T. Appuhamillage, V. Bokil, E. Thomann, B.Wood,

Depts of Mathematics, Chemical, Biological, & Environmental Engineering, Oregon State University

ED WAYMIRE

Pacific Northwest Probability Seminar - OCTOBER 2010

Interfacial Phenomena & Skew Diffusion

Example 1: Dispersion in Heterogeneous Media

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y #

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y # D+ D!

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y # D+ D!

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y #

x z

!"

!x

!!!!x=0+

=!"

!x

!!!!x=0!

!"

!y

!!!!x=0+

=!"

!y

!!!!x=0!

!c

!t= ! · (D(x)!c) + U

!c

!x

c |I+ = c |I!

D+ !c

!x

!!!!I+

= D!!c

!x

!!!!I!

x z

!"

!x

!!!!x=0+

=!"

!x

!!!!x=0!

!"

!y

!!!!x=0+

=!"

!y

!!!!x=0!

!c

!t= ! · (D(x)!c) + U

!c

!x

c |I+ = c |I!

D+ !c

!x

!!!!I+

= D!!c

!x

!!!!I!

c = concentration of injected dye

D(x) =

!

D+ x ! 0D! x < 0.

Fickean Dispersion Model:

!c

!t=

1

2! · (D(x)!c) " v

!c

!x

v

B. Wood, CBEE LAB, OSU

Figure 2.!!Experiments showing differences in

breakthrough behavior for coarse to fine (C-F)

and fine-to-coarse (F-C) directions of flow at

flow rates of 0.3, 0.4, and 1 mL/min. The tracer

input pulse times are 5, 3.75, and 1.5 min, and

the sample collection intervals are 5, 3.75, and

1.5 min, respectively. Solid symbols represent

C-F direction, and open symbols represent F-C

direction. Two experiments in each direction

were measured for the 0.3 and 0.4 mL/min flow

rates. Each point represents an average of three

measurements. The vertical axis shows

electrical conductivity (EC), which is directly

proportional to concentration.

B. Berkowitz, A. Cortis, I Dror and H Scher - 2009., B.Wood -2010.

PROBLEM: Explain the asymmetry from the Fickian Dispersion Model ?

Experimentally Observed Asymmetry

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y # D+ D!

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y #

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y # D+ D!-1 1

Q: Assume that D- < D+. Which is more likely removed first, a particle injected at -l and removed at l, or a particle injected at l and removed at -l ?

Example 2: Physical Oceanography - Upwelling of the Malvinas Current off the coast of Argentina

Highlighted region points to flotilla of ‘fishing factories’

Concentration of ships on the continental break where upwelling occurs.

Acknowledgment: Ricardo Matano - COAS - Oregon State University

Arrested Topographic Wave Model

Quasi geostrophic balance - Hydrostatic Approximation.

h(x)

!fv = !g!"

!x

fu = !g!"

!y! rv

h

0 =!(uh)

!x+

!(vh)

!y

h(x)

!fv = !g!"

!x

fu = !g!"

!y! rv

h

0 =!(uh)

!x+

!(vh)

!y

"(x , y) = free surface, h(x) = ocean depth

h(x)

!fv = !g!"

!x

fu = !g!"

!y! rv

h

0 =!(uh)

!x+

!(vh)

!y

"(x , y)

!"

!y+

r

f

!!h

!x

"!1 !2"

!x2= 0

Eliminate velocity in terms of free surface

f < 0 in Southern

hemisphere

y

x

1

2

d2

dx2

f ! C(R) " C2(R\{0})

!f !(0+) = (1# !)f !(0")

B!(t)

h(x)

Transport = velocity times height of water column

Height of water column =

= slope of continental shelf (-)and slope (+)

Arrested Topographic Wave Model:

S(B!(t)) =!

D+B!(t)1(B!(t) > 0) +!

D!B!(t)1(B!(t) < 0)

∂η/∂y > 0 ∂η/∂y < 0 ∂η/∂x < 0

η0 > η1 > η2

∂n

∂t=

1

2D±

∂2η

∂x2

D+ =2r

|f |s+, D+ =

2r

|f |s!

H±

S(B!(t)) =!

D+B!(t)1(B!(t) > 0) +!

D!B!(t)1(B!(t) < 0)

!"/!y > 0 !"/!y < 0 !"/!x < 0

"0 > "1 > "2

!n

!t=

1

2D±

!2"

!x2

D+ =2r

|f |s+, D+ =

2r

|f |s!

s±

S(B!(t)) =!

D+B!(t)1(B!(t) > 0) +!

D!B!(t)1(B!(t) < 0)

!"/!y > 0 !"/!y < 0 !"/!x < 0

"0 > "1 > "2

!"

!t=

1

2D±

!2"

!x2

D+ =2r

|f |s+, D+ =

2r

|f |s!

s±

h(x) + !(x, y)

Arrested Topographic Wave Model

(Shelf Break Interface)-

Lupinus sulphureus ssp. kincaidii at Oak Basin, 2009

1

IN T R O DU C T I O N

This report documents work conducted on

the population of !"#$%"&'()*+,"#-).Lupinus sulphureus ssp. kincaidii; Figure 1) located

at Oak Basin, a site managed by the Eugene

District Bureau of Land Management. The

proposed Oak Basin ACEC represents the

largest known lupine population in the

Upper Willamette Resource Area.

Species status !"#$%"&'()*+,"#-, a member of the

legume family (Fabaceae), is listed by the

Oregon Department of Agriculture and the

U.S. Fish and Wildlife Service as a

threatened species (ORNHIC 2007).

!"#$%"&'()*+,"#-)(-/0-()%()the primary host

plant for larvae of 1-#&-/'()2*+-)2+33-/4*5)

(Icaricia icarioides fenderi; Figure 2), an

endangered species.

Background information !"#$%"&'()*+,"#- is found in native

prairie remnants in the Willamette Valley

and southwestern Washington and in forest

openings in Douglas County, Oregon.

Because !"#$%"&'()lupine serves as the

primary host for 1-#&-/'()2*+-)2+33-/4*5)

larvae, conservation of the lupine is a

common goal for the protection of both

species.

The majority of sites known to

(+,,6/3)!"#$%"&'()*+,"#-)%/-)smaller than 1

hectare (Wilson et al. 2003) and located on

privately held land (U.S. Fish and Wildlife

Service 2000), which is exempt from

protections provided by state and federal

listing. Management by state and federal

agencies on public land is therefore critical

for survival of this species.

Oak Basin is an important site for

!"#$%"&'()*+,"#- because of its ownership

and size. In addition to the protection listed

species receive on public lands, Oak Basin

F igure 1. !"#$%"&'()*+,"#- (Lupinus sulphureus ssp. kincaidii).

F igure 2. 1-#&-/'()2*+-)2+33-/4*5).Icaricia icarioides fenderi).

Fender’s Blue Kincaid’s Lupin Patch Distribution

Example 3: FENDERS BLUE BUTTERFLY

1880 CHERYL B. SCHULTZ AND ELIZABETH E. CRONE Ecology, Vol. 82, No. 7

these studies rarely investigate movement using meth-

ods from which mechanistic ‘‘rules’’ can be described

to predict an organism’s responses to new habitat loss

or creation. Pattern-oriented approaches such as inci-

dence function models often assume simple rules about

dispersal processes, assumptions which are rarely ex-

plicitly tested and may lead to misleading interpreta-

tions of population dynamics (Crone et al. 2001).

An additional complication is that dispersal studies

often assume that organisms’ responses to habitat are

uniform—as if the habitat were homogeneous—be-

cause the studies only occur within high quality habitat

(e.g., Stenseth and Lidicker 1992; but see Zalucki and

Kitching 1982, Miyatake et al. 1995, Kindvall 1999).

Fragmented habitat is, by definition, heterogeneous.

Assuming a homogeneous landscape may produce mis-

leading predictions if organisms change movement

rates between habitats. For example, if organisms in-

crease their movement rates to move quickly through

hostile habitats (as was observed by Miyatake et al.

1995, and Schultz 1998a), models assuming homoge-

neous movement will underestimate organisms’rate of

dispersal through a landscape. Several insect studies

have measured features of dispersal behavior that are

useful in modeling dispersal processes (movement

rates, turning angles, and subsequent ‘‘diffusion rates’’)

and have shown that these features vary between hab-

itats (Kareiva and Odell 1987, Morris and Kareiva

1991, Turchin 1991, Schultz 1998a, Haddad 1999,

Kindvall 1999) and are affected by population density

(Kindvall et al. 1998). When investigators have mea-

sured sensitivity to habitat quality, they have found that

these differences influenced the expected spatial dis-

tributions of insects (see review by Turchin 1991).

Finally, to understand how organisms move across

fragmented landscapes, we need to understand the

‘‘permeability’’ of habitat–nonhabitat interfaces—the

degree to which organisms leave high quality habitat

(Wiens et al. 1985, Stamps et al. 1987, Fagan et al.

1999, Kindvall 1999). To some organisms, certain hab-

itat edges form an absolute barrier (e.g., the ocean to

a nonflying terrestrial mammal). To other organisms, a

habitat edge may merely slow dispersal a bit. Edge

permeability influences responses to landscape struc-

tures, such as how well a corridor serves as a conduit

connecting habitat patches (Lidicker and Koenig 1996,

Schultz 1998a, Haddad 1999). This view of an edge is

distinct from the literature on ‘‘edge effects.’’ Edge

effects usually refer to changes in an organism’s density

at the edge, such as changes in density due to biotic

or abiotic effects (Yahner 1988, Meffe and Carroll

1997). These views examine the edge as a distinct hab-

itat type. Instead, we view edge as a habitat boundary

that influences organisms’ behaviors. Fagan et al.

(1999) refer to this as an ‘‘edge-mediated’’ process. In

this context, animals may respond to clearly defined

edges, such as a clearcut boundary, or less discrete

boundaries, such as a meadow edge.

Here, we present an investigation of edge-mediated

dispersal behavior in a prairie butterfly, the Fender’s

blue. The Fender’s blue system is a useful model system

in which to investigate this behavior. From past re-

search we know that movement behavior and diffusion

rates differ inside preferred lupine habitat vs. outside

lupine habitat and that butterflies are not randomly dis-

tributed over the landscape (Schultz 1998a). In addi-

tion, based on coarse-scale experiments, we know that

the butterflies released near patch edges are more likely

to stay in habitat than to leave habitat (Schultz 1998a).

Also, in anecdotal observations of butterflies within

habitat near patch boundaries, it appeared that after

butterflies left the patch, they markedly changed their

behavior by stopping less often and flying in a more

directed fashion. However, these field observations do

not provide quantitative insight into how butterflies

respond to patch boundaries, or a mechanism for pre-

dicting the consequences of habitat loss or creation.

The Fender’s blue is an appropriate study system be-

cause techniques to quantify dispersal behavior are well

established and because patch boundaries are easy to

distinguish in the field.

Given past research on the Fender’s blue and the

potential to investigate response to patch boundaries

in this system, we ask two central questions. First, how

do organisms respond to habitat edges? Second, what

are the implications of this behavior for residence time

(the number of days butterflies spend in their natal

patch before leaving)? For butterflies like the Fender’s

blue, residence time is a key feature of the effects of

habitat fragmentation because if residence time varies

with patch size, individuals will leave smaller patches

and contribute fewer young to future generations in

that patch, decreasing population viability (Crone and

Schultz 2001). To answer the first question, we develop

a straightforward methodology to consider the butter-

fly’s response to habitat edge. This model is based on

empirical parameter estimation of butterfly dispersal in

the field. Second, we incorporate the results of this

analysis into simple models of dispersal to ask how

edge-mediated behavior could influence residence

time.

METHODS

Fender’s blue butterfly biology and habitat

The Fender’s blue butterfly (Icaricia icarioides fen-

deri) resides in remnant native prairies of western

Oregon. It is an indicator species for Willamette Valley

upland prairies and was recently listed as endangered

under the U.S. Endangered Species Act (Anonymous

2000). At the same time, its primary larval host plant,

Kincaid’s lupine (Lupinus sulphureus spp. kincaidii),

was listed as threatened (Anonymous 2000). Currently

only about three thousand of these butterflies remain,

confined to a dozen isolated patches across the Wil-

lamette Valley (Hammond 1998, Schultz and Fitzpat-

DISPE

RSAL

INTER

FACE

Local time

e!ect ?

1879

Ecology, 82(7), 2001, pp. 1879–1892! 2001 by the Ecological Society of America

EDGE-MEDIATED DISPERSAL BEHAVIOR IN A PRAIRIE BUTTERFLY

CHERYL B. SCHULTZ1 AND ELIZABETH E. CRONE2

1National Center for Ecological Analysis and Synthesis, University of California, 735 State Street #300,Santa Barbara, California 93101 USA

2Ecology Division, Biological Sciences Department, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Abstract. Animal responses to habitat boundaries will influence the effects of habitatfragmentation on population dynamics. Although this is an intuitive and often observedanimal behavior, the influences of habitat boundaries have rarely been quantified in thefield or considered in theoretical models of large scale processes. We quantified movementbehavior of the Fender’s blue butterfly (Icaricia icarioides fenderi) as a function of distancefrom host-plant patches. We measured the butterfly’s tendency to move toward habitatpatches (bias) and their tendency to continue to move in the direction they were alreadygoing (correlation). We found that butterflies significantly modify their behavior within 10–22 m from the habitat boundary.

We used these data to predict large scale patterns of residence time as a function ofpatch size, using three dispersal models: homogeneous response to habitat, heterogeneousresponse to habitat, and heterogeneous response to habitat with edge-mediated behavior.We simulated movement for males and females in eight patch sizes (0.1–8 ha) and askedhow residence time varies among the models. We found that adding edge-mediated behaviorsignificantly increases the residence of Fender’s blue butterflies in their natal patch. Onlythe model with edge-mediated behavior for females was consistent with independent mark–release–recapture (MRR) estimates of residence time; other models dramatically under-estimated residence times, relative to MRR data.

Key words: biased correlated random walk; dispersal; edge behavior; Fender’s blue butterfly;habitat fragmentation.

INTRODUCTION

Ecologists face a fundamental challenge in under-

standing how populations respond to habitat fragmen-

tation. Meffe (1997:148) defines fragmentation as ‘‘the

breakup of extensive habitats into small, isolated patch-

es.’’ From an applied perspective, understanding hab-

itat fragmentation is central to interpreting the effects

of habitat loss on endangered species (Noss and Coop-

errider 1994, Wiens 1997, Gascon and Lovejoy 1998).

From a basic perspective, responses to fragmentation

are influenced by the ability of animals to disperse

among patches, the population dynamics in habitat

patches of different sizes, and the factors that control

whether animals remain in or disperse from suitable

habitat (Forman and Godron 1986, Turchin 1998, Fa-

gan et al. 1999). Recent theoretical studies highlight

fragmentation effects and potential consequences for

population processes (e.g., Holmes et al. 1994, Pulliam

and Dunning 1995, Turner et al. 1995, Hanski and Gil-

pin 1997, Tilman and Kareiva 1997, Fagan et al. 1999).

The most common obstacle to linking models to pre-

dictions of large-scale processes is lack of understand-

ing of the basic ecology of dispersal behavior—how

and when animals move through heterogeneous land-

scapes (Wennergren et al. 1995, Lima and Zollner

1996, Ruckelshaus et al. 1997).

Manuscript received 1 November 1999; revised 30 May 2000;accepted 30 June 2000; final version received 24 July 2000.

This obstacle persists, at least in part, because of a

wide gap between many of the analytical models used

to study spatial ecology and on-the-ground studies of

dispersal behavior (Turchin 1998). Analytical spatial

models are often so abstract that it is difficult for field

ecologists to estimate model parameters, to test models

or simply to know how or when to use them (but see

Kareiva and Shigesada 1983, Turchin 1991). For ex-

ample, the connection between parameters in partial

differential equation models (e.g., Othmer et al. 1988)

and empirical data is not often clear. Similarly, too

often empirical estimates of dispersal are not done in

ways that can be translated into parameters for models.

For example, measures of tortuosity and fractal di-

mension are useful in describing the movement behav-

ior of animals in heterogeneous landscapes (Stapp and

Horne 1997, Etzenhouser et al. 1998) but such descrip-

tions of movement patterns are difficult to translate into

movement parameters that could predict population

distributions and dynamics in novel landscapes. A

number of experimental studies have investigated the

effects of fragmentation phenomenologically by ex-

perimentally fragmenting a system and reporting the

responses (e.g., Bierregaard et al. 1997, Andreassen et

al. 1998). However, in many systems (e.g., endangered

species), the experimental approach is not possible. Fi-

nally, studies documenting immigration and coloni-

zation have been done in the context of metapopulation

dynamics (see review by Ims and Yoccoz 1997), but

``

?’’

INTERFACES AND PROBABILITY

Interface conditions: Continuity of c and of fluxExample 1:

x z

!"

!x

!!!!x=0+

=!"

!x

!!!!x=0!

!"

!y

!!!!x=0+

=!"

!y

!!!!x=0!

!c

!t= ! · (D(x)!c) + U

!c

!x

c |I+ = c |I!

D+ !c

!x

!!!!I+

= D!!c

!x

!!!!I!

!"

!x

!!!!x=0+

=!"

!x

!!!!x=0!

!"

!y

!!!!x=0+

=!"

!y

!!!!x=0!

!c

!t= ! · (D(x1)!c) + U

!c

!x1

c |I+ = c |I!

D+ !c

!x1

!!!!I+

= D!!c

!x1

!!!!I!

Example 2:

Interface conditions: Continuity of c and of derivative

!|I+ = !|I!

"!

"x

!!!!I+

="!

"x

!!!!I!

!|I+ = !|I!

"!

"x

!!!!I+

="!

"x

!!!!I!

Example 3: ?

Stochastic Versions of Interface Conditions ?

INTERFACES AND PROBABILITY

Interface conditions: Continuity of c and of fluxExample 1:

x z

!"

!x

!!!!x=0+

=!"

!x

!!!!x=0!

!"

!y

!!!!x=0+

=!"

!y

!!!!x=0!

!c

!t= ! · (D(x)!c) + U

!c

!x

c |I+ = c |I!

D+ !c

!x

!!!!I+

= D!!c

!x

!!!!I!

!"

!x

!!!!x=0+

=!"

!x

!!!!x=0!

!"

!y

!!!!x=0+

=!"

!y

!!!!x=0!

!c

!t= ! · (D(x1)!c) + U

!c

!x1

c |I+ = c |I!

D+ !c

!x1

!!!!I+

= D!!c

!x1

!!!!I!

Example 2:

Interface conditions: Continuity of c and of derivative

!|I+ = !|I!

"!

"x

!!!!I+

="!

"x

!!!!I!

!|I+ = !|I!

"!

"x

!!!!I+

="!

"x

!!!!I!

Example 3: ?

Stochastic Versions of Interface Conditions ?

General Interface Conditions:

!f !(0+) ! (1 ! !)f !(0") = 0 0 " ! " 1

FELLER’S CLASSIFICATION : Given msble coefficients

there is a unique diffusion process with generator

A =

d2

dsdm

X

where

!2(x) > 0, µ(x)

ds(x) = e!

!x

·

2µ(y)

!2(y)dy

dx

dm(x) =2

!2(x)e

!x

·

2µ(y)

!2(y)dy

dx

=1

2!2(x)

d2

dx2+ µ(x)

d

dx

DA = {f ! C(J) :d2f

dmds! C(J)}

STROOCK-VARADHAN MARTINGALE: Given measurable coefficients

!2(x) > 0, µ(x)

there is a unique continuous process such that X

Mt(f) = f(Xt) !

!t

0

Af(Xs)ds, t " 0

is a martingale for all .f ! C!

STROOCK-VARADHAN MARTINGALE: Given measurable

!2(x) > 0, µ(x)

there is a unique martingale such that X

Mt(f) = f(Xt) !

!t

0

Af(Xs)ds, t " 0

is a martingale for all .f ! C!

c(x, t) = Exc0(Xt)

c0 ! DA "!c

!x(0+, t) =

!c

!x(0!, t), t # 0.

! =1

2i.e.

WHERE’S THE WIGGLE ROOM FOR INTERFACES ?

Let

Define

if

c0 ! C(R), !c!0(0+) " (1 " !)c!0(0

") = 0

Then

!(x) =

!

!x if x ! 0(1 " !)x if x < 0.

c0 ! ! " I 1

2

Write c0 ! I!

Analytic Remedy

X! ! A!Get Adjusted S-V Process:

Adjust Coefficients (Time Change)

Then

(Space Change)

A! :

(Ouknine 1987)x ! E!!1(x)c0(!(X!(t))) " I!

D(x) =

!

(1 ! !)2D! if x < 0.!2D+ if x " 0!

On Brownian Motion Observations``The trajectories are confused and complicated so often and so rapidly that it is impossible to follow them; the trajectory actually measured is very much simpler and shorter than the real one. Similarly, the apparent mean speed of a grain during a given time varies in a wildest way in magnitude and direction, and does not tend to a limit as the time taken for an observation decreases, as may be easily shown by noting, in the camera lucida, the positions occupied by a grain from minute to minute, and then every five seconds, or, better still, by photographing them every twentieth of a second, as has been done by Victor Henri Comandon, and de Broglie when kinematographing the movement. It is impossible to fix a tangent, even approximately, at any point on a trajectory, and we are thus reminded of the continuous underived functions of the mathematicians.''

- Jean Baptiste Perrin, Atoms 1913

On Brownian Motion Observations``The trajectories are confused and complicated so often and so rapidly that it is impossible to follow them; the trajectory actually measured is very much simpler and shorter than the real one. Similarly, the apparent mean speed of a grain during a given time varies in a wildest way in magnitude and direction, and does not tend to a limit as the time taken for an observation decreases, as may be easily shown by noting, in the camera lucida, the positions occupied by a grain from minute to minute, and then every five seconds, or, better still, by photographing them every twentieth of a second, as has been done by Victor Henri Comandon, and de Broglie when kinematographing the movement. It is impossible to fix a tangent, even approximately, at any point on a trajectory, and we are thus reminded of the continuous underived functions of the mathematicians.''

- Jean Baptiste Perrin, Atoms 1913

Q: So what would Perrin see when there is an interface ?

SKEW BROWNIAN MOTION

Infinitesimal Generator1

2

d2

dx2

f ! C(R) " C2(R\{0})

!f !(0+) = (1# !)f !(0")

B(!)(t)

1

2

d2

dx2

f ! C(R) " C2(R\{0})

!f !(0+) = (1# !)f !(0")

B(!)(t)

1

2

d2

dx2

f ! C(R) " C2(R\{0})

!f !(0+) = (1# !)f !(0")

B!(t)

Ito-McKean (1963):

A =1

2

d2

dx2

DA :

0 ! ! ! 1

Construct diffusion defined by:

Infinitesimal Generator1

2

d2

dx2

f ! C(R) " C2(R\{0})

!f !(0+) = (1# !)f !(0")

B(!)(t)

1

2

d2

dx2

f ! C(R) " C2(R\{0})

!f !(0+) = (1# !)f !(0")

B(!)(t)

1

2

d2

dx2

f ! C(R) " C2(R\{0})

!f !(0+) = (1# !)f !(0")

B!(t)

X = {(Xt ,Yt) : t ! 0}

dXt = U(t)dt +!

D(Yt)dB1

Yt = f (B!!(t)),!! =

"D+

"D+ +

"D"

B! ! Jn(t) An

B!(t) =#"

n=1

1Jn(t)An|B(t)|

pi ,i±1 =1

2, si i != 0

p0,1 = !, p0,!1 = 1" !.

Z!k#B!

t

f (Z!k )#f (B!

t )

P(An = 1) = !

Ito-McKean (1963):

A =1

2

d2

dx2

DA :

0 ! ! ! 1

IID Valued±1

) (1 )t

B( )(t)

0J

A =–1

A =+1

A =–1

J

J

t

B(t)

0J J J

(b)

(a)

Jn Excursion Intervals

B(!)(t) =

!!

n=1

1Jn(t)An|B(t)|

Construct diffusion defined by:

SKEW BROWNIAN MOTION

GENERAL MARTINGALE FORMULATION

Re-scaled Skew Brownian Motion:

1

2

d2

dx2

f ! C(R) " C2(R\{0})

αf !(0+) = (1# α)f !(0")

B!(t)

h(x)

s(B!(t)) =$

D+B!(t)1(B!(t)) +$

D"B!(t)1(B!(t))

Martingale Problem Given D± and 0 % λ % 1, determine α so that

f (s(B!(t)))# 1

2

! t

0D(s(B!)(u))f ”(s(B!(u))du

is a martingale for all f ! D" where

D" = {g ! C2(R\{0}) " C(R) : λg !(0+) = (1# λ)g !(0")}

1

2

d2

dx2

f ! C(R) " C2(R\{0})

!f !(0+) = (1# !)f !(0")

B!(t)

h(x)

s(B!(t)) =$

D+B!(t)1(B!(t)) +$

D"B!(t)1(B!(t))

Martingale Problem Given D± and 0 < " < 1, determine ! so that

f (s(B!(t)))# 1

2

! t

0D(s(B!)(u))f ”(s(B!(u))du

is a martingale for all f ! D" where

D" = {g ! C2(R\{0}) " C(R) : "g !(0+) = (1# ")g !(0")}

!c

!t=

D±

2

!2c

!x2

with interface condition

c(0+) = c(0!), "c "(0+) = (1! ")c "(0!)

Consider

with the interface condition

!c

!t=

D±

2

!2c

!x2

with interface condition

c(0+) = c(0!), "c "(0+) = (1! ")c "(0!)

SOLUTION

! = 1/2, " =

!D!!

D+ +!

D!

Example 1:Example 2:

! =D+

D+ + D!, " =

!D+

!D+ +

!D!

dB! = dB + (2"" 1)dl = dB +2"" 1

2"dA+

! (Le Gall 1982)

local time from the rightA+! = lim

"!0

1

!

! "

01[0 ! B!(u) < !]du

Ito-Tanaka applied to Y (t) = s(B!(t)) gives

dY ="

D(Y )dB +1

2

#1 +

"D"

"D+

#"

D"

"

$dA+

Y

For f $ D#, and Z (t) = f (Y (t)) one has

dZ = f #(Y )"

D(Y )dB +1

2D(Y )dt

+1

2f #(0")

#"D"

"D+

#"

D"

"+

#

1# #

$dA+

Y

" =#"

D" + (1# #)"

D+"

D"

A+! = lim

"!0

1

!

! "

01[0 ! B!(u) < !]du

Ito-Tanaka applied to Y (t) = s(B!(t)) gives

dY ="

D(Y )dB +1

2

#1 +

"D"

"D+

#"

D"

"

$dA+

Y

For f $ D#, and Z (t) = f (Y (t)) one has

dZ = f #(Y )"

D(Y )dB +1

2D(Y )dt

+1

2f #(0")

#"D"

"D+

#"

D"

"+

#

1# #

$dA+

Y

" =#"

D" + (1# #)"

D+"

D"

A+! = lim

"!0

1

!

! "

01[0 ! B!(u) < !]du

Ito-Tanaka applied to Y (t) = s(B!(t)) gives

dY ="

D(Y )dB +1

2

#1 +

"D"

"D+

#"

D"

"

$dA+

Y

For f $ D#, and Z (t) = f (Y (t)) one has

dZ = f #(Y )"

D(Y )dB +1

2D(Y )f ”(Y )dt

+1

2f #(0")

#"D"

"D+

#"

D"

"+

#

1# #

$dA+

Y

" =#"

D"

#"

D" + (1# #)"

D+

A+! = lim

"!0

1

!

! "

01[0 ! B!(u) < !]du

Ito-Tanaka applied to Y (t) = s(B!(t)) gives

dY ="

D(Y )dB +1

2

#1 +

"D"

"D+

#"

D"

"

$dA+

Y

For f $ D#, and Z (t) = f (Y (t)) one has

dZ = f #(Y )"

D(Y )dB +1

2D(Y )f ”(Y )dt

+1

2f #(0")

#"D"

"D+

#"

D"

"+

#

1# #

$dA+

Y

" =#"

D"

#"

D" + (1# #)"

D+

!c

!t=

D±

2

!2c

!x2

with interface condition

c(0+) = c(0!), "c "(0+) = (1! ")c "(0!)

f (Y (t))!! t

0D(Y (s))f ""(Y (s))dx "#

martingale

A+! = lim

"!0

1

!

! "

01[0 ! B!(u) < !]du

Ito-Tanaka applied to Y (t) = s(B!(t)) gives

dY ="

D(Y )dB +1

2

#1 +

"D"

"D+

#"

D""

D+"

$dA+

Y

For f $ D#, and Z (t) = f (Y (t)) one has

dZ = f #(Y )"

D(Y )dB +1

2D(Y )f ##(Y )dt

+1

2f #(0")

#"D"

"D+

#"

D""

D+"+

1# #

#

$dA+

Y

" =#"

D"

#"

D" + (1# #)"

D+

A+! = lim

"!0

1

!

! "

01[0 ! B!(u) < !]du

Ito-Tanaka applied to Y (t) = s(B!(t)) gives

dY ="

D(Y )dB +1

2

#1 +

"D"

"D+

#"

D""

D+"

$dA+

Y

For f $ D#, and Z (t) = f (Y (t)) one has

dZ = f #(Y )"

D(Y )dB +1

2D(Y )f ##(Y )dt

+1

2f #(0")

#"D"

"D+

#"

D""

D+"+

1# #

#

$dA+

Y

" =#"

D"

#"

D" + (1# #)"

D+Proposition:

A+!(t) = lim

"!0

!t

01[0 ! B

(!)(s) < !]ds

f(Y (t)) !1

2

!t

0

D(Y (s))f !!(Y (s))ds

DA

Back To Examples

Example 2 - Continuity of Derivative

Example 1 - Continuity of Flux

! = 1/2, " =

!D!!

D+ +!

D!

Example 1:Example 2:

! =D+

D+ + D!, " =

!D+

!D+ +

!D!

dB! = dB + (2"" 1)dl = dB +2"" 1

2"dA+

!

! ! !!

=

"D+

"D+ +

"D"

! =D+

D+ + D!

A+! = lim

"!0

1

!

! "

01[0 ! B!(u) < !]du

Ito-Tanaka applied to Y (t) = s(B!(t)) gives

dY ="

D(Y )dB +1

2

#1 +

"D"

"D+

#"

D"

"

$dA+

Y

For f $ D#, and Z (t) = f (Y (t)) one has

dZ = f #(Y )"

D(Y )dB +1

2D(Y )f ”(Y )dt

+1

2f #(0")

#"D"

"D+

#"

D"

"+

#

1# #

$dA+

Y

" =#"

D"

#"

D" + (1# #)"

D+

Back To Examples

Example 2 - Continuity of Derivative

Example 1 - Continuity of Flux

! = 1/2, " =

!D!!

D+ +!

D!

Example 1:Example 2:

! =D+

D+ + D!, " =

!D+

!D+ +

!D!

dB! = dB + (2"" 1)dl = dB +2"" 1

2"dA+

!

! ! !!

=

"D+

"D+ +

"D"

! =D+

D+ + D!

Walsh 1978: Discontinuity of Local Time for SBM

A+! = lim

"!0

1

!

! "

01[0 ! B!(u) < !]du

Ito-Tanaka applied to Y (t) = s(B!(t)) gives

dY ="

D(Y )dB +1

2

#1 +

"D"

"D+

#"

D"

"

$dA+

Y

For f $ D#, and Z (t) = f (Y (t)) one has

dZ = f #(Y )"

D(Y )dB +1

2D(Y )f ”(Y )dt

+1

2f #(0")

#"D"

"D+

#"

D"

"+

#

1# #

$dA+

Y

" =#"

D"

#"

D" + (1# #)"

D+

Back To Examples

Example 2 - Continuity of Derivative

Example 1 - Continuity of Flux

! = 1/2, " =

!D!!

D+ +!

D!

Example 1:Example 2:

! =D+

D+ + D!, " =

!D+

!D+ +

!D!

dB! = dB + (2"" 1)dl = dB +2"" 1

2"dA+

!

! ! !!

=

"D+

"D+ +

"D"

! =D+

D+ + D!

Walsh 1978: Discontinuity of Local Time for SBM

Proposition: For in Example 1, the modified local timeis (spatially) continuous if and only if

!A

+!

! = !!

A+! = lim

"!0

1

!

! "

01[0 ! B!(u) < !]du

Ito-Tanaka applied to Y (t) = s(B!(t)) gives

dY ="

D(Y )dB +1

2

#1 +

"D"

"D+

#"

D"

"

$dA+

Y

For f $ D#, and Z (t) = f (Y (t)) one has

dZ = f #(Y )"

D(Y )dB +1

2D(Y )f ”(Y )dt

+1

2f #(0")

#"D"

"D+

#"

D"

"+

#

1# #

$dA+

Y

" =#"

D"

#"

D" + (1# #)"

D+

Back To Examples

Example 2 - Continuity of Derivative

Example 1 - Continuity of Flux

! = 1/2, " =

!D!!

D+ +!

D!

Example 1:Example 2:

! =D+

D+ + D!, " =

!D+

!D+ +

!D!

dB! = dB + (2"" 1)dl = dB +2"" 1

2"dA+

!

! ! !!

=

"D+

"D+ +

"D"

! =D+

D+ + D!

Walsh 1978: Discontinuity of Local Time for SBM

Proposition: For in Example 1, the modified local timeis (spatially) continuous if and only if

!A

+!

! = !!

``MODIFICATION’’ -- Integrate w.r.to Lebesgue in place of QV

A+! = lim

"!0

1

!

! "

01[0 ! B!(u) < !]du

Ito-Tanaka applied to Y (t) = s(B!(t)) gives

dY ="

D(Y )dB +1

2

#1 +

"D"

"D+

#"

D"

"

$dA+

Y

For f $ D#, and Z (t) = f (Y (t)) one has

dZ = f #(Y )"

D(Y )dB +1

2D(Y )f ”(Y )dt

+1

2f #(0")

#"D"

"D+

#"

D"

"+

#

1# #

$dA+

Y

" =#"

D"

#"

D" + (1# #)"

D+

APPLICATION TO EXAMPLE 1

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y # D+ D!

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y #

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y # D+ D!

Q: Assume that D- < D+. Which is more likely removed first, a particle injected at and removed at or a particle injected at and removed at ?

1!1

!11!1

1

D!

< D+

APPLICATION TO EXAMPLE 1

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y # D+ D!

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y #

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y # D+ D!

Q: Assume that D- < D+. Which is more likely removed first, a particle injected at and removed at or a particle injected at and removed at ?

A: The experiments show that in this configuration, a particle injected at arrives faster at than when the particle is injected at and removed at .

1!1

!1

1!1

1

!1

!1 1

1

D!

< D+

i.e. FINE TO COARSE IS FASTER THAN COARSE TO FINE

APPLICATION TO EXAMPLE 1

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y # D+ D!

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y #

Ley de Conservacion

!c

!t=

1

2! · (D!c)"! · (Uc)

!c

!n= 0 !G

[c] = 0

!D

!c

!nI

"= 0

U(y) I x y # D+ D! 1!1

D!

< D+

Impact on asymmetry of break-through curves. Fine to Coarse corresponds to and thus <<> 1/2. Coarse to Fine corresponds to>>< 1/2.

A+! = lim

"!0

1

!

! "

01[0 ! B!(u) < !]du

Ito-Tanaka applied to Y (t) = s(B!(t)) gives

dY ="

D(Y )dB +1

2

#1 +

"D"

"D+

#"

D""

D+"

$dA+

Y

For f $ D#, and Z (t) = f (Y (t)) one has

dZ = f #(Y )"

D(Y )dB +1

2D(Y )f ##(Y )dt

+1

2f #(0")

#"D"

"D+

#"

D""

D+"+

1# #

#

$dA+

Y

" =#"

D"

#"

D" + (1# #)"

D+

A+! = lim

"!0

1

!

! "

01[0 ! B!(u) < !]du

Ito-Tanaka applied to Y (t) = s(B!(t)) gives

dY ="

D(Y )dB +1

2

#1 +

"D"

"D+

#"

D""

D+"

$dA+

Y

For f $ D#, and Z (t) = f (Y (t)) one has

dZ = f #(Y )"

D(Y )dB +1

2D(Y )f ##(Y )dt

+1

2f #(0")

#"D"

"D+

#"

D""

D+"+

1# #

#

$dA+

Y

" =#"

D"

#"

D" + (1# #)"

D+D

!

< D+

Proposition: Assume!

D! <!

D+, !" =!

D+/(!

D+ +!

D!)Let Y = s(B!!), T "

y = inf{t " 0 : Yt = y}. Then for eacht > 0, y > 0,

P!y (T "y > t) < Py (T "

!y > t)

Scaling properties B!ct

d=!

cB!t

T "0 =P1!dist 1

D+T0 T "

0 =P"1!dist 1

D!T0

T "1 =P0!dist 1

D+T!!

1 T "!1 =P0!dist 1

D!T (1!!!)1

SKEW BROWNIAN MOTION

!c

!t=

D±

2

!2c

!x2

with interface condition

c(0+) = c(0!), "c "(0+) = (1! ")c "(0!)

f (Y (t))!! t

0D(Y (s))f ""(Y (s))dx "#

Proposition: Let T!a = inf{t $ 0 : B!(t) = a}. If # > 1/2, then

for any t > 0, a > 0, P!a(T!a > t) < Pa(T!

!a > t).

Pf: Note T0 = T 1/20 and T!

0d= T0 under Pa, a != 0. so

Pa(T!0 > t) = Pa(T0 > t) = P!a(T0 > t) = P!a(T!

0 > t).Strong Markov property of SBM

Pf: Note T0 = T 1/20 and T!

0d= T0 under Pa, a != 0. so

Pa(T!0 > t) = Pa(T0 > t) = P!a(T0 > t) = P!a(T!

0 > t).

Strong Markov property of SBM

P!a(T!a > t)) =

! t

0P0(T

!y > t " s)P!a(T0 # ds)

Pa(T!!a > t)) =

! t

0P0(T

1!!y > t " s)P!y (T0 # ds)

B!(t) =""

m=1

1Jm(t)[21[0,!)(Um)" 1]|B(t)|

Um are iid uniformly distributed on [0, 1].

Pf: Note T0 = T 1/20 and T!

0d= T0 under Pa, a != 0. so

Pa(T!0 > t) = Pa(T0 > t) = P!a(T0 > t) = P!a(T!

0 > t).

Strong Markov property of SBM

P!a(T!a > t)) =

! t

0P0(T

!y > t " s)P!a(T0 # ds)

Pa(T!!a > t)) =

! t

0P0(T

1!!y > t " s)P!y (T0 # ds)

B!(t) =""

m=1

1Jm(t)[21[0,!)(Um)" 1]|B(t)|

Um are iid unif dist on [0, 1].

Pf: Note T0 = T 1/20 and T!

0d= T0 under Pa, a != 0. so

Pa(T!0 > t) = Pa(T0 > t) = P!a(T0 > t) = P!a(T!

0 > t).

Strong Markov property of SBM

P!a(T!a > t)) =

! t

0P0(T

!a > t " s)P!a(T0 # ds)

Pa(T!!a > t)) =

! t

0P0(T

1!!a > t " s)P!a(T0 # ds)

B!(t) =""

m=1

1Jm(t)[21[0,!)(Um)" 1]|B(t)|

Um are iid unif dist on [0, 1].

Pf: Note T0 = T 1/20 and T!

0d= T0 under Pa, a != 0. so

Pa(T!0 > t) = Pa(T0 > t) = P!a(T0 > t) = P!a(T!

0 > t).

Strong Markov property of SBM

P!a(T!a > t)) =

! t

0P0(T

!a > t " s)P!a(T0 # ds)

Pa(T!!a > t)) =

! t

0P0(T

1!!a > t " s)P!a(T0 # ds)

B!(t) =""

m=1

1Jm(t)[21[0,!)(Um)" 1]|B(t)|

Um are iid unif dist on [0, 1].

Pf: Note T0 = T 1/20 and T!

0d= T0 under Pa, a != 0. so

Pa(T!0 > t) = Pa(T0 > t) = P!a(T0 > t) = P!a(T!

0 > t).

Strong Markov property of SBM

P!a(T!a > t)) =

! t

0P0(T

!a > t " s)P!a(T0 # ds)

Pa(T!!a > t)) =

! t

0P0(T

1!!a > t " s)P!a(T0 # ds)

B!(t) =""

m=1

1Jm(t)[21[0,!)(Um)" 1]|B(t)|

Um are iid unif dist on [0, 1].Note if 1 > ! > 1/2,

[T!a > t] $ [T 1!!

a > t]

Strong Markov Property of Skew BM

Natural Coupling:

P0(T!

a> t ! s) "

1 ! !

!P0(T

1!!

a> t ! s) < P0(T

1!!

a> t ! s)

[T 1!!

a! t] " [T!

a! t]

Proof of Ordering

Proposition: Assume!

D! <!

D+, !" =!

D+/(!

D+ +!

D!)Let Y = s(B!!), T "

y = inf{t " 0 : Yt = y}. Then for eacht > 0, y > 0,

P!y (T "y > t) < Py (T "

!y > t)

Scaling properties B!ct

d=!

cB!t

T "0 =P1!dist 1

D+T0 T "

0 =P"1!dist 1

D!T0

T "1 =P0!dist 1

D+T!!

1 T "!1 =P0!dist 1

D!T (1!!!)1

Proposition: Assume!

D! <!

D+, !" =!

D+/(!

D+ +!

D!)Let Y = s(B!!), T "

y = inf{t " 0 : Yt = y}. Then for eacht > 0, y > 0,

P!y (T "y > t) < Py (T "

!y > t)

Scaling properties B!ct

d=!

cB!t

T "0 =P1!dist 1

D+T0 T "

0 =P"1!dist 1

D!T0

T "1 =P0!dist 1

D+T!!

1 T "!1 =P0!dist 1

D!T (1!!!)1

Proposition: Assume!

D! <!

D+, !" =!

D+/(!

D+ +!

D!)Let Y = s(B!!), T "

y = inf{t " 0 : Yt = y}. Then for eacht > 0, y > 0,

P!y (T "y > t) < Py (T "

!y > t)

Scaling properties B!ct

d=!

cB!t

T "0 =P1!dist 1

D+T0 T "

0 =P"1!dist 1

D!T0

T "1 =P0!dist 1

D+T!!

1 T "!1 =P0!dist 1

D!T (1!!!)1

P!1(T"1 > t) =

! t

0P0(

1

D+T!!

1 > t ! s)P0(1

D!T1 " ds)

P1(T"!1 > t) =

! t

0P0(

1

D+T1 > t ! s)P0(

1

D!T (1!!!)1 " ds)

P!1(T"1 > t) =

! t

0P0(

1

D+T!!

1 > t ! s)P0(1

D!T1 " ds)

P1(T"!1 > t) =

! t

0P0(

1

D+T1 > t ! s)P0(

1

D!T (1!!!)1 " ds)

P!1(T"1 > t) =

! t

0P0(

1

D+T!!

1 > t ! s)P0(1

D!T1 " ds)

P1(T"!1 > t) =

! t

0P0(

1

D+T1 > t ! s)P0(

1

D!T (1!!!)1 " ds)

S<

P!1(T"1 > t) =

! t

0P0(

1

D+T!!

1 > t ! s)P0(1

D!T1 " ds)

P1(T"!1 > t) =

! t

0P0(

1

D+T1 > t ! s)P0(

1

D!T (1!!!)1 " ds)

S<

!! + "1[b,!)(x)

"u(x) =

1

2

d2u

dx2+ 1[b,!)

x ! R " {0, b}

#u"(0+)" (1" #) u"(0#) = $u(0)

{B(!)t > b}

P0(B(!)t > b, L0

t (B(!)) ! dl , !!

+(t) ! d%) =

(1" #)l

2&(t " %)3/2%1/2exp

#"((1" #)l)2

2(t " %)" (b + #l)2

2%

$dld%

P0(B(!)t ! db, L0

t (B(!)) ! dl) =

2#(l + b)#2&t3

exp

#"(l + b)2

2t

$dbdl

Flux Averaged Breakthrough Curves

0 50 100 150 200 250 300 350 400 450 5000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Delta initial injection at !20 for 5 time units and observe at 20

Time

Avera

ge F

lux

fine to corse

corse to finev = 0.447

v = 0.179

v = 0.134

Time to peak

v = 0.447 b 87 r 89v = 0.179 b 213 r 218v = 0.134 b 284 r 291

!

(D

2

!c

!t! vc)

v

! t

0

U2

2D(X (!!)s )

ds = tU2

2D!+ µ

! t

01

[X (!!)s ]

ds" #$ %

!!!+ (t)

c(x , t) = exp

&! U2

2D!t +

U

D(x)x

'"

Ex

(exp

&! U

D(X!!)X!!

t

'c0(X

!!t ) exp

)!µ!!!

+ (t)*+

A

U

&Uc(b, t) + D(b)

!c

!x

'= Q(t)

Alternative: Resident Concentration Curve

KEY:

PROBLEM: First passage time distribution for skew Brownian motion

SOLUTION:

Appuhamillage, T., D. Sheldon (2010): ArXiv1008.2989

Skew Brownian Motion with Drift - OPEN

•T. Appuhamillage, V. Bokil, E. Thomann, E. Waymire, B. Wood, Ann. of Appld. Probab., 2010 (to appear).•T. Appuhamillage et al - Water Resour. Res., 2009.• T. Appuhamillage, D Sheldon - Math ArXiv, 2010.• J. Ramirez, R. Haggerty, E. Thomann, E. Waymire, B Wood, SIAM Jour. Multiscale Mod., 2006.• J. Ramirez, et al, Water Resour. Res. 2008.• J. Ramirez, Proc. of the AMS, 2010 (to appear).

REFERENCES