be found in ax
iomatic form in the books by Soong [1) or Papoulis [2). Nevertheless for the reader's
convenience the basic concepts and properties of stochastic processes are here re
called.
A basic concept of a probability theory is a probability space. A probability
space is a triple (O, F, P) such that
• O is a non empty set of elementary events.
• F is a u-algebra of subsets of O, i. e. F is a family of subsets of O which
satisfies
a) O E F,
b) A E F ~ O \ A E F,
• P is a probability measure on (O, F), i. e. P: F -+ [0,1) satisfies
d) P(O) = 1, P(0) = O,
e) P(U:=o An) = E~o P(An), whenever An, nE N C Fis a family of pair
wise disjoint sets.
191
192 _____________ N.ON LINEAR STOCHASTIC EVOLUTION EQUATION
A function e: n --+ IR is called a random variable if it is measurable when the
Borel u-algebra 8(IR) is considered on IR This condition is equivalent to the following
{e<a}={wEn:e(w)<a}EF forevery aEIR.
Given T, any non empty set of parameters, a function e = {e(t,w} defined on
T X n is called a stochastic process indexed by T iff, for any t E T, e(t,·): n --+ IR
is a random variable.
For fixed t E T, the random variable e(t,·) is called a realization of e at the time
t. For fixed wEn, a function T :3 t --+ e(t,w) is called a trajectory or path of e. Usually a random process as above is denoted by {ethET or simply et, if the set of
indexes is not ambiguous.
Very often, the set T is some subset of the set of real numbers lR, usually lN,
[0,00) and sometimes IR or Z. In those cases a natural interpretation of such a
parameter is the time, continuous (if T = [0,00) or IR) or discrete (in the other
cases ).
Some definitions can now be given.
At any particular time t E T, let the process e( t, w) be a random variable. Then
it has a Distribution Function F(x; t) defined by the following formula
F(x;t) = P{w E n,e(t,w) ~ x}, x E IR.
In other words F(x, t) is the probability that the random variable ((t,.) takes
values not greater then a real number x.
If there exists a measurable function f( x, t) of the variable x E IR such that
l'oof(y,t)dy=F(x,t) V xEIR,
then f(', t) is called the Probability Density Function of the variable (t.
APPENDIX __________________________ 193
If two elements t 1 , t2 E T are chosen, then we define the Second Order Distribu
tion Function, (or simply Two-Dimensional Distribution Function) the function
In a similar way, a measurable function f(:Cl,:C2jt1 ,t2) is called a Second Order
Probability Density if it is a probability density of the lIt-valued random function
In other words, the following holds
This procedure can be easily generalized by choosing n elements of T, h, ... ,tn
so that we obtain the n-Order Distribution Function
and the n-Order Probability Density
It is worth recalling that the knowledge of all the higher order distribution func
tions (or probability densities, if they exist) implies the knowledge of the lower order
distribution functions. In fact the following property holds
194 _____________ NON LINEAR STOCHASTIC EVOLUTION EQUATION
and more in general, if m < n then
If there exists the n-Order Probability Density,
then also
exists for m < n and the last formula takes the form
Moreover, a permutation of any arguments in a n-order distribution function (or
probability density if it makes sense) does not change the function itself, i. e. if 0-
is an n-permutation then
The last two properties are actually known as the Kolgomorov's compatibility
conditions.
All the definitions which have been given until now are related to a single stochas
tic process ~t. If ~t is a JR'-valued stochastic process, so that n stochastic processes
APPENDIX _________________________ 195
e: , ... ,e~ are given, then again the joint distributions can be considered. Accord
ingly one can define the n-Order Joint Distribution, together with the n-Order Joint
Probability Density.
Statistical Measures
When finite dimensional distributions of a stochastic process up to a certain order
are known, then some the statistical properties of the stochastic process can be
characterized. For example, by computing its moments (if they exist).
In particular the first order distribution function F(x, t) is known, then the n
Order Moment of et at every tEl can be computed by
or, if there exists the first order probability density function,
The First Order Moment is called Mean Value
moreover the n-th Order Central Moment of et is defined by
An obvious modification of this formula is necessary if one uses the density
function. The second order central moment, called the Variance of the Stochastic
Process et, is given by
196 ____________ NON LINEAR STOCHASTIC EVOLUTION EQUATION
Higher order statistics can be derived by the knowledge of the second order prob
ability density function F(:l:l, :1:2; tlo t2). In fact the Joint Moments can be computed
for the stochastic process at two different instants of time, t} and t2
This last formula gives the expression of the elements Rnm(t}, t 2 ) of the matrix
R(t}, t2). In particular the element Rl1(tl, t2) is called the Autocorrelation Function.
Analogously to the first order statistics the Central Joint Moments can be defined
as follows
Again the matrix
can be then computed and its element Cll (t}, t 2) is called Autocovariance Function.
In this case, i. e. for n = m = 1 we simply write C(t},t2). Then one can verify
that
A measure of the correlation between the events at tl and t2 is given by the
Autocorrelation Coefficient Function p(t}, t2), which is defined by
APPENDIX ____________________________________________________ 197
Until now we have defined the joint moments of an autocorrelation statistics,
in the sense that all operations have been made upon the same stochastic process
et. If we consider two stochastic processes, say et and "It, then a cross-correlation
statistics can be defined by the second-order distribution function F(x, y; t l , t2).
In particular, it is immediate to define the Cross Joint Moments
with Ri7 is the Cross-Correlation Function.
Moreover, the Cross Central Joint Moments are
where Cfi is the Cross-Covariance Function.
Obviously the cross-correlation statistics takes into account moments calculated
for realizations of the stochastic processes et and "It at the same instant of time
tl = t2 = t, i. e. R~':n(t, t) and C!::'(t, t).
It should be plain to the reader that higher order statistics, which involve n-order
probability densities, can be defined in the same fashion. However, in practice, as
we shall see in the examples which will be given in what follows, the knowledge of
the second order statistics is often sufficient to characterize a stochastic process.
Stationary Processes
A stochastic process et, which may be vector valued, defined on a complete proba
bility space (0, F, P) is defined Stationary Process iff the distribution functions are
198 _____________ NON LINEAR STOCHASTIC EVOLUTION EQUATION
invariant with respect to an arbitrary translation in time. In other words this type
of process is characterized by the property
for all n E IN, t l , ... , tn E T , T E T , Xl, ..• , Xn E IR.
It is assumed that T is a translation invariant subset of IR. This means that
stationary processes depend only on a time parameter T which establishes the dif
ferences between different times.
Another, however different, notion of a stationary process, which is in general
called Stationary in Wide Sense can be stated in L2. A stochastic process ~t is called
stationary in a wide sense iff its second moments are finite,
E{~t} = m = constant
and
for some function C( s).
One can show that a stationary process with finite second moments is stationary
in a wide sense. On the other hand, there is a class of stochastic processes, the so
called Gaussian Processes, for which these two notions are equivalent.
Starting from the autocorrelation function it is possible to define a statistical
functional which plays an important role in several stochastic processes, namely
the Power Spectral Density S(A). This quantity is defined as the inverse Fourier
transform of the autocorrelation function,
APPENDIX __________________________ 199
Conversely, we have
These last two relationships are known as the Wiener-Khintchine formulae.
Moreover from the fact that C ( T) is an even function of T, the Wiener-Khintchine
formulae reduce to
1100 S(A) = - COS(AT)C(T) dT, 71' 0
and
Consequently, for T = 0, we have
An important property of stationary processes is Ergodicity. Let {etheT with
T c lR, be a stationary process and let g : IR----t IR be a given function. Then we can
define the pathwise time averages as
Therefore, a process et is defined Ergodic Process iff for every suitable function
9 the pathwise time averages are constant functions for wEn and
200 _____________ NON LINEAR STOCHASTIC EVOLUTION EQUATION
or, In other words, if the time average of the process equals its mean value for
every time t. One should point out, that there are other definitions of ergodicity,
which more general and such that our definition is a consequence (by means of
ergodic theorems of different types). For practical purposes, the definition above is
sufficient.
Examples of Stochastic Processes
In general a stochastic process is completely determined by its finite order distribu
tion functions. By this we mean that all its statistical measures can be calculated
by using only these functions. Nevertheless, in many cases, stochastic processes
are well characterized by lower order probability densities. Some examples of these
processes, which are often met in applications, are given here.
Processes with Independent Increments
A family {Xa} of random variables is called independent iff for any finite subset
aI, ... ,ak of indexes and any Borel set Ai E JR
p ({w En: Xa;{W) E Ai, i = 1, ... , k}) = IT P ({W En: Xa;{w) E A;}) . i
This is equivalent to the following condition
k
F{X1, ... , Xki a1,···, an) = IT F{Xii a;) for all Xi E JR,
where F{ Xl, ... ,xki a1> ... ,an) is ajoint distribution function of k variables Xa1 , ••• ,Xak •
A stochastic process ~t is said to have Independent Increments iff for any finite
number of elements t1 :S t2 :S ... tk of T the random variables
APPENDIX __________________________________________________ 201
are independent.
Markov Processes
Classically, a Markov process is a process {eth~o such that for each t the behaviour
of e. for s > t depends only on et and is independent of er for r < t. This rough
picture has the following more rigorous version.
Assume that there exists a family P( s, x, t, r) which satisfies the conditions:
i) P( s, x, t, .) is a probability measure on (0, F, P), for all s :s t E T and x EX,
where X is a metric space, usually ill. or II{'.
ii) P( s, ., t, A) is measurable for all s :s t E T and A E F
iii) P(s,x,s,·) = 6., for all sET, x E X.
We say that a stochastic process et is a Markov Process iff there exists a function
P( s, x, t, A) which satisfies conditions i)-iii) and, in addition, for any pair s :s t E T
p(et E Ale. = x) = P(s,x,t,A),
where on the left hand side one has the so called Conditional Probability.
The last condition can also be expressed in the equivalent manner
p(et E A Ie.) = P(s, e., t, A) for all s:S t, A a.e. In o.
This last equality, in turn, has the following meaning: For any Borel set B in ill.
we have
p({et E A,e. E B}) = r p(s,e.(w),t,A)dP(w). Je;l(B)
202 ____________ --'NON LINEAR STOCHASTIC EVOLUTION EQUATION
Gaussian Process
A stochastic process is called Gaussian if the random vector «(tl , ... , (t,,) is Gaussian,
for any finite set of tb' .. , tn E T. This means that there exist n-order density
functions which are given by the following formulae
an n x n matrix.
One can compute
and
Therefore m(ti) is the mean value of (ti) and B is the correlation matrix of
(t. It follows that a Gaussian process is completely characterized by its mean value
m( t) and by the autocorrelation function.
Wiener Process
A Wiener Process is a stochastic process {w(t)h~o, which is Gaussian and, in addi
tion, satisfies the following properties
i) w(O) = 0,
ii) m(t) = E(w(t)) = 0 for all t ~ 0,
iii) Rn(s, t) = E (w(s( w(t)))) = Ds 1\ t = D min{ s, t},
where D > 0 is a given constant. When D = 1 then w(t) is called a Standard Wiener
Process. Generally, the word Standard is not even mentioned.
There is an equivalent definition od a Wiener process, which we give for complete
ness. A stochastic process w(t) is called a Wiener process iff it satisfies conditions
APPENDIX ____________________________________________________ 203
1) and 2) above, for each t ;?: o. w{t) is a Gaussian random variable with zero mean
and variance Dt.
Rice Noise
The Rice Noise is a stochastic process, which is characterized by a probability density
function of an order greater than two, and which can be written in the form
n
{t{w) = E aj(w) cos(Vj(w)t + 4>j(w», j=l
where aj, Vj and 4>j are 3n given random variables.
It is worth remarking that a Rice process is characterized by the 3n-Order Joint
Distribution Functions
Random Fields
This book is mainly devoted to partial differential equations where the unknown
functions are stochastic processes which depend either upon the time and upon
spatial variable x E nr , i. e. u(t, x, w).
In the literature, (see, for instance, the book [3]), a stochastic process which
depends only on the position x E nr and not on the time, i. e. u( x, w) , is called
a Random Field. If the random field depends also on the time, then one generally
speaks of a Time-Dependent Random Field. A few peculiarities of the random fields
will be summarized in this Appendix. First of all if a finite set of times t l , ... , tn
is chosen, the corresponding functions u(tl,X,.), ... ,u(t,x,.) form a finite set of
random fields. In the same way if we select a finite set of positions, Xl, ... , Xn , then
the functions u(t, Xl, .), •.. , u(t, xn ,·) form a finite family of stochastic processes.
Conversely a selection of finite sets of position and time transforms the random field
into a sequence u(tl , xd, . .. , u(tn , xn) of random variables.
Accordingly the statistical measures can be computed either with respect to the
time, as in the standard stochastic processes, or with respect to the position. In other
words one can study the correlation between the two random variables u( t l , Xl, W)
204 ____________ -lNON LINEAR STOCHASTIC EVOLUTION EQUATION
and u(t2 ,:l:l,w) at different times but in the same space position, or, vice-versa,
compute the correlations ofu(t1,:l:l,w} and U(t1,:l:2,W) at the same time instant t
for two different space positions. As a consequence, we can define two families of
n-order probability density functions, as
and compute the statistical moments of any order with respect to the time or the
position, respectively, as shown above.
Ito's Integral
The relevance of a Wiener process consists mainly in its connection with the anal
ysis of stochastic differential equations. The basic notion associated with a Wiener
process is the so called Ito '8 Integral. We will give here its definition and then list its
basic properties. Let w(t) be a d-dimensional Wiener process and Ft be its natural
filtration, i. e. Ft = u(w. : 8 ~ t}, that is Ft is the smallest u-algebra with respect
to which all the random variables w., 8 ~ t are measurable. In other words, it is
a u-algebra of events observable up to time t. A stochastic process et is said to be
adapted, if for each t E T the function
e Tn(-oo,t]xS13(s,w}--+lR
is measurable.
The Ito's Integral is defined only for adapted processes. First, it is defined for
simple processes, i. e. for processes of the form
APPENDIX __________________________________________________ 205
where 0 = to < h < ... < tn = T, e. is JR""xd-valued :Ft. measurable random variable
with finite second moment. For such a process we simply put
n-l
I(e) = L: < e., il.w > .=0
where il.w = W(ti+l) - W(ti) and for WEHr, e E JR""xd, < e,w > is JR"" vector
whose i-th coordinate is equal to L:eijWj. The properties of the Wiener process j
together with the fact that e(t) is adapted yield the following equality
again only for a simple process e. But it is sufficient to define I(e) for any process
e(t) which is a limit of simple processes in the norm
It can be proved that any progressively measurable process with finite norm has
this property. Hence, there is a linear map I that maps the space M2(0, T; JR""Xd)
of all JR""xd-valued progressively measurable processes into L2(n; JR""). Finally we
define the Ito's integral by putting
re(s)dw(s) = I(l[ot)e)· 10 .
In particular, we see that
loT e(s) dw(s) = I(e).
This procedure can be followed for any T > 0, so we will omit T in the sequel.
Now we list the basic properties of Ito's Integral:
206 ___________ NON LINEAR STOCHASTIC EVOLUTION EQUATION
1. Linearity: For all 0, (3 E ~ and e,7J E M2(0, T),
2. Isometry property: For all e E M2(0, T),
3. Ito's Formula: Assume that e and." are progressively measurable processes,
respectively firxd and fir valued and z(t) is defined by
z(t)=zo+ le(s)dw(s) + l7J(s)ds,
where the second integral is in the Lebesgue sense. We say that z(t) has a
stochastic differential and write
dz(t) = e(t) dw(t) + 7J(t) dt.
Assume that F : fir -+ IR be a function of class C2 • Then the stochastic
process F(z(t)) has a stochastic differential and
d m of . m of . dF(z(t)) = L L a(e(t))eij dw3 (t) + L aWt))7Ji dw3 (t)+
j=1i=1 Xi i=1 Xi
(A.I)
APPENDIX ____________________________________________________ 207
4. Doob-Burholder inequality: For e E M2(O, T) and r E (1,00),
E (sup I! 1e(s) dw( s )Ir) ::; ( __ r __ ) r sup{ Elf 1e( s) dw(s W}~ . t~T r - 1 t~T
Stochastic Differential Equations
A very useful and important notion is that of Stochastic Differential Equations. Let
us assume that, as before, w(t) is a d-dimensional Wiener process on a probability
space (0, F, P). Consider two Lipschitz functions
u : JR' ---+ :urxn ,
where n E IN and we assume global Lipschitz properties of these functions for the
simplicity of exposition. We are interested in solving the following ordinary stochas
tic differential equation (but one might consider a system of ordinary stochastic
differential equations)
de(t) = b(e(t)) dt + uWt)) dw(t), (A.2)
subjected to the initial condition
e(O) = eo,
where eo E L2(O; JR') is Fo-measurable. Very often we will simply take eo = :z: for
some :z: E JR'. It is known that under the above assumptions a unique global solution
to the initial value problem exists. The proof of this theorem can be obtained by
several different methods, one of them is the fixed point method. We fix T > 0 and
208 ____________ -.lNON LINEAR STOCHASTIC EVOLUTION EQUATION
observe that a process e E M2(0, T) is a solution to the initial value problem iff e is a fixed point of the mapping
where
Next, we observe that if c)(e) = e then, in view of Doob's inequality
Therefore we may look for a fixed point of C), considered as a function in XT . But
endowing X T with a suitable norm (equivalent to the original, sometimes called
Bielecki's norm), i.e.
we prove that for >. sufficiently large, c) is a strict contraction in X T and so by
Banach fixed point theorem has a unique fixed point.
The Ito's formula provides a very efficient way of deriving the Fokker-Planck
Equation. With this in mind, assume that the unique solution e(t) to the stochastic
differential equation exists. Assume that the functions band (j are sufficiently regular
and finally aussume that the density function of the stochastic process e( t, z), the
unique solution to the stochastic differential equation (A.2) with initial condition
eo = z, exists for t > o. Let u(t, z) be a function defined by
u(t,x) = E{4>(e(t, x))} ,
APPENDIX ________________________ 209
where cPis a C2 functin with compact support. By Ito's formula, the process cP(e(t, z))
satisfies
Taking the mean value of this equality we find that u( t, z) satisfies
dv(t,z) = Av(t,x)dt,
v(O,x) = cP(x),
where A is a second order differential operator given by
n, 8 n ~
A = tt b'( x) 8x; + ;~1 a;j( x) 8x;8xj ,
where ai;( x) = L:k U;k( x )Ujk( z). This allows the following argument, If p( t, y) is the
probability density of e( t, x) then
d J d d dt cP(y)p(t,y)dy= dtE{cP(e(t,z))} = dtu(t,x) = Au(t,x) =
_ ~b;( )8u(t,x) ~ ,,82u(t,x) - ~ Z + ~ a" .
'-I 8x; , '-I 8x;8x,' 1- 1,3-
Taking into account that Au( t, x) is equal to the solution of
8u -=Au 8t
210 ___________ NON LINEAR STOCHASTIC EVOLUTION EQUATION
with initial condition A4> yields
J J { n {)4>( x ) 1 n {)2 4>( x ) } Au(t,x) = A4>(y)p(t,y)dy = Lbi(x)-{)-. +"2 L aiifiT. p(t,y)dy =
i=1 x, ',j=1 x, X3
where in the last equality the formula of integration by parts has been used.
We conclude the proof by observing that since 4> can be taken arbitrarily, p( t, y)
satisfies the following
which is the Fokker Plank Equation.
APPENDIX __________________________ 211
References to Appendix
1. Soong T.T., Random Differential Equations in Science and Engineer
ing, Academic Press, New York, (1973).
2. Papoulis A., Probability Random Variables and Stochastic Processes,
McGraw Hill, New York, (1965).
3. Ivanov A.V. and Leonenko N.N., Statistical Analysis of Random Fields,
Mathematics and Its Applications, (Soviet Series), Ed. M. Hazewinkel,
Kluwer Academic Publishers, Dordrecht, (1989).
AUTHORS INDEX
Abramowitz M. 31, 60, 151, 153, 165.
Adomian G. 2, 9, 11, 20, 22, 34, 60, 68, 82, 92, 98, 150, 154, 163, 164, 169, 188.
Akilov G. 3, 22, 50, 60.
Arnold L. 2, 20, 22, 34, 60, 101, 122, 130.
Ash R. 66, 98.
Baker G. A. 43, 44, 6l.
Beck J. 179, 189.
Becus G. 78, 98, 181, 188.
Bellomo N. 1, 2, 11, 12, 20, 22, 34, 57, 60, 61, 62, 72, 78, 82, 84, 89, 98, 99,
102, 123, 127, 129, 130, 137, 138, 144, 145, 147, 155, 156, 165, 169, 184, 186,
188, 189.
Bellman R. 34, 60, 68, 77, 98, 169, 188.
Bensoussan A. 102, 108, 132.
Bernard P. 102, 122, 123, 131, 164, 165.
Bharucha Reid A. 35, 38, 61, 68, 98.
Blackwell B. 179, 189.
Bonzani 1. 57, 62, 137, 147, 152, 163, 164, 165.
Brzezniak Z. 102, 108, 109, 111, 130.
Cabannes H. 14, 15, 16, 23.
Cannon J. 185, 186, 189.
Canuto C. 76, 99.
Capinski M. 102, 108, 109, 111, 130.
Carlomusto L. 156, 159, 165.
Cashef J. 169, 188.
213
214 ____________ NON LINEAR STOCHASTIC EVOLUTION EQUATION
Casti J. 34, 35, 59.
Chorin A. 106, 107, 133.
Colton D. 185, 189.
Conti R. 34, 39, 59.
Courant R. 3, 22, 61.
Da Prato G. 102, 108, 130.
De Blasi G. 102, 131.
de Socio L. 11, 22, 34, 39, 45, 60, 61, 72, 78, 84, 92, 98, 99, 156, 159, 165, 180,
184, 186, 189.
Du Chateau P. 185, 186, 189.
Elworthy K. 100, 131.
Evans J. 169, 188.
Ewing R. 185, 186, 189.
Fitzgibbon W. 169, 187.
Fitz Hugh R. 172, 173, 188.
Flandoli F. 61, 102, 108, 109, 111, 123, 126, 129, 130, 132.
Friedman A. 20, 23, 34, 60, 100, 131.
Gabetta E. 138, 147, 156, 164.
Gardner M. 66, 98.
Gatignol R. 14, 15, 23.
Graves Morris P. 43, 44, 61.
Gualtieri G. 39, 45, 61, 92, 99.
Hilbert D. 3, 22, 61.
Hille B. 172, 186.
Hodgkin A. 169, 170, 188.
Hussaini M. 76, 99.
Huxley A. 169, 170, 188.
Kampe de Feriet J. 181, 188.
Kantorovic L. 3, 22, 50, 60.
Kashef B. 34, 59.
Kazimierzik P. 101, 130.
Keener J. 172, 173, 181, 188, 189.
AUTHORS' INDEX __________________ 215
Kotulski Z. 101, 102, 130.
Kreener A. 102, 130.
Kustnezov P. 155, 165.
Ichikawa A. 102, 130.
Ito S. 102, 131.
Ivanov A. 206, 211.
Yakhono V. 185, 186, 188.
Lachowicz M. 39, 49, 50, 51, 61, 174, 189.
Lavrent'ev M. 185, 188.
Leonenko N. 206, 211.
Lions J. L. 6, 11, 22, 61, 102, 113, 132.
Lieberstein H. 169, 188.
Lobry C. 102, 130.
Malakian K. 154, 164.
Marsden J. 106, 107, 133.
McShane E. J. 20, 23, 34, 60, 67, 98, 101, 121, 130.
Mikhailov V. 3, 22.
Myjak J. 102, 131.
Monaco R. 34, 39, 49, 50, 51, 60, 61, 72, 78, 83, 98, 156, 158, 166, 174, 189.
Nelson J. 101, 130.
Padgett W. 68, 98.
Payne L. 185, 186, 187, 189.
Papageorgiu N. 102, 131.
Papoulis A. 12, 14, 22, 191, 211.
Pardoux E. 102, 108, 131.
Pianese A. 154, 159, 165.
Pistone G. 137, 144, 147, 164.
Preziosi L. 72, 78, 92, 98, 99, 156, 158, 165, 182, 184, 186, 189.
Pugachev V. 150, 164, 165.
Quarteroni A. 76, 99.
Repaci A. 182, 189.
Reznitskaya K. 183, 184, 188.
216 ____________ NON LINEAR STOCHASTIC EVOLUTION EQUATION
Riganti R. 1, 2, 12, 20, 22, 34, 60, 98, 138, 150, 152, 154, 164, 165.
Rybinski L. 102, 122, 131.
Roozen H. 102, 120, 131.
Rundell W. 185, 186, 188, 189.
Sambandham M. 35, 38, 61.
Sansone G. 34, 39, 60.
Saraiyan D. 89, 99.
Satofuka A. 39, 45, 46, 61.
Shannon C. 142, 164.
Schuss Z. 102, 131.
Sinitzin T. 150, 165.
Smale S. 6, 23.
Smart D. 12, 23.
SobczykK. 20,23,57,61,62,99,101,102,121,130.
Soize C. 102, 121, 123, 131, 164, 165.
Soong T. 20, 23, 34, 60, 68, 98, 101, 121, 130, 138, 144, 146, 164, 191, 211.
St. Claire G. 179, 189.
Stegun 1. 31, 60, 151, 153, 165.
Steube K. 185, 186, 189.
Stratonovich 1. 144, 154, 165.
Sussman H. 102, 111, 131.
Temam R. 6, 23, 51, 102, 108, 130.
Teppati G. 72, 78, 98, 184, 186, 189.
Tichonov 1. 144, 154, 165.
Tsokos C. 68, 98.
Vacca M. T. 169, 188.
Weaver W. 142, 164.
Wehr A. 143, 164.
Zang T. 76, 99.
Zavattaro M. G. 135, 164.
SUBJECT INDEX
Autoccorrelation 12, 13, 196.
Autocovariance 12-14.
Autonomous equations 6.
Banach spaces 10.
Boundary conditions 2-6, 27, 69-70.
Boltzmann equation 14-16, 26, 28, 48-53, 156-158.
Bernstein polynomials 36.
Brownian motion 9, 101, 103-106, 108.
Burgers' equation 29,52-56, 107-108.
Chebychev collocation 37.
Classification of PDE 3-9.
Classification of models 3-7.
Coloured noise 110-114.
Conditional probability 200.
Correlation function 12-15.
Covariance function 13-14, 197.
Distribution function 192-194.
Doob-Burholder inequality 207.
Entropy function 140-146, 156-158.
Ergodic process 199.
Euler (stochastic) integration 120-121.
Evolution of the Probability Density 143-147.
Faedo-Galerkin approximation 111.
Fixed point theorems 11, 50-54, 55.
217
218 _____________ NON LINEAR STOCHASTIC EVOLUTION EQUATION
Fokker-Plank-Kolmogorovequation 121-125, 150, 163,209-210.
Fourier random expansion 42.
Function spaces 9-10.
Gaussian process 202.
GronwaJ1s'Lemma 37.
Hermite polynomials 152-154.
Karunen-Loeve expansion 66.
Kata- Trotter's tbeorem 113.
Kolgomorov's compatibility conditions 194.
Kronecker delta 31.
lli-posed problems 6,174-181.
Initial value problem 4,25,136-141, 174.
Initial-boundary value problem in tbe balf-space 4, 32, 75-77, 84-88.
Initial-boundary value problem 5, 25, 138, 174.
Integra-differential equations 173-174.
Interpolation tecbniques 38-48, 149-155.
Inverse problems 174-181.
Ito's differential 100.
Ito's differential equation 207.
Ito's integral 204-207.
Lagrange polynomials 30-31, 71-72.
Laguerre polynomials 151.
Lipschitz condition 34, 38.
Markov process 201.
Mean value 12, 17.
Moments 12-14, 34-35, 195-197.
Moment approximantion 149-154, 156-160.
Moving boundary problems 91-95, 181-185.
Navier-Stokes (stocbastic) equations 107-108.
Nonautonomous equations 6.
Pade's approximants 43-44.
Periodic polynomials 41-43.
SUBJECTS' INDEX __________________ .219
Picard iterative schmeme 90.
Power spectral density 189
Probability density 135-154, 192-194.
Probability space 191.
Processes with independent increments 200.
Power spectral density 198.
Random fields 203-204.
Random heat equation 68-70, 77-90.
Random variables 191-192.
Rice noise 78, 203.
Runge-Kutta (stochastic) integration 121.
Semilinear equations 6-7.
Separable stochastic processes 8, 66-67.
Sobolev imbedding theorem 55.
Solution to the initial-boundary value problem 9-10.
Splines 39-41.
Stationary Processes 197-200.
Statistical measures 12-13, 195-197.
Stochastic calculus 11.
Stochastic differential equations 207-210.
Stochastic operators 67.
Weakly nonlinear equations 7.
Well-posed problems 6.
Well-specified problems 6.
White noise 106, 110-114, 116.
Wiener-Khintchine formulae 199.
Wiener process 106, 110-114, 116, 202.
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