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Unilateral small deviations of processes related to the fractional
Brownian motion (FBM)
Molchan G
E-mail: molchan@mitp.ru
St. Petersburg, September 12-18, 2005
1 Problem: find
 lim log pT / log T    , T , T  
where
pT  P  (t )  1, t   T ,
 (t ),  (0)  0 is a random process/field and  T    T is a convex domain, 0   .
The interesting cases: self-similar (H-SS) processes with parameter of similarity H, HSS processes with stationary increments (H-SSSi); T  (0, T ) and T  (T , T ) .
2 Motivation
Example 1.
Sediment deposition model in geology [3].
The thickness of the sedimentary layer as a function of time is modeled as H-SSSi
process  (t ) . Due to erosion the sedimentary record is described as follows:
h(t )  inf  (s); s  t.
The question: what is Hausdorff dimension of episodes of deposition represented in
the sedimentary record, i.e.
dimsupport of dh(t )  D  ?
The answer: if  ( s)  FBM ( s)  bs , b > 0 then
D  1   ( FBM , T  (0, T )) .
Example 1(a). If M (t )  max(  ( s) : 0  s  t ) is the record function of FBM then
dimsupport of dM (t )  1   ( FBM , T  (0, T )) .
Example 2.
1-D Inviscid Burgers Equation
ut  uu x   u xx ,   0
.

u (0, x)   ( x)
Solution: u (t , x)  ( x  a(t , x)) / t .
Initial positions of particles which have not collisions up to moment t  1 form set


S  a : a(1, x), x  R1 .
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Question: dim S-?
t

Hopf ‘s algorithm: if c(a )  convex hull    ( s ) ds  t 2 / 2
 0

then a (1, x) is inverse function to c' (a ) and S  supp dc' (a).
Conjecture (see [4]): if  ( x)  FBM then
 t

1  dim S      ( s ) ds,  T  (T , T ) 


0

Results
M T  sup (t ), t  T ,
3 Statistics:
ZT  sup t :  (t )  0, t  T ,
GT  arg sup  (s), s  T ,
S T   1 ( s ) 0 ds .
T
A typical trajectory of H-SS process  (t ) with small value of M T perhaps have a
small values of ZT , GT and ST . Therefore the following events AT ,
M T




 1, ZT  1,  |T  0 ,  GT  1, ST  1, GT  T ,
are considered as characteristic in our problem. The following
 ( AT )  lim  log PrAT / log T
T 
is the index of AT .
An example of strong correlation of characteristic events:
if  is Levy stable process, P( (1)  0)   , 0    1 , then
P(GT  x | M T  1)  F ( x), T  
where F (x) is non-trivial distribution, [2].
4 Theorem 1. If

 (t ),  (0)  0 is H-SS, stochastically continuous Gaussian process

 T  H (T ) : T (t )  1, t  1;  T
2
 o(ln T ) .
( H  is the reproducing kernel Hilbert space associated with  (s), s  T  )
then the indexes  (M T ),  (ZT ),  (GT ) exist simultaneously and are equal.
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t
Examples: 1. Theorem 1 hold for FBM, IFBM   FBM ( s) ds with intervals
0
T  (0, T ) and ( T , T ) . In these cases one has
 ( A(M T ))   ( A(Z T ))   ( A(GT ))   ( A(ST ))
and T (s)  1, s  1, T (s)  0, s  1/ 2 , [1, 4].
2. If  T  0 is a compact convex domain in R d then Theorem 1 holds for the
H-fractional Levy field but not for the H-fractional Brownian sheet.

3. If  (t )  t  e ts dw( s) , w is Brownian motion and T  (0, T ) then
0
T (t )  t (1   T ) /(t   T ),  T  1/ ln T and  T
2
 O ( ln T ) .
See (Dembo et al., J. A. M. S., 15, 857-892,2002) for applications of  (t ) to random
polynomials.
Proof of Theorem 1 (sketch)

pT  P(M T  1, T )  P( (s)  T  1  T , T )  E 1  1-  ( )
where
 ( )  d (  T ) / d ( )
and  is Gaussian measure associated with FBM on  T .Applying Hölder’s inequality
one has
pT  P(  1   T , t  1,  T )1 exp (  T
where  2   T
2
2
/ 2 )  P( A( Z T ))1 T  / 2
/ ln T .

P( A(ZT ))  P( GT  1) ,

P( GT  1)  P( GT  1, M 1  cT )  P(M 1  cT )  P(M T  cT )  P(M 1  cT ) .
Due to H-SS property of  one has P(M T  cT )  P(M T '  1)
where T '  TcT1 / H . If cT  2a ln T then P(M 1  cT )  o(T  ac ) and T '  T 1o(1) .
5 H-SSSi Processes, T  (T , T ) .
Theorem 2, [1,4]. Let  (t ),  (0)  0 be a cadlag (right-continuous with limits to the
left) H-SSSi-process and G  arg sup  (s), s  (0,1). Then

G has a continuous probability density  (t ) in (0,1),
 ( x) x , (1  x) ( x) , 0  x  1
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i. e.   0 or   0 ,
if   0 then

P GT  1, T  (Ta, T (1  a)   (a) /( 2T )  (1  o(1)), T   ,
where 0  a  1 .
Consequence:
 (H-FBM, T  (T , T ))  1, 0  H  1 .
Hint to Theorem 2 ().
 ()   (t 0 ),  T    (),  T
law
~
 t 0    (Si property)
law
~
GT ( )  GT (   (t 0 ))  GT ( )  t 0
If t 0  aT and T  (aT , T (1  a)) then  T  t 0  (0, T ) . Therefore
~
~
P(| GT ( ) | 1)  P(| GT ( )  aT | 1)  P(| G ( ,   (0,1))  a | 1 / T )   (a) / 2T
The second equality is the result of SS-property of .
Theorem 3, [1]. If  (t ), t  R d is H-fractional Levy field i. e.  is Gaussian and
E |  (t )   (s) | 2 | t  s | 2 H then  ( , T   t  T )  d .
6 H-SSSi processes, T  (0, T ) .
Lemma. Let  (t ),  (0)  0 be a cadlag SSSi process and for some   0

E max  (t )
1
( 0,1)

 ,
E exp ( max  (t ))   ,
( 0,1)
then
T

I T  E   e  ( s ) ds 
 0

1
 HT (1 H ) E max  ( s)  (1  o(1)), T   .
(0,1)
(see [8] for the case of Brownian Motion).
Theorem 4 [1]. If  (t )  H - FBM, T  (0, T ) then
P(ZT  1) LT  I T  P(M T  1) LT
where LT and LT are slow variables.
Therefore  ( A(ZT ))  1  H   ( A(M T )) and
 (H-FBM, T  (0, T ))  1  H .
Consequences.

If  (t )  H-FBM then dim (supp dM (t ))  H .
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 duality of h-FBM with h = H and 1 – H:
pTc  P( (t )  0, 1  t  T |  (s)  0,  T  s  1)  T  H o(1)
pT  P( (t )  0, 1  t  T )  T (1 H )o (1)
and persistence: pTc  pT if H  1 / 2 , pTc  pT if H  1 / 2 .
Conjecture. If  (t1 ...t d ) is H-fractional Levy field and
 T  (0  t i  T , i  1,..., k ; | t j | T , k  j  d ) then  ( , T )  d  kH .
This conjecture includes Th. 3, Th. 4 and it holds for the case H = 1
i. e. for  (t )   t i i with i. i. d. Gaussian variables { i } .
t
7 The integrated fractional Brownian motion,  (t )   FBM ( s) ds .
0

The index  ( ,  T ) exists for the cases
T  (0, T ), 0  H  1,
T  (T , T ), 1/ 2  H  1 .
The proof is based on Li & Shao idea: if pT  P( A(ZT )) then one has
pTS  pT p S , T , S  0 ,
due to positivity of E  (t )  (s), t , s  T , Slepian Lemma and SS-property of  .

 ( , T  (T , T ))  1  H , ([4]).
Sketch of proof.
Let c (t ) be the convex hull of  (t )  t 2 / 2 then dim (supp dc (t ))  H .
Indeed, if t1 , t 2  S  {supp dc} and | t i | k then
c (t1 )  c (t 2 )   (t1 )   (t 2 )  t1  t 2  t1  t 2
H 
,   0 , a. s.
Therefore dim( S )  H due to the Frostman’s Lemma.
On the other hand if  is index of A(M T ) for  and T  (T , T ) then
dim ( S )  1   .

If H  1 / 2 then (see [6, 7])
1 / 2,  T  (T , T )
.
1 / 4,  T  (0, T )
 ( IFBM )  

Numerical results [4, 5].
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 T  (T , T )
1  H ,
.
H (1  H ),  T  (0, T )
 ( IFBM )  
Conjecture: If   IFBM , T (0, T ) then
P(M T  1)  cT  H (1 H ) (log T ) ( H )
where  ( H )   (1  H ) .
8 Non-power asymptotic.
Theorem 5. If

 (t )   ( k cos kt   k sin kt) f k exp( k / b)
k 1
where 0  c1  f k  c2  , b  0 and { k }, { k } are i. i. d. standard
Gaussian variables then
 lim | log  | 2 log P( (t )   t )  b .
 0
This example came from analysis of Burgers equation with random force.
9 References.
1. Molchan G.M., Maximum of fractional Brownian motion: probabilities of small
values. Commun. Math. Phys., 1999, 205:1, 97-111.
2. Molchan G.M., On a maximum of stable Levy processes. Theory of Prob. and its
Applications, 2001, v.45, 343-349.
3. Molchan G.M., D.L. Turcotte, A stochastic model of sedimentation: probabilities
and multifractality. Euro. Jln. Applied Math., v.13, part 4, 371-383 (2002).
4. Molchan G.M., Khokhlov A.V., Small values of the maximum for the integral of
fractional Brownian motion, J. Stat. Phys. 114:3-4 (2004), 923-946.
5. Molchan G.M., Khokhlov A.V., Unilateral small deviations for the integral of
fractional Brownian motion. (http://arXiv.org/math.PR/0310413, 2003).
6. Sinai, Ya.G., Distribution of some functionals of the integral of Brownian motion.
Theor. Math. Phys., 90:3, 323 (1992) (in Russian).
7. Isozaki, Y., Watanabe, S., An asymptotic formula for the Kolmogorov diffusion
and a refinement of Sinai’s estimates for the integral of Brownian motion. Proc.
Japan Acad., 70:A-9 (1994).
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8. Kawazu, K. & Tanaka, H., On the maximum of a diffusion in a drifted Brownian
environment. Seminaire de Probabilites, 27, Lecture Notes in Math. 1557, Berlin:
Springer, 1993, 78-85.