Skew Brownian motion with dry friction: The Pugachev Sveshnikov

UDC 519.217.4
Skew Brownian motion with dry friction: The
PugachevSveshnikov equation approach
∗
Sergey Berezin , Oleg Zayats
∗
∗
Department of Applied Mathematics,
Peter the Great St.Petersburg Polytechnic University,
Polytechnicheskaya str. 29, St.Petersburg, 195251, Russia
The Brownian motion with dry friction is one of the simplest but
very common stochastic processes, also known as the Brownian motion with
two valued drift, or the CaugheyDienes process. This process appears in many
applied elds, such as physics, mechanics, etc. as well as in mathematics itself.
In this paper we are concerned with a more general process, skew Brownian
motion with dry friction. We study the probability distribution of this process
and its occupation time on the positive half line. The PugachevSveshnikov
equation approach is used.
Abstract.
Keywords: dry friction, skew Brownian motion, piecewise linear diusion, integral functional, occupation time, local time, Pugachev equation, Pugachev
Sveshnikov equation, characteristic function, Riemann boundary value problem.
1.
Introduction
Brownian motion plays an important role in statistical physics and
other applied areas of science. The classical physical theory of Brownian
motion was developed by Einstein and Smoluchovwski [1] in the beginning
of 20th century. Their mathematical model is based on the assumption
that a Brownian particle is weightless. That leads to the formulation in
terms of the Wiener process. The major drawback of this model is that
the trajectories are only continuous but nowhere dierentiable, so velocity
cannot be dened. A rened theory that takes into account particles'
inertia was developed by Ornstein and Uhlenbeck [2] later on. That leads
to the OrnsteinUhlenbeck process, which trajectories now have the rst
continuous derivative.
In both of the theories a Brownian particle is driven by the random
force resulting from collision of molecules, and the viscous friction takes
place, that is, the force is proportional to the velocity of the particle.
Although, it is not always the case. A well-known example of that is the
dry (Coloumb) friction of macroscopic materials. For this type of friction
the resistive force is independent of the speed but depends on the direction
of motion.
Behavior of the mechanical systems with dry friction under random excitation was studied rst by Caughey and Dienes [3] in 60s. The Caughey
Dienes process is similar to the OrnsteinUhlenbeck process up to replacement of the viscous friction by the dry one, and its applications include control theory [4], seismic mechanics [5], communication systems theory [6],
radio physics [7], and nonlinear stochastic dynamics [8]. It should also
be mentioned that this process appears as well in pure mathematical papers [11, 12]. In 2000s, there was another wave of interest in studying
the CaugheyDienes process [9], that lasts till the present time. Some
additional publications on the subject can be found in author's work [10].
In the present paper we investigate so-called skew CaugheyDienes
process, or skew Brownian motion with dry friction, which generalizes
Brownian motion with dry friction in the same way the Wiener process
generalizes skew Brownian motion [13].
2.
Main section
For η ∈ (−1, 1) we dene skew Brownian motion with dry friction X(t)
as a unique strong solution [13] of the following equation
√
dX(t) = −2µ sign(X(t)) dt + ηdL0X (t) + 2dW (t), t > 0, X(0) = 0.
By W (t) we denote a standard Wiener process starting at zero, and L0X (t)
is the symmetric local time of the semimartingale X(t) at the level zero
Z t
1
0
LX (t) = lim
1(−ε,+ε) (X(s)) d[X]s ,
ε→+0 2ε 0
where [X]s = 2s is the quadratic variation of X(s). Further, we will be
interested in the positive half-line occupation time of X(t)
Z t
1(0,+∞) (X(s)) ds.
I(t) =
0
One can think of X(t) and I(t) as of the components of the vector
process (X(t), I(t)) governed by the system of SDEs
(
√
dX(t) = −2µ sign(X(t)) dt + ηdL0X (t) + 2dW (t),
(1)
dI(t) = 1(0,+∞) (X(s)) ds, X(0) = I(0) = 0.
In what follows, we derive explicit formulas for the probability density function of (1) following ideas from [14]. Usually for this purpose
one uses the FokkerPlanckKolmogorov equation or random walks approximation. We use an alternative approach based on the characteristic
function method that manifests in the PugachevSveshnikov singular integral dierential equation.
It can be shown [14] that the characteristic function E(z1 , z2 ; t) of the
process (X(t), I(t)) satises equation
∂E
+(z12 −iz2 /2)E+(2µz1 −z2 /2)Ê−2iηz1 Ψ0 = 0,
∂t
E(z1 , z2 ; 0) = 1, (2)
where we use the short notation Ê(z1 , z2 ; t) and Ψ0 (z2 , t):
+∞
Z
1
Ê = v.p.
π
−∞
E|z1 =s
ds,
s − z1
1
Ψ0 =
v.p.
2π
+∞
Z
E|z1 =s ds.
(3)
−∞
For Im ζ 6= 0 let us introduce the Cauchy-type integral Φ(ζ, z2 ; t) and
its limit values Φ± (z, z2 ; t) on the real axis from upper and lower halfplanes (with respect to the rst argument):
1
Φ(ζ, z2 ; t) =
2πi
+∞
Z
E|z1 =s
ds, Φ± (z, z2 ; t) = lim Φ(ζ, z2 ; t), Im z = 0.
ζ→z±i0
s−ζ
−∞
It is well known that Φ± is analytic when Im ζ 6= 0, and that Φ± satisfy
SokhotskiPlemelj formulas
Φ+ − Φ− = E, Φ+ + Φ− = −iÊ.
(4)
Clearly, one can rewrite (2) in terms of Φ± , that gives a Riemann boundary
value problem. Applying then Laplace transform with respect to t, and
denoting its argument by p, we get to the formula
(z12 +2µiz1 +p−iz2 )Φ̃+ −iηz1 Ψ̃0 −
1
1
= (z12 −2µiz1 +p)Φ̃− +iηz1 Ψ̃0 + . (5)
2
2
The Laplace transforms are labeled with the tildes above the functions.
Note that the left-hand side of (5) can be analytically continued for
all z1 ∈ C such that Im z1 > 0, also the right-hand side can be analytically continued for all z1 ∈ C such that Im z1 < 0. Since they match
when Im z1 = 0, they turn out to be elements of the same entire function
1
) when z → ±∞
of argument z1 ∈ C . Assuming that Φ± (z, z2 ; t) = O( |z|
for Im z ≷ 0, by generalized Liouville's theorem one can realize that this
entire function is actually linear: G0 (z2 , t) + z1 G1 (z2 , t). This leads to the
equality
G0 + z1 G1 ± iηz1 Ψ̃0 ± 1/2
Φ̃± = 2
.
(6)
z1 ± 2iµz1 + p − (1 ± 1)iz2 /2
p
±
Note that the denominator
in
(6)
has
zeros
iν
=
i(−µ±
µ2 + p − iz2 )
p
±
±
2
and iκ = i(µ ± µ + p) such that Im(iν ) ≷ 0 and Im(iκ ± ) ≷ 0. At
the same time, Φ̃± should be analytic in upper and lower half-planes,
therefore, the singularities iν + and iκ − should be removable. This gives
a system of linear equations to determine G0 and G1 . Also, taking into
account the denition of Ψ0 in (3) and performing integration in (6) one
can get that Ψ̃0 = −iG1 . After that the system of linear equations for G0
and G1 can be written in the following form
G0 + iν + (1 + η)G1 + 1/2 = 0, G0 + iκ − (1 − η)G1 − 1/2 = 0.
(7)
The nal expression for the Laplace transform of the characteristic function Ẽ then follows from the rst of the SokhotskiPlemelj formulas (4).
Now we start with the process X(t) itself. Let z2 = 0 in (6). After
substituting G0 and G1 from (7) into (6) and necessary simplications one
obtains the Laplace transform of the characteristic function of X(t)
1+η
1−η
1
−
ẼX (z, p) = Ẽ(z, 0; p) =
.
2iκ − z + iκ +
z − iκ +
Cumbersome but trivial in nature computations will immediately lead us
to the nal result, the probability density function,
α,
x > 0,
(|x|+2µt)2
|x| − 2µt
1
√
fX (x, t) = √ e− 4t
+ µe−2µ|x| Erfc
·
1 − α, x < 0,
πt
2 t
where Erfc (·) is the complementary error function, and α = (1 + η)/2.
When t → +∞, we get
∞
fX
(x) = 2µe−2µ|x| (α1(0,+∞) (x) + (1 − α)1(−∞,0] (x)).
Now, let z1 = 0 in (6). This leads to the expression for the Laplace
transform of the occupation time characteristic function
ẼI (z, p) = Ẽ(0, z; p) =
(1 + η)κ + − (1 − η)ν −
.
ν − κ + ((1 − η)κ − − (1 + η)ν + )
This corresponds to the following probability density function (0 < y < t):
2
4e−µ t
fI (y, t) = p
π y(t − y)
+∞ Z
+∞
Z
0
√
2
2
√
χ(2 ys1 , 2 t − ys2 ) s1 s2 e−s1 −s2 ds1 ds2 ,
0
η+3
1 − η −µ(s1 + η−3
1 + η −µ( η−1
s1 +s2 ) −
η+1 s2 ) χ+ (s , s ) +
χ (s1 , s2 ),
χ(s1 , s2 ) =
e
e
1 2
1+η
1−η
χ+ (s1 , s2 ) = 1(0,+∞) ((1 + η)s1 − (1 − η)s2 ), χ− (s1 , s2 ) = 1 − χ+ (s1 , s2 ).
3.
Conclusions
We derived explicit formulas for the probability density function of
the Brownian motion with dry friction and its occupation time on the
positive half-line, that generalizes known results for the regular Caughey
Dienes process. In fact, more general result was obtained for the Laplace
transform of the joined characteristic function. Essentially our approach is
based on the reduction to a Riemann boundary value problem, and clearly
it can be used to nd characteristics of more general SDEs with piecewise
linear coecients and local time.
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