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Wiener Sausage and Sensor Networks

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Wiener Sausage and Sensor Networks Wiener sausage and sensor networks Evgeny Spodarev Joint research with R. Cerˇ ny, D. Meschenmoser, S. Funken, J. Rataj and V. Schmidt Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.1 Overview Sensor networks Wiener sausage Mean volume and surface area Almost sure approximation and convergence of curvature measures Boolean model of Wiener sausages Capacity functional and volume fraction Contact distribution function and covariance function Specific surface area Outlook Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.2 Motivation: sensor networks Finding an object in a medium using a totally random strategy (G. Kesidis et al. (2003), S. Shakkottai (2004)): a set of sensors with reach r > 0 move randomly in the medium. Target detection area of a sensor = Wiener sausage Sr Paths of two sensors with reach r = 20 and their target detection areas Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.3 Motivation: sensor networks Sensor network: Sensors are located “at random” in space moving according to the Brownian motion. Target detection area of such network: the Boolean model of Wiener sausages E Target detection probability: p = 1 e−λ jSrj, where λ is the − intensity of the Poisson process of initial positions of sensors and E S is the mean volume of the Wiener sausage j rj Other specific intrinsic volumes? = find other mean intrinsic ) volumes of Sr E.g., the mean surface area E d−1(@S ) H r Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.4 Polyconvex sets and sets of positive reach Polyconvex sets: K = n K , where K are convex bodies. [i=1 i i Sets of positive reach: For a closed subset A Rd, define ⊆ reach A = sup r 0 : x Rd; ∆ (x) < r = card Σ (x) = 1 ; f ≥ 8 2 A ) A g where ∆ (x) is the distance from x Rd to A and A 2 Σ (x) = a A : x a = ∆ (x) A f 2 j − j A g is the set of all metric projections of x on A. A is said to be of positive reach if reach A > 0: Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.5 Polyconvex sets and sets of positive reach K (x) A r A x r (x) K reach A x A A polyconvex set A set of positive reach The parallel set of A, r > 0: A = A B (o) = x Rd : ∆ (x) 6 r r ⊕ r f 2 A g Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.6 Intrinsic volumes Steiner's formula (H. Federer (1959)): If A is a set of positive reach then d A = ! riV (A); 0 r < reach A; j rj i d−i ≤ i=0 X where V ( ) = is the Lebesgue measure in Rd (volume), d · j · j !i is the volume of the unit i-ball Vi is the i-th intrinsic volume of A, i = 0; : : : ; d. In particular, V0 is the Euler-Poincaré characteristic and Vd−1 is one half of the surface area. Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.7 Curvature measures Local version of Steiner's formula: For any Borel subset F Rd ⊂ d (A A) ξ−1(F ) = ! riC (A; F ); 0 r < reach A; j r n \ A j i d−i ≤ i=1 X where ξA(x) is the nearest point of A to x the signed Radon measure C (A; ) concentrated on @A is i · the ith curvature measure of A, 0 i d 1 ≤ ≤ − If @A is compact then C (A; Rd) = V (A), i = 0; : : : ; d 1. i i − Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.8 Brownian motion Wiener process W (t) : t 0 : a random process with f ≥ g continuous paths defined on (Ω; F; P ) such that W (0) = x R a.s., 2 W has independent increments, W (t) W (s) N(0; σ2(t s)), 0 s < t. − ∼ − ≤ d Brownian motion in R initiated at x = (x1; : : : ; xd): X(t) = (W (t); : : : ; W (t)) ; t 0; 1 d ≥ where W1; : : : ; Wd are independent Wiener processes starting at x ; : : : ; x R. 1 d 2 Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.9 Wiener sausage Let S(T ) = X(t) : t [0; T ] be the path of X up to time T > 0. f 2 g Wiener sausage Sr of radius r > 0: S = S(T ) B (o) = x Rd : ∆ (x) 6 r : r ⊕ r f 2 S(T ) g r = 10 r = 40 A realization of Sr Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.10 Intrinsic volumes of the Wiener sausage Intrinsic volumes V0(Sr); : : : ; Vd(Sr) are well–defined a.s. for d 6 3, r > 0; Vd(Sr) = VS(r) 2V (S ) = d−1(@S ) d−1 r H r V (S ) = ( 1)d−i−1V Rd S , i = 0; : : : ; d 2, where @S is i r − i n r − r a Lipschitz manifold with reach Rd S > 0 a.s. n r Compute E Vi(Sr), i = 0; : : : ; d. It is proved that E V (S ) < , i = d; d 1 for all d > 2 and i r 1 − E V (S ) < for d = 2 (RSS 09, RSM 09). 0 r 1 Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.11 Mean volume of the Wiener sausage Explicit formulae d = 2: A. Kolmogoroff and M. Leontowitsch (1933) d = 3: F. Spitzer (1964) d > 4: A. Berezhkovskii et al. (1989) Asymptotics of the volume R. Getoor (1965) M. Donsker und S. Varadhan (1975) J.-F. Le Gall (1988): CLT for shrinking Wiener sausage (T or r 0) ! 1 ! M. van den Berg and E. Bolthausen (1994) : : : Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.12 Other mean curvature measures Mean surface area: RSS (2009) Support measures and mean curvature functions: G. Last (2006) Other mean intrinsic volumes E V (S ), i = 0; : : : ; d 2 : an i r − open problem. Approximations can be obtained numerically (RSM (2009)) Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.13 Mean volume of the Wiener sausage A. Berezhkovskii, Yu. Makhnovskii, R. Suris (1989) : for d > 2 d(d 2) E S = ! rd + − ! σ2rd−2T j rj d 2 d 1 σ2y2T d − 2 4d !d r 1 e 2r + 2 3 2− 2 dy ; π y (Jν (y) + Yν (y)) Z0 where Jν(y) and Yν(y) are Bessel functions of the first and second kind of order ν = (d 2)=2 and ! = πd=2=Γ(1 + d=2) is the volume − d of the unit d-ball. Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.14 Mean volume of the Wiener sausage Bessel functions Jν(y) and Yν(y) are linearly independent solutions of the Bessel diff. equation y2f 00(y) + yf 0(y) + (y2 ν2)f(y) = 0, and − 1 ( 1)k(y=2)2k+ν J (y) = − ; ν k!Γ(ν + k + 1) Xk=0 J (y) cos(νπ) J (y) Y (y) = ν − −ν : ν sin(νπ) In three dimensions, the above formula for the mean volume of Sr simplifies to 4 E S = πr3 + 4σr2p2πT + 2πσ2rT : j rj 3 Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.15 Mean surface area of the Wiener sausage d Theorem 1. Let Sr be the Wiener sausage in R , d > 2. Then, it holds 2 2 1 − σ y T 2 d−1 2 E d−1 d−1 4d !d r 1−e 2r (@Sr) = d!dr + 2 3 2 2 dy π y (Jν (y)+Yν (y)) H 0 2 2 1 R − σ y T 2 d−3 (d−2)2 4 e 2r2 + d!dσ r T 2 2 2 dy 2 π y(Jν (y)+Yν (y)) − 0 ! R for almost all radii r > 0. For d = 2; 3, this formula holds for all r > 0. In the case d = 3, it simplifies to E 2(@S ) = 4πr2 + 8rσp2πT + 2πσ2T : H r Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.16 Mean surface area of the Wiener sausage Asymptotic behaviour (RSS 09) πσ2T r−1 log−2 r if d = 2; d−1 2 E (@Sr) 8 2πσ T if d = 3; as r 0: H ∼ > 2 ! <> 2 (d−2) d−3 > d!dσ T 2 r if d 4 > :> Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.17 Idea of the proof of Theorem 1 Theorem 2. For d 3 and any r > 0, it holds ≤ d E V (r) E d−1(@S ) = S ; H r d r where V (r) = S . For dimensions d > 4, this relation holds for almost all S j rj r > 0. follows from the dominated convergence theorem and Lemma 1. It holds d−1(@S ) a:s:= V 0 (r) H r S for all r > 0 if d = 2; 3 and almost all r > 0 if d > 4. Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.18 Idea of the proof of Theorem 1 A r A critical points r - C(A) The point x Rd is called critical for A if 2 x conv a A : x a = d(A; x) : 2 f 2 j − j g The radius r > 0 is critical for A if a critical point x for A with 9 d(A; x) = r. Let C(A) be the set of critical dilation radii r for A. Evgeny Spodarev, St. Petersburg, 2.04.2010 – p.19 Idea of the proof of Theorem 1 Lemma 1 follows from Theorem 3. Let A Rd be any compact set. If r (0; ) C(A) then ⊂ 2 1 n V 0 (r) exists and equals d−1(@A ). Here A = x Rd : d(A; x) 6 r is A H r r f 2 g the parallel neighborhood of A of radius r > 0 and C(A) is the set of critical dilation radii for A.
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