Survival exponents for fractional Brownian motion with multivariate time

Transcription

1 Survival exponents for fractional Brownian motion with multivariate time G Molchan Institute of Earthquae Preiction heory an Mathematical Geophysics Russian Acaemy of Science 84/3 Profsoyuznaya st 7997 Moscow Russian Feeration aress: molchan@mitpru Abstract Fractional Brownian motion -FBM of inex 0 with -imensional time is consiere in a spherical omain that contains 0 at its bounary he main result : the logasymptotics of probability that -FBM oes not excee a fixe positive level is -log+o >> is raius of the omain ey wors Fractional Brownian motion; One-sie exit problem; Survival exponent Introuction Fractional Brownian motion -FBM of inex 0 with multivariate time centere Gaussian ranom process w t with correlation function Ew t w s / t s t s -FBM is -self-similar -ss isotropic an has stationary increments si ie { w 0 0 t t R is a t t w t } { w } hols in the sense of the equality of finite-imensional istributions for any fixe t 0 0 an rotation aroun 0 he one-sie exit problem for a ranom process t an its characteristics the so-calle survival exponents: lim log P t t are the subject of intensive analysis in applications ere / is an increasing sequence of omains of size an is a suitable slowly varying function typically log for ss-processes he greatest progress in this area has been achieve for processes with oneimensional time See surveys by Bray et al 03 in the physics literature an by Aurzaa an Simon 05 in the mathematical one

2 -FBM was one of the first non-trivial examples of non-marovian processes for which the survival exponents have been foun exactly Molchan999 Namely for log the survival exponents are 0 an w w Recently Aurzaa et al 06 consierably refine the asymptotics of probability p P w t t 0 3 an showe that the exponent is universal for a broa class of -ss processes with stationary increments he ieas of this wor have prove useful in analysis of the conjunction that for w t in w ] [ 0 ] [ Molchan 0 he case 0 correspons to the right part of he case is supporte by the result which we iscuss below: for fractional Brownian motion in is w a unit ball that contains 0 at its bounary o estimate w we moify the famous result by Aurzaa et al 06 which says that for a broa class of si-processes: t 0 0 with iscrete time t Z P t t \ {0} E max t t 4 [ 0 ] an For -ss processes with continuous time the right part of 4 is proportional to an therefore the exponent for 3 is owever the result by Aurzaa et al 06 essentially uses the -D nature of time Consiering as the volume of relation 4 is foun to be in formal agreement with the conjunction for but not for ; in aition 4 is very crue for 0 see 3 his means that the analysis of the cases nees in aitional ieas he lower boun Proposition Let t 0 0 t R be a centere isotropic ranom process with stationary increments hen P t t t c E max t t 5 is a ball of raius that contains 0 at its bounary

3 3 Consequence If t is fractional Brownian motion of inex 0 exponent has the lower boun w in then the survival Remar Proposition hols for [0 ] [ ] as well Proof Let { x n ; } be a subset of ball B of raius in of N points such that R ; consists x r x xm N C r r 6 Consier the following increasing sequence of subsets of : x x : x r } { i i Fix { t : t e } e 00 Let O be a rotation transferring x in x r e Setting O one has B x \ {0} \ herefore using the notation M A sup t t A we get p 7 : P t t \ B P M x \ {0} 8 By the si-property of t we can continue P M \ x x P M x 9 he last equality hols because t is rotation invariant he event { M x } is measurable relative to the sequence x x ; x 0 n x n an means that x is a recor which excees the previous one by at least Let be the number of such recors in 0 hen by 89 N p P M x E E M x { x } n x n x Finally by 6 p E M / N c Esup{ t t Suppose that t is fractional Brownian motion of inex 0 in By the stanar proceure we can compare p with

4 p Pw t t \ B For this purpose we can fin a continuous function t such that 4 t t const 3 is the norm of the ilbert space with the reproucing ernel Ew t w s t s see for this fact Molchan999 or Appenixhen h p Pw t t t \ B Accoring to Aurzaa&Dereich 00 ln / p ln/ p Т / 4 From the self-similarity of -FBM an one has Combining 3-5 one has p с EM 5 w [ln/ P w t t ] / / ln O/ ln 6 ie w 3 he upper boun Below we use notation M A sup w t t A an A #{ t : t A} Proposition Let w t be -FBM in R the is a boune omain an 0 Consier a finite -net of ie a subset { x N 0} } { such that N N an B xr B x is unite ball with center x hen for 0 an 0 q P M c ln qp M 0 7

5 5 c v v is the Fernique constant In aition EM EM o EM o 8 Proof One has c P M 0 P M 0 A P A 9 A {max maxt w t w x t B x b } We can continue the previous inequality P M b Pmax w t w x t B x b 0 P M b N P M B b : p p Applying the Fernique 975 result to w t we have u / P M B r c c e u r r / 4 From here settingb 4 ln c one has p C / ln C / ln o show p o p note that B D is iameter of D herefore p P M b P M B b P M B D D/ 3 / b By Molchan 999 P M B o Due to 3 we have p / p / / Oln =o 4 Relations 9 0 an 4 imply 7 o prove relation 8 note that

6 M M max max w t w x t B x : M 5 t 6 As above using the event A {max maxt w t w x t B x b } one has E b E b N EM B [ M B b ] 6 A b 4 ln c an N C herefore the - term in 6 is o By 5 6 we obtain 8 because EM EM E EM с ln o EM с ln o Proposition 3 Let w t t be -FBM 0 hen R is a unite ball an P M c ln ie the survival exponent for -FBM in has the upper boun w Consequence Due to Propositions 3 the survival exponent for -FBM in exists an is equal to Proof As in proof of Proposition we consier again the subset { x n ; } elements of are numerate in such way that of ball B R : { 0} In aition to the properties 6 we suppose that the x B x an x B x n 7 As before x x : x r } : 0 { ; i i { t : t e } e 00 ; O is a rotation transferring x in x r e Setting O one has B x \ {0} \ Due to 7 x is -net in herefore by 7 for 0 P M ln c qp M x \ {0} 0

7 qp M w x 0 qp M w x 7 As a result n n P M c ln q P M w 8 x [ ] an [ ] Similarly to the proof of Proposition we conclue that the right part of 8 is equal to E is a number of recors in the following sequences: M w x w x n ; w x w x n Let be the maximum increment between ajacent elements of the sequence w x w x w x n ; w x w x n n hen M M Т b R 9 R \ Т [ b ] Due to 7 ER t \ max Ew t[ w t b ] Setting b n ln an we obtain ER с n c n 30 By 9 E EM ER EM b accoring to 8 Setting n EM EM o one has

8 8 E с n ln с o 3 Now we can continue 8 as follows: qe n P M c ln 3 Due to ss-propery of -FBM P M c ln P M / / c ln an therefore the probability term ecreases as function of ence 3 implies qe \ P M c ln C P M 33 / /c ln or ln / c o Finally by 3 an P M c ln Appenix Example from Proposition Consier -FBM in omains 0 ; then a suitable function t t can be chosen as follows: t f t / f t f x x R is a finite smooth function such that t f for x / an f t 0 for x ere is the iameter of his can be seen as follows Molchan 999 Due to the spectral representation of -FBM the ilbert space with the reproucing ernel Ew t wh s t s Aronszajn 950 is closure of smooth functions t 0 0 relative to the norm inf с ˆ Where t is a finite function such that t t t ; ˆ R is the Fourier transform of t Obviously we have 0 0 for t \ B an f f t / f t f t / f f

9 9 References Aronszajn N heory of reproucing ernels ransamer Math Soc Aurzaa F an Dereich S niversality of the asymptotics of the one-sie exit problem for integrate processes Ann Inst Poincar é Probab Statist 49: Aurzaa FGuillotin-Plantar N Pene F Persistence probabilities for stationary increment processes Preprint https: arxiv: Aurzaa F; Simon Persistence probabilities an exponents L évy matters V p 83- Lecture Notes in Math 49 Springer 05 Bray A J; Majumar S N; an Schehr G Persistence an first-passage properties in nonequilibrium systems Avances in Physics 63: FerniqueX Regularite es trajectories es functions aleatoires gaussiannes Lecture Notes in Mathematics 480 Berlin-eielberg-New Yor : Springer Verlag975 Molchan G Maximum of fractional Brownian motion: probabilities of small values Comm Math Phys05: Molchan G nilateral small eviations of processes relate to the fractional Brownian motion Stoch Proc Appl Molchan G Survival exponents for some Gaussian processes IntJ of Stochastic Analysis 0 Article ID 377 inawi Publishing Corporationoi:055/0/377