Brownian motion and the heat equation .

�1�

Prep : Fix 9 :# R"

nice"

,let vk.tl = E

,.[ g ( BH ) ) ]

.

Then o¥v( x. f) =ov=:tzf÷zv( x. t)(

[ 2,

bounded

Lemmas : Let Plt , x. g) =

fzgte" " " " "

be B. M.

transition density . Then

ftp.x.ykoplt.x.y ) .

Proof pH , x. y ) =P H,

x -

y , 0 ) so can assume y- 0 .§

PH ," ,d=- ztfeiiht + 1- e-

" 42?

¥,= pit , x. o ) (2¥ - zt )

Fhtt

§=

plt.x.ot-PH.x.dz#o2z.pH.x.d=pH.x.dfz2g+4zI)=2fpH.x.o:

Proof of Proposition

¥ vk.ti.gg/ngcy)pH.x.y)dy=/ngcy)dqPH.x.y)dy=f.gcyzg,2gpH.x.y)dydt

'

= £ Ifngcyipct, x. g) dy=0VC' '

.t )

⇐-

°

0 Exlg CBHID

Branching Brownian Motion

yEx →

starting position x ( default ; so )�2�

- Time 0 : Particle at

xeR ( Rdl '

- Time t :

Dt=# particles at time t .

Particles at positions (

X.HI, ...

,

Xottl)

. Particles branch at rate 1 C i. e. Time for a particle to branch is EXPN - distributed ;

or,

PC fixed particle branches eftttdt ) ) - dt )Tt

- Particles more as Brownian Motion

- Particles move and branch indep . from one - another.

wvmar4NB When

k particles ,time to branch is m ;n( Ei

,

ieiek ) where E, ,

. .Et are independent Expa ) .

So,

( Dt ,t 30 ) is a pure birth process with rates 7 ,.=k

Filtration of BBM is Feokxicstie Ds ) ,set ) = all info

. aboutBBM up to time t

Pap ( Fisher - KPP ) Fix fR→lR nice,

write ZHI :

II. FCXICH) �3�

ulx.tl = IEXZH ).

Then osfucxtl= On ( x. t ) - ucxt ) ( I - uc at )).

proof" Backward equation

"- Think about what happens in time ( ahl

,h

smallu( x.

tth) = IE

,fZ(t.tt/=lE,c/7(t+h)/

no branching in 6. h) ) .IN No branching in (

ah) ]

+ E×[ Z ( tth ) / branching in (ah )|P[ branching inCah )) (#

Hold t fixed,

let gcx ) = ul x. t ),

v( g. s ) - IEY( g ( Bcs ) )).

Then ffv ( g. s ) =

ovcy,s )

Exffltth)|no branching in 6.

41= InEYIZLHIPH,x.g) dy =),nuly.t)p(hx,y)dy=v(x,

h)

# =v ( x. at hDv( x. o ) + 01h ) =ubstltHOUCx. t ) + 01h )

P( No branching in ( ah) ] = I - h + och )

E ,dZ(tth ) / branching in ( ah) ] = uln , f) 2+0 ( h) P[ branching in ( ah))= htolh )

: ( x. tth ) = ( i - htolh ) ) @( x , D thou( x. Htoch ) ) + ( htolh ) ) ( U ( x. f)2+0 ( h ) )

= ( 1- buthow + hit + och )

sodycx.tl = DUGST ) - ucx ,

f) ( tucx ,t )) Dat

�4�

Important special case : fcx )= 1 no

Then

Egftp.fkiltl ) =

Pg( min ( Xict )

,ie Dt ) >

OlyPo ( max ( XICH ,

is

DDEY)

X. (

H=y- Xia ) gives BBM from 0 ,

and min ( Xilt ),

is Dt ) > 0 < ⇒ max ( Xi ( t ),

isDDEYThe ( Neveu ) tiny t'

'm. IF

I

NB.

This implies that from Heaviside initial condition o

Ul x. 0 ) = Tfxzo x=o

f t > o, ulx ,t ) is increasing in x ( and CT in fact )

,

and too, tianya.nu#=otlIId?k+atu"

' t' - l'

Prop :

IEDC= et for all t �5�

Proof :

EDt+h=µ§MDt=klE[Dt+hIDEK]

=

§,PlDt=k) . ( Ktkhtockh ) ) =( lthtoch )

)ED£so ddqEDT =

EDT,

and

EDo=Do=l. D

Corollary : Let Mt =

imax,

Xilt ) = maximal position of a particle at timetD

Thenflimsy My gfz almost surely

Proof If lineup Ptt > Fate,

eecon ) then can find times ( tn )na ,tn+ ,

? tntl,

St . My > F. + £ for all n.

IfMtzfktfttn

then there is i st.

Xiltn ) 3152+ E) tn .

For a Bmstarted fromany

ice R,

P,c( BH ) > x ) =L.

so

PolMrt .it#EHnl2+n.Mtn3Cr+EItn) ?± .

3 ( if + E)ftn7 - i ) > ( f+§ ) # for n large

�6�

So if lizards, if + ethen can find

anincreasing seq

. ( Sk ) # 1 of integer times

s.t.MS, ¢ + f) V. k

Sk

Each time - n particle has positionBut for any

of > 0,

NEN , with dist Ncanl ± if Z,

Z~N( al )

P( Mn 's (52+8) n ) =

§P( Mink 'S)n/Dn=k)P(Dn=k)K

s E H P ( AZ > ( Eton ) p(Dn=k ).

+ E I . P(Dn=k )KerenKnier

s menextpc

. (( rzts)rn)% ) + PC Dn ? n' en )

s nze' on

+ IE

Dnlrien= n2 @-

on+ n

- 2

Son,§P( Mn ? fztrh ) < a

0%

Lenya For Z ~ Ncol ), @

PCZ = x ) >

rffexe' '%

for xs ,016 )=¥e -

%> ¥Fe e-

542for se t.tt

proof PCZ . x ) ± IPIZEK ,xt±H=k" Finds ¥f⇒e×e⇒2o

Proof of Neveu 's Thm : Already showed lifeaupmtlttif. Remains toprove life,sgpMtA¥ik .

f. i. recoil) For ( E - d) t ? I,

e-( rzrttk

,et ECFEH

.

-

p( tkz > ( if . d) t ) = P( Z 3 ( E - d) th ) 7 rate . ( rent't 14 eoet

' '

Since ED+= et and or - E > %

lE[ # { is Dt : XIH ) > ( rz . dlt } ]= et . P( tkz > ( F . of t )

-ITt->

, @

onYruttot' "

Fix T = TW ) big enough that

eottyrttert"z > I

. Icp ICT )

Then IEIIOI> I.

For each jetolet

Ij = { it DKT ) : Xi(2T ) a descendant ofXJCT)

,X:( ZT ) -Xj ( T) 3 ( if - d) T }

.

Then ( Ij , je Io ) are conditionally indep . given Io ,and all have dist

.

of I¢ .

�8�

Continuing this way gives a brandingprocess :

Ij , , . → jµ ,

= { is D*+, )t:X :( HADa descendant of Xjw ( ttl , Xi CKTDTI - X ;kCkT) > af . d) T }

Any particle in gen . g of this process has X ; ( 9T ) > ( rz . on )gT .

So if the B. P.

survives then fiminaf M¥ > ligninaf lgng¥> rz - or

.

And the process survives with positive probability since supercritical !

This gives PC timid Mitt > R - o ) > e > o . But

P ( timid Mitt < R - o ) =P ( timid Mitt < R - or )2 ( consider first branch time ) .

so =o

so PC lining Mitt > R . o ) = I as