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. 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