Lemma 1. Consider a positive denite symmetric matrix A.
1
(Det A) 2 = cn
S n 1
ds
n
[< s, As >] 2
Proof:
n
1
e
(Det A) 2 = (2 ) 2
<x,Ax>
2
dx
Rn
n
er
= (2 ) 2
2 <s,As>
2
r n1 dsdr
S n 1
= cn
S n 1
ds
n
[< s, As >] 2
It is therefore sucient to estimate for
We now look at the nonlinearly perturbed version
dx(t) = Ax(t)dt + F (x(t)dt + Bd (t)
where F : H H is a Lipschitz map, i.e.
F ( x) F ( y ) C x y
A mild solution of the equation is one that satises which is almost surely in C [0, T ], H]
and satises
t
X (
The subject of stochastic partial dierential deals with the study of solutions of partial dierential equations perturbed by noise. Just like the study of ordinary dierential
equations perturbed by noise leads to the study of Stochastic Dierential Equation
Nov 2, 2009.
Polar Coordinates.
A point in the plane can be specied by its coordinates x and y coordinates. There is an
origin, an x-axis going left to right and a y -axis perpendicular to it going up and down.
There are positive directions chosen by conv
Nov 4, 2009.
Areas and lengths in Polar Coordinates.
Given a little arc r = f ( ) between and + , the area of the wedge from the origin to
2
the arc is roughly r so that of a wedge formed by r = f ( ) between 1 and 2 is
2
A=
1
2
2
[f ( )]2d
1
The arc leng
For any noise if the viscosity is large enough there is a unique invariant measure. Let
the equation be
du(t, x) = u P (u D)u + dW (t, x)
u(0, x) = u0 (x)
If we take two solution u(t), v (t) with dierent initial data, then the dierence w(t) =
u(t) v (t) s
Multidimensional version of Kolmogorovs Theorem.
Let us do d = 2. d > 2 is not all that dierent. We need to interpolate a function from the
four corners of a square to its interior. Pretending the square to be [0, 1]2 , the function
will be of the form
f
Hormanders Theorem
Let L denote the operator
L=
k
1
2
2
Xi + Y
i=1
where X1 , , , Xk , Y are C vector elds in Rn . Assume that the Lie Algebra generated
by X1 , , , Xk , Y span Rn at every x. Then, if u is a distribution such that
Lu = f
and f is C in an
1. Gaussian Model.
We will be dealing with Gaussian Models. The central object is the Cameron-Martin space
which will be a real Hilbert Space H. There is a probability space (X, F , P ), that can be
thought of as a vector space containing H and there is a
Let (s) and b(s) be smooth progressively measurable functions of . Then so are
t
x( t ) =
(s)dk (s)
0
and
t
y ( t) =
b(s)ds
0
In fact
t
L x( t ) =
[L (s) (s)]d (s)
0
and
t
Lb(s)ds
L y ( t) =
0
It is easily proved by approximating the integrals by a sum.
Oct 26, 2009.
Area Between Curves.
Given two functions f (x) and g (x) with f (x) > g (x) on [a, b] the area betwen the curves is
b
a
[f (x) g (x)]dx
Often the points a, b are not specied and we take it as the points where f (x) = g (x) so
that in [a, b]
Oct 28, 2009.
Arc length.
If we consider the graph of y = f (x) as a curve, the area under the curve between x = a
and x = b is calculated by the integral
b
Area =
f (x)dx
a
But the length of the curve has to be computed dierently. Because the length also