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First passage time statistics for some stochastic processes with superimposed shot noise
Physica 149A (1988) 395-405 North-Holland, Amsterdam
FIRST PASSAGE TIME STATISTICS FOR SOME STOCHASTIC PROCESSES WITH SUPERIMPOSED SHOT NOISE Jaume MASOLIVER* Department of Chemistry (B-014) and Institute for Non-linear Science (R-W2), University of California, San Diego, La Jolla, CA 92093, USA
George H. WEISS Division of Computer Research and Technology, National Institutes of Health, Bethesda, MD 20892, USA
Received 22 October 1987
We present a method for finding statistical properties of the first passage time to exit an interval of general diffusion processes subject to random delta function impulses. Exact solutions are found for the mean first passage time for Brownian motion. Other special cases, detailed in the text, can also be solved in some generality.
1. Introduction While there are several general techniques available for the solution of first passage time problems for Markov processes’.2), there is not a single approach to the solution of these problems for non-Markovian problems. The analysis of these problems is usually tackled on an ad hoc basis not readily extended to any larger category. Recently, a number of investigations has been reported which allows one to find the solution of first passage time problems involving dichotomous3) and shot noise4) by enumerating all possible trajectories of the dynamical system. It was later shown that the solution of a number of these problems could be analyzed in terms of the solution to an evolution equation in a uniform way, which effectively eliminated combinatoric problems that arise in the method of trajectory enumeration5,6). The class of problems studied in the cited references are those in which the evolution of the system is deterministic in the absence of the noise. In the present note we discuss the first * Address after 1 September 1987: Department Diagonal 647, 08028 Barcelona, Spain.
de Fisica Fonamental, Universitat de Barcelona,
0378-4371/88/ $03.50 @ Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)
J. MASOLIVER
396
AND
G.H.
WEISS
passage time problem for diffusion processes subject to superimposed shot noise. Similar problems have been studied in connection with the growth of populations subject to catastrophic changes’).
2. General formulation The class of problems considered physical systems in one dimension expressed as i(t) = f(x) + n(t) 3 where f(x) is a deterministic process:
in refs. 4 and 6 concerns the evolution of in which the equation of motion can be
(1) function and n(t) is a delta function shot noise
(2) where the ‘y,and Ai = t; - t,_, (i = 1,2, . . .) are independent, identically distributed random variables and 8(t) is a delta function. In our analysis the {ti} are assumed to constitute a regenerative process*) and the {A,} a renewal process. We will consider the problem of finding statistical properties of the time at which a diffusing particle first leaves a one-dimensional interval Z = (a, b) when eq. (1) is replaced by i(t) = r(t) + n(t) )
(3)
where r(t) is some random process other than shot noise. A quite general formulation is possible at this point; later we will specialize to the case of zero-mean Brownian motion in which r(t) dt = dW(t) where IV(t) is a Wiener process. The novelty in our formulation is that there are now two competing ways in which the diffusing particle can leave I; either by random drift, or because an impulse pushes it across one of the exit boundaries. Let us list the assumptions specific to our problem. 1) The probability density of the shot noise amplitudes will be denoted by h(y). 2) The probability density for the interarrival time between two impulses will be denoted by @(A) and the cumulative probability that the interarrival time exceeds A will be denoted by ??(A) so that a
P(A) = 1 $(u) du . A
(4)
FIRST PASSAGE TIME FOR STOCHASTIC PROCESSES
397
3) The evolution of the process in the absence of impulses will be described by a probability density p(x, t\ x,,, to). If the (random) state of the system is denoted by X(t) then the definition of p(x, t) x0, to) is ~(x,t(x,,,t~)=Prob{x
Ax, 4 x0, to)= w
- x
(5)
of the shot noise problem corresponds
(6)
to>) 7
where X(t lx,,, to) is the solution to the deterministic
to
equation
x =f(X)
(7)
which satisfies the initial condition X(t,) = x0. In order to describe statistical properties of the impulses we follow ref. 6 and introduce two probability densities to describe the state of the system, one being valid at any time at which an impulse does not occur and the second just after the occurrence of an impulse. Specifically we define the probability densities 17(x t) and V(x, t) by U(x, t) dx = Prob{x < X(t) < x + dx 1no impulse at t} , V(x, t) dx = Prob{x < lim X(t) < x + dx 1impulse at t} . In addition describes the assume, for impulse, and
(8)
to these we will need one further probability density which evolution of the system at any time before the first impulse. We simplicity, that t = 0 does not coincide with the time of this that X(0) = x,,. Then we can write, explicitly,
WY 0 =p(x, tlx,,O>W *
(9)
The function of principal interest to us is S(t I x,,), the probability that no absorption has taken place before time t conditional on x0 being the initial value. This function can be expressed in terms of V(x, t), +(t) and p(x, t 1x,,, 0) as
Wlxo) =
Wjix,tbo,Wx n b
b
t
(10)
J. MASOLIVER
398
The
first term
impulse
gives
the contribution
and the second
occurrence
AND G.H. WEISS
of at least
represents one
from
the time
contributions
impulse.
Eq.
function
V(x, t) in order
to calculate
equation
for this function
can be found
(10)
the
interval
before
from all times shows
survival
by starting
that
the first
following
one
must
probability.
An
the
find the integral
from the pair of equations
for U and V: c
U(x, t) = U,(x, t) +
I
V(x, t) =
I 0
h
dr
I dy V(Y> T)P(~, tl y, r)+(t 0
- T),
(114
dy U(Y, t)h(x - Y>
(lib)
These can be combined into a single integral equation for V(x, t) by substituting eq. (llb) into (lla) which yields a single equation for U(x, t), and then substituting the result back into eq. (llb). In this way we find that
V(x, t) = I’, (x, t) +
d5 h(x - Y)V( i, T)P( Y, t 1l, ~)(cl(t - 7)
(12) in which
(13)
Eqs.
(10)
and
(12)
constitute
a reduction
of the original
first passage
time
problem to a mathematical problem that can be solved numerically in the most general cases. One can simplify the formalism in the particular case in which the random process
in the
probability
absence
densities
of impulses
is stationary
in time.
In this case
the
p(x, t 1y, T) satisfy
P(XY t I Y 7) = p(x, t 9
7
1y, 0) )
(14)
which allows us to introduce the Laplace transform with respect to t of eq. (12). We denote the Laplace transform of a function by that same function with a caret and a transform parameter s in place of t. Thus, for example,
v(x, s)
=
I
e-“V(x,
t) dt .
(15)
FIRST PASSAGE TIME FOR STOCHASTIC
399
PROCESSES
When eq. (14) is valid, one finds for the Laplace transform of eq. (12)
where the transformed
kernel &x, y; S) is m
b
Furthermore, the Laplace transform of the survival probability can be expressed in terms of m
I
8x,s 1y) = e-%x,t1y, O)?P(t)dt
(18)
0
as b
b
b
(19)
~(~Ixo)=IB(x,sl~~)dx+ldxl~(y,~)~(~,~Iy)dy.
n
a
a
When g(s I x0) is known one can find the mean first passage time by using the relation
Wo)) = two) f
(20)
Higher moments of the first passage time are calculable in terms derivatives of i(s I x0). When the transformed kernel has a particular structure one can (16) in closed form. One case in which this is possible is that I@, y; S) is separable, i.e., the simplest version of this assumption qx,
y;
s) = 4x;
s)P(r;
(21) into eq. (16) one finds b
b s) = q
s) +
&x> a
solve eq. in which is
s>.
When this expression is substituted
P(x;
of higher
Y; &,(Y;
4 dy
IA I 1-
&x,x;
a
s)dx
1,
(22)
J. MASOLIVER
400
AND
G.H.
WEISS
which can then be substituted into eq. (19) to give the desired expression for L?((s(x0). This analysis is readily generalized to deal with more general separable kernels of the form
Rx, y; s) = c “k(x; s)P!Ay; s) >
(23)
k
provided that the number of terms is finite.
3. Brownian
motion subject to random
impulses
As an example of the above formalism for which some results can be found in closed form, we analyze the case of Brownian motion together with superimposed non-negative impulses for which the probability density of the ‘y, is negative exponential. The interval Z in the present example is chosen to be (-co, 0). In the absence of shot noise it is known that the mean first passage time to exit Z starting from any y
h(x) = y epY”B(x) ,
(24)
where e(x) is the Heaviside step function, 13(x)= 0, x < 0, /3(x) = 1, x > 0. The function p(x, t 1y, 0) in the presence of an absorbing barrier at x = 0 is given by
z4-G4 Y, 0) =
v4;Dr(e-(~-~)294D~
_
em(x+y)2/4Dr,
(25)
Since we are dealing with Brownian motion we can use eq. (16) to find c((x; s). The kernel appearing in that equation is found to be
zqx,y;
s) =
T&c, + ecr -
_ y)p-.-(e”‘Y-~)_ ey(Y-q x)
e4-Y)
-Y+fl where the parameter CT =
((A +
l y+ff
e @(x-Y)
I>
+
-2-_ ..(Y-q (26)
CTis defined by
s)/o)“2 .
(27)
401
FIRST PASSAGE TIME FOR STOCHASTIC PROCESSES
After substituting this expression p((x; S) satisfies the equation
into eq. (16) one finds that the function x
-“(X-Y)e( y; S) dy
-y(X-y)c( y; S) dy 0
0
e”(X-Y)~( y;s) 1.
(28)
dy
+
This equation is shown, in the appendix, differential equation d3 d2 dn3fYz-o
to be equivalent
to the third order
2 d z+Y
(29)
which is to be solved subject to the boundary conditions at x = -00, Q-w;
s) =
dp(x; s) = d2% s) =0 dx x=-cc &,-Cc dx2
(30)
We will not discuss the solution of eq. (29) in any generality but rather confine ourselves to the simpler problem of calculating the mean first passage time to reach the boundary at x = 0. On setting s = 0 in eq. (29) we find that the function dp(x; 0) ldx satisfies the second order equation 2 A
-$+Y&-D
I
de(x;O) dx
together with the boundary
hy = - z 6(x - x0)
(31)
conditions
dti(x; 0) =o d2%; 0) x=-m * dx X=--m= dX2
(32)
Once d?(x; 0) Idx is known one finds c(x; 0) by integration,
qx;
0)
=
& _
e:(s-Y)x
0(x 2r)
Xo)Y
[h+y)e
t(v-Y)(x-x,)
+
(71
_
y)
e-f(s+Ywxo)
-
24
(33)
402
J. MASOLIVER
in which
n is the combination
AND G.H. WEISS
of parameters
77= (y2 + 4AID)‘l’ and c is a constant
(34)
which is determined
We now have all of the functions
by introducing
required
to evaluate
eq. (33) into eq. (28). (t(x,,)),
which is found
to be
;
(1
_
e”xcl)
+
27’
Y(;;4@)_
v+y A(q - A) (T + ~ C 2
+ 1, Da-
(e”L” - yx,,)
J
(35) This average
time increases
roughly
exponentially
The formalism for extending the preceding sions for $(t) and h(y) is quite complicated. analysis suggest a number of further questions
with xg as x0 -+ -m. result to more general expresThe results of the preceding on models related to the one
analyzed here. For example, does the change in ( t,,(.xo)) from an infinite to a finite value depend on whether h(-y) allows for all possible values of y or can it occur when the largest possible impulse is finite? We conjecture that the former is the case, but have not been able either to prove or disprove it. A second, equally difficult, problem is to determine how the transition from an infinite to a finite value of (t(x,)) depends on the properties of Q(t). For example, we conjecture that whenever $(t) is such that x
I
0
t+(t) dt = ~2
(36)
FIRST PASSAGE TIME FOR STOCHASTIC
PROCESSES
403
that (t(x,)) will be infinite. This also seems quite hard to prove. Finally, we mention that the methodology developed here does not depend on dimension, but in practice one would encounter considerable difficulties in applying it to problems in higher dimensions. The reason why the development of an analytic theory is at all possible is because of the simple form of the shot noise in which the impulses are taken to be delta functions. Any relaxation of this assumption leads to apparently intractable mathematical problems.
Acknowledgement The work of one of us (JM) was supported in part by DOE grant DE-FG0386ER13060 and by the U.S. Defense Advance Research Projects Agency through the University Projects Initiative under Contract N00014-86-K-0758 administered by the U.S. Office of Naval Research.
Appendix Derivation of eqs. (29) and (30) The first and second derivatives of eq. (28) with respect to n are
$
[c(x; s) - VI@;s)] =
& [&
1
e-“(x-y)~(y;
s) dy
-m +
2oy 2 Y -(+
--
ff
X e -y(X-y)~( y; s) dy + -& i --m 0
?+a__ I
e v(X-y)ti( y; s) dy
/e”(x-y)~(y; X
1
s) dy
(A.11
and
$
[9(x; s) - Qx;
s)] =
cT2[V(x; s) - Qx;
s)]
-y(x-y)p( y; s) dy .
64.2)
J. MASOLIVER
404
AND G.H. WEISS
Finally, taking the third derivative with respect to n yields the identity d2 d3 dX3+Y&?-(T (A.3)
The detailed expression for pi@; s) is found to be {@ _ X,,[ &
9,(X; s) = & + -&
(eC(Wx) _ ev(X”-G) + L&
[e(x, - x) eu(x-xo) - e-ix+%)l} .
eYC’“-‘,]
(A.4)
On introducing eq. (A.4) into (A.3) we are led to the result shown in eq. (29). Next we consider the boundary conditions. If we take into account that x lim
x+-cc
I
err('-Y)
P( y; s) dy = ; P(-m; s)
(A.5)
-cc
we find from eq. (28)
Q-w;
s) =
Q-y
s) +
wg + r> 2D02(rr + y) .
G4.6)
Since CY> 0 we have from eq. (A.4) that C1(-m; s) = 0
(A.3
and therefore q-w); s) = 0.
Starting from eqs. (A.l) and (A.2) and taking eqs. (A.5)-(A.8) we are finally led to eq. (30).
64.8)
into account
FIRST PASSAGE TIME FOR STOCHASTIC
PROCESSES
References 1) 2) 3) 4j 5) 6) 7) 8) 9)
G.H. Weiss, Adv. Chem. Phys. 13 (1967) 1 C.W. Gardiner, Handbook oi Stochastic Methods, 2nd ed. (Springer, New York, 1985). J. Masoliver. K. Lindenbere and B.J. West. Phvs. Rev. A 34 (1986) \ , 2351. J. Masoliver; Phys. Rev. A-35 (1987) 3918.’ ’ G.H. Weiss, J. Masoliver, K. Lindenberg and B.J. West, Phys. Rev. A 36 (1987) 1435. J. Masoliver and G.H. Weiss, J. Stat. Phys., in press. P.J. Brockwell, Adv. Appl. Prob. 17 (1985) 42. W.L. Smith, Proc. Roy. Stat. Sot. B20 (1958) 243. G.H. Weiss, R.J. Rubin, Adv. Chem. Phys. 52 (1983) 363.
405
FIRST PASSAGE TIME STATISTICS FOR SOME STOCHASTIC PROCESSES WITH SUPERIMPOSED SHOT NOISE Jaume MASOLIVER* Department of Chemistry (B-014) and Institute for Non-linear Science (R-W2), University of California, San Diego, La Jolla, CA 92093, USA
George H. WEISS Division of Computer Research and Technology, National Institutes of Health, Bethesda, MD 20892, USA
Received 22 October 1987
We present a method for finding statistical properties of the first passage time to exit an interval of general diffusion processes subject to random delta function impulses. Exact solutions are found for the mean first passage time for Brownian motion. Other special cases, detailed in the text, can also be solved in some generality.
1. Introduction While there are several general techniques available for the solution of first passage time problems for Markov processes’.2), there is not a single approach to the solution of these problems for non-Markovian problems. The analysis of these problems is usually tackled on an ad hoc basis not readily extended to any larger category. Recently, a number of investigations has been reported which allows one to find the solution of first passage time problems involving dichotomous3) and shot noise4) by enumerating all possible trajectories of the dynamical system. It was later shown that the solution of a number of these problems could be analyzed in terms of the solution to an evolution equation in a uniform way, which effectively eliminated combinatoric problems that arise in the method of trajectory enumeration5,6). The class of problems studied in the cited references are those in which the evolution of the system is deterministic in the absence of the noise. In the present note we discuss the first * Address after 1 September 1987: Department Diagonal 647, 08028 Barcelona, Spain.
de Fisica Fonamental, Universitat de Barcelona,
0378-4371/88/ $03.50 @ Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)
J. MASOLIVER
396
AND
G.H.
WEISS
passage time problem for diffusion processes subject to superimposed shot noise. Similar problems have been studied in connection with the growth of populations subject to catastrophic changes’).
2. General formulation The class of problems considered physical systems in one dimension expressed as i(t) = f(x) + n(t) 3 where f(x) is a deterministic process:
in refs. 4 and 6 concerns the evolution of in which the equation of motion can be
(1) function and n(t) is a delta function shot noise
(2) where the ‘y,and Ai = t; - t,_, (i = 1,2, . . .) are independent, identically distributed random variables and 8(t) is a delta function. In our analysis the {ti} are assumed to constitute a regenerative process*) and the {A,} a renewal process. We will consider the problem of finding statistical properties of the time at which a diffusing particle first leaves a one-dimensional interval Z = (a, b) when eq. (1) is replaced by i(t) = r(t) + n(t) )
(3)
where r(t) is some random process other than shot noise. A quite general formulation is possible at this point; later we will specialize to the case of zero-mean Brownian motion in which r(t) dt = dW(t) where IV(t) is a Wiener process. The novelty in our formulation is that there are now two competing ways in which the diffusing particle can leave I; either by random drift, or because an impulse pushes it across one of the exit boundaries. Let us list the assumptions specific to our problem. 1) The probability density of the shot noise amplitudes will be denoted by h(y). 2) The probability density for the interarrival time between two impulses will be denoted by @(A) and the cumulative probability that the interarrival time exceeds A will be denoted by ??(A) so that a
P(A) = 1 $(u) du . A
(4)
FIRST PASSAGE TIME FOR STOCHASTIC PROCESSES
397
3) The evolution of the process in the absence of impulses will be described by a probability density p(x, t\ x,,, to). If the (random) state of the system is denoted by X(t) then the definition of p(x, t) x0, to) is ~(x,t(x,,,t~)=Prob{x
Ax, 4 x0, to)= w
- x
(5)
of the shot noise problem corresponds
(6)
to>) 7
where X(t lx,,, to) is the solution to the deterministic
to
equation
x =f(X)
(7)
which satisfies the initial condition X(t,) = x0. In order to describe statistical properties of the impulses we follow ref. 6 and introduce two probability densities to describe the state of the system, one being valid at any time at which an impulse does not occur and the second just after the occurrence of an impulse. Specifically we define the probability densities 17(x t) and V(x, t) by U(x, t) dx = Prob{x < X(t) < x + dx 1no impulse at t} , V(x, t) dx = Prob{x < lim X(t) < x + dx 1impulse at t} . In addition describes the assume, for impulse, and
(8)
to these we will need one further probability density which evolution of the system at any time before the first impulse. We simplicity, that t = 0 does not coincide with the time of this that X(0) = x,,. Then we can write, explicitly,
WY 0 =p(x, tlx,,O>W *
(9)
The function of principal interest to us is S(t I x,,), the probability that no absorption has taken place before time t conditional on x0 being the initial value. This function can be expressed in terms of V(x, t), +(t) and p(x, t 1x,,, 0) as
Wlxo) =
Wjix,tbo,Wx n b
b
t
(10)
J. MASOLIVER
398
The
first term
impulse
gives
the contribution
and the second
occurrence
AND G.H. WEISS
of at least
represents one
from
the time
contributions
impulse.
Eq.
function
V(x, t) in order
to calculate
equation
for this function
can be found
(10)
the
interval
before
from all times shows
survival
by starting
that
the first
following
one
must
probability.
An
the
find the integral
from the pair of equations
for U and V: c
U(x, t) = U,(x, t) +
I
V(x, t) =
I 0
h
dr
I dy V(Y> T)P(~, tl y, r)+(t 0
- T),
(114
dy U(Y, t)h(x - Y>
(lib)
These can be combined into a single integral equation for V(x, t) by substituting eq. (llb) into (lla) which yields a single equation for U(x, t), and then substituting the result back into eq. (llb). In this way we find that
V(x, t) = I’, (x, t) +
d5 h(x - Y)V( i, T)P( Y, t 1l, ~)(cl(t - 7)
(12) in which
(13)
Eqs.
(10)
and
(12)
constitute
a reduction
of the original
first passage
time
problem to a mathematical problem that can be solved numerically in the most general cases. One can simplify the formalism in the particular case in which the random process
in the
probability
absence
densities
of impulses
is stationary
in time.
In this case
the
p(x, t 1y, T) satisfy
P(XY t I Y 7) = p(x, t 9
7
1y, 0) )
(14)
which allows us to introduce the Laplace transform with respect to t of eq. (12). We denote the Laplace transform of a function by that same function with a caret and a transform parameter s in place of t. Thus, for example,
v(x, s)
=
I
e-“V(x,
t) dt .
(15)
FIRST PASSAGE TIME FOR STOCHASTIC
399
PROCESSES
When eq. (14) is valid, one finds for the Laplace transform of eq. (12)
where the transformed
kernel &x, y; S) is m
b
Furthermore, the Laplace transform of the survival probability can be expressed in terms of m
I
8x,s 1y) = e-%x,t1y, O)?P(t)dt
(18)
0
as b
b
b
(19)
~(~Ixo)=IB(x,sl~~)dx+ldxl~(y,~)~(~,~Iy)dy.
n
a
a
When g(s I x0) is known one can find the mean first passage time by using the relation
Wo)) = two) f
(20)
Higher moments of the first passage time are calculable in terms derivatives of i(s I x0). When the transformed kernel has a particular structure one can (16) in closed form. One case in which this is possible is that I@, y; S) is separable, i.e., the simplest version of this assumption qx,
y;
s) = 4x;
s)P(r;
(21) into eq. (16) one finds b
b s) = q
s) +
&x> a
solve eq. in which is
s>.
When this expression is substituted
P(x;
of higher
Y; &,(Y;
4 dy
IA I 1-
&x,x;
a
s)dx
1,
(22)
J. MASOLIVER
400
AND
G.H.
WEISS
which can then be substituted into eq. (19) to give the desired expression for L?((s(x0). This analysis is readily generalized to deal with more general separable kernels of the form
Rx, y; s) = c “k(x; s)P!Ay; s) >
(23)
k
provided that the number of terms is finite.
3. Brownian
motion subject to random
impulses
As an example of the above formalism for which some results can be found in closed form, we analyze the case of Brownian motion together with superimposed non-negative impulses for which the probability density of the ‘y, is negative exponential. The interval Z in the present example is chosen to be (-co, 0). In the absence of shot noise it is known that the mean first passage time to exit Z starting from any y
h(x) = y epY”B(x) ,
(24)
where e(x) is the Heaviside step function, 13(x)= 0, x < 0, /3(x) = 1, x > 0. The function p(x, t 1y, 0) in the presence of an absorbing barrier at x = 0 is given by
z4-G4 Y, 0) =
v4;Dr(e-(~-~)294D~
_
em(x+y)2/4Dr,
(25)
Since we are dealing with Brownian motion we can use eq. (16) to find c((x; s). The kernel appearing in that equation is found to be
zqx,y;
s) =
T&c, + ecr -
_ y)p-.-(e”‘Y-~)_ ey(Y-q x)
e4-Y)
-Y+fl where the parameter CT =
((A +
l y+ff
e @(x-Y)
I>
+
-2-_ ..(Y-q (26)
CTis defined by
s)/o)“2 .
(27)
401
FIRST PASSAGE TIME FOR STOCHASTIC PROCESSES
After substituting this expression p((x; S) satisfies the equation
into eq. (16) one finds that the function x
-“(X-Y)e( y; S) dy
-y(X-y)c( y; S) dy 0
0
e”(X-Y)~( y;s) 1.
(28)
dy
+
This equation is shown, in the appendix, differential equation d3 d2 dn3fYz-o
to be equivalent
to the third order
2 d z+Y
(29)
which is to be solved subject to the boundary conditions at x = -00, Q-w;
s) =
dp(x; s) = d2% s) =0 dx x=-cc &,-Cc dx2
(30)
We will not discuss the solution of eq. (29) in any generality but rather confine ourselves to the simpler problem of calculating the mean first passage time to reach the boundary at x = 0. On setting s = 0 in eq. (29) we find that the function dp(x; 0) ldx satisfies the second order equation 2 A
-$+Y&-D
I
de(x;O) dx
together with the boundary
hy = - z 6(x - x0)
(31)
conditions
dti(x; 0) =o d2%; 0) x=-m * dx X=--m= dX2
(32)
Once d?(x; 0) Idx is known one finds c(x; 0) by integration,
qx;
0)
=
& _
e:(s-Y)x
0(x 2r)
Xo)Y
[h+y)e
t(v-Y)(x-x,)
+
(71
_
y)
e-f(s+Ywxo)
-
24
(33)
402
J. MASOLIVER
in which
n is the combination
AND G.H. WEISS
of parameters
77= (y2 + 4AID)‘l’ and c is a constant
(34)
which is determined
We now have all of the functions
by introducing
required
to evaluate
eq. (33) into eq. (28). (t(x,,)),
which is found
to be
;
(1
_
e”xcl)
+
27’
Y(;;4@)_
v+y A(q - A) (T + ~ C 2
+ 1, Da-
(e”L” - yx,,)
J
(35) This average
time increases
roughly
exponentially
The formalism for extending the preceding sions for $(t) and h(y) is quite complicated. analysis suggest a number of further questions
with xg as x0 -+ -m. result to more general expresThe results of the preceding on models related to the one
analyzed here. For example, does the change in ( t,,(.xo)) from an infinite to a finite value depend on whether h(-y) allows for all possible values of y or can it occur when the largest possible impulse is finite? We conjecture that the former is the case, but have not been able either to prove or disprove it. A second, equally difficult, problem is to determine how the transition from an infinite to a finite value of (t(x,)) depends on the properties of Q(t). For example, we conjecture that whenever $(t) is such that x
I
0
t+(t) dt = ~2
(36)
FIRST PASSAGE TIME FOR STOCHASTIC
PROCESSES
403
that (t(x,)) will be infinite. This also seems quite hard to prove. Finally, we mention that the methodology developed here does not depend on dimension, but in practice one would encounter considerable difficulties in applying it to problems in higher dimensions. The reason why the development of an analytic theory is at all possible is because of the simple form of the shot noise in which the impulses are taken to be delta functions. Any relaxation of this assumption leads to apparently intractable mathematical problems.
Acknowledgement The work of one of us (JM) was supported in part by DOE grant DE-FG0386ER13060 and by the U.S. Defense Advance Research Projects Agency through the University Projects Initiative under Contract N00014-86-K-0758 administered by the U.S. Office of Naval Research.
Appendix Derivation of eqs. (29) and (30) The first and second derivatives of eq. (28) with respect to n are
$
[c(x; s) - VI@;s)] =
& [&
1
e-“(x-y)~(y;
s) dy
-m +
2oy 2 Y -(+
--
ff
X e -y(X-y)~( y; s) dy + -& i --m 0
?+a__ I
e v(X-y)ti( y; s) dy
/e”(x-y)~(y; X
1
s) dy
(A.11
and
$
[9(x; s) - Qx;
s)] =
cT2[V(x; s) - Qx;
s)]
-y(x-y)p( y; s) dy .
64.2)
J. MASOLIVER
404
AND G.H. WEISS
Finally, taking the third derivative with respect to n yields the identity d2 d3 dX3+Y&?-(T (A.3)
The detailed expression for pi@; s) is found to be {@ _ X,,[ &
9,(X; s) = & + -&
(eC(Wx) _ ev(X”-G) + L&
[e(x, - x) eu(x-xo) - e-ix+%)l} .
eYC’“-‘,]
(A.4)
On introducing eq. (A.4) into (A.3) we are led to the result shown in eq. (29). Next we consider the boundary conditions. If we take into account that x lim
x+-cc
I
err('-Y)
P( y; s) dy = ; P(-m; s)
(A.5)
-cc
we find from eq. (28)
Q-w;
s) =
Q-y
s) +
wg + r> 2D02(rr + y) .
G4.6)
Since CY> 0 we have from eq. (A.4) that C1(-m; s) = 0
(A.3
and therefore q-w); s) = 0.
Starting from eqs. (A.l) and (A.2) and taking eqs. (A.5)-(A.8) we are finally led to eq. (30).
64.8)
into account
FIRST PASSAGE TIME FOR STOCHASTIC
PROCESSES
References 1) 2) 3) 4j 5) 6) 7) 8) 9)
G.H. Weiss, Adv. Chem. Phys. 13 (1967) 1 C.W. Gardiner, Handbook oi Stochastic Methods, 2nd ed. (Springer, New York, 1985). J. Masoliver. K. Lindenbere and B.J. West. Phvs. Rev. A 34 (1986) \ , 2351. J. Masoliver; Phys. Rev. A-35 (1987) 3918.’ ’ G.H. Weiss, J. Masoliver, K. Lindenberg and B.J. West, Phys. Rev. A 36 (1987) 1435. J. Masoliver and G.H. Weiss, J. Stat. Phys., in press. P.J. Brockwell, Adv. Appl. Prob. 17 (1985) 42. W.L. Smith, Proc. Roy. Stat. Sot. B20 (1958) 243. G.H. Weiss, R.J. Rubin, Adv. Chem. Phys. 52 (1983) 363.
405