On countably skewed Brownian motion

Transcription

1 On countably skewed Brownian motion Gerald Trutnau (Seoul National University) Joint work with Y. Ouknine (Cadi Ayyad) and F. Russo (ENSTA ParisTech) Electron. J. Probab. 20 (2015), no. 82, 1-27 [ORT 2015] G. Trutnau (SNU) CSBM 1 / 25

2 For given initial condition X 0 = x R, construct X t = x + W t + k Z(2α k 1)l z k t (X ), (t 0, a.s.) (W t ) t 0 standard BM in R starting from 0 (α k ) k Z (0, 1) (z k ) k Z unbounded sequence in R (may have an accumulation point) For z R, (l z t (X )) t 0 unique increasing continuous process, with: (1) l z 0 (X ) = 0 (2) t 0 1 {X s=z}dl z s (X ) = l z t (X ) (uniqueness up to mult. const.) G. Trutnau (SNU) CSBM 2 / 25

3 For given initial condition X 0 = x R, construct X t = x + W t + k Z(2α k 1)l z k t (X ), (t 0, a.s.) (W t ) t 0 standard BM in R starting from 0 (α k ) k Z (0, 1) (z k ) k Z unbounded sequence in R (may have an accumulation point) For z R, (l z t (X )) t 0 unique increasing continuous process, with: (1) l z 0 (X ) = 0 (2) t 0 1 {X s=z}dl z s (X ) = l z t (X ) (uniqueness up to mult. const.) G. Trutnau (SNU) CSBM 3 / 25

4 The multiplicative constant is usually fixed by means of the symmetric Tanaka formula, i.e. (l z t (X )) t 0 is given through t l z t (X ) = X t z x z sgn(x s z)dx s 0 = symmetric semimartingale local time of X at z (l z t (X )) t 0 analytically related to const δ z G. Trutnau (SNU) CSBM 4 / 25

5 Some remarkable properties: It may hold {k z k U 0 } 2α k 1 = U 0 =neighborhood of an accumulation point of (z k ) k Z, but X still semimartingale ([ORT 2015], see later). CSBM is not automatically non-explosive, e.g. z k = k l=1 ([ORT 2015], see later). 1 l, α k ε for k N 0 G. Trutnau (SNU) CSBM 5 / 25

6 Strong uniqueness of CSBM in special cases: ([Harrison/Shepp 1981]) α k = 1 2 k 0, α 0 = α (0, 1), z 0 = 0 α-skew BM. ([Le Gall 1984]) (z k ) k Z R arbitrary k Z (2α k 1)l z k t (X ) = R la t (X )µ(da) µ of bounded total variation k Z 2α k 1 < ([Takanobu 1986]) CSBM occurs first time as explicit equation, but in dimension one under Le Gall s condition and with z k < z k+1 : 0 < inf k Z (z k+1 z k ) < sup k Z (z k+1 z k ) < but also conditions for uniqueness in the multidimensional setting G. Trutnau (SNU) CSBM 6 / 25

7 Definitive (?) answer in [ORT 2015]: {k z k U 0 } 2α k 1 < U 0 =neighborhood of an accumulation point of (z k ) k Z + non-explosion condition (Feller s test) = strong uniqueness (and Le Gall) G. Trutnau (SNU) CSBM 7 / 25

8 Applications to advection-diffusion in layered media: ([Waymire/Ramirez et al. 2006, 2013]) (z k ) k Z no accumulation points: Ψ continuous, piecewise linear, Ψ(0) = 0: Z t := Ψ(X t ) (similar to CSBM but with dispersion coeff. σ = Ψ Ψ 1 ) Z t := σ(z t )db t + D(Z t )dt (σ, D (loc.) bounded) where B t is independent of W t. G. Trutnau (SNU) CSBM 8 / 25

9 p t (x, dy) = concentration of solute at time t, immersed at time t = 0 at x Want to determine asymptotic behavior of p t (x, dy) or (Z t, Z t ) as t need ergodic properties (recurrence, positive recurrence) of CSBM G. Trutnau (SNU) CSBM 9 / 25

10 Special case α-skew BM, α (0, 1) [Harrison/Shepp 1981] α k = 1 2 k 0, α 0 = α, z 0 = 0: Existence? Let with E(f, g) := 1 2 Ibp gives E(f, g) = X t = x + W t + (2α 1)l 0 t (X ) R f (x)g (x)ρ(x)dx, f, g C 0 (R) ρ(x) = β1 (,0) (x) + γ1 (0, ) (x), β, γ > 0. R 1 2 f (x)g(x)ρ(x)dx + R γ β f (x)g(x)δ 0 (dx). 2 G. Trutnau (SNU) CSBM 10 / 25

11 By general Dirichlet form Theory: associated process X and it solves X t = x + W t + γ β l 0 t 2 where l 0 t δ 0 by the Revuz correspondence. Comparing Fukushima s decomposition for X t with sym. Tanaka formula for X t one obtains l 0 t = 2 γ + β l0 t (X ), and so One gets X t = x + W t + γ β l 0 t (X ). γ + β }{{} =2 γ γ+β 1=2α 1 αβ = (1 α)γ. G. Trutnau (SNU) CSBM 11 / 25

12 Uniqueness? Let h(x) = βx1 (,0) (x) + γx1 (0, ) (x), β, γ > 0 (strictly increasing, difference of two convex functions) Apply sym. Tanaka formula Y t := h(x t ) = h(x) + = h(x) + t 0 t 0 + γ β l 0 t (X ) 2 = h(x) + t 0 h (X s )dx s R l a t (X )h (da) h h 1 (h(x s ))dw s + γ + β (2α 1)l 0 t (X ) 2 h } {{ h 1 }(Y s )dw s bdd above and away from zero if β = α and γ = 1 α. = existence of a strong solution and pathwise uniqueness for (Y t ) t 0 by [Le Gall 1986]. G. Trutnau (SNU) CSBM 12 / 25

13 Construction and semimartingale property of CSBM Let E(f, g) := 1 2 R f (x)g (x)ρ(x)dx, f, g C 0 (R), where ρ : R (0, ) locally integrable and determined by Then (E, C0 (R)) is closable in L2 (R, ρdx) and the closure (E, D(E)) is a regular symmetric Dirichlet form. By Fukushima (Nineteen Seventies) there exists a diffusion ((X t ) t 0, (P x ) x R ) up to lifetime ζ associated to (E, D(E)). G. Trutnau (SNU) CSBM 13 / 25

14 Assume: (1) ζ = + (2) Cap({x}) > 0 for all x R Then: Theorem (ORT 2015) For x R, ((X t ) t 0, P x ) is a semimartingale, iff k 0 γ k+1 γ k + k 0 γ k+1 γ k < In this case ( ) lim γ k =: γ [0, ) and lim γ k =: γ [0, ) k k and ((X t ) t 0, P x ) solves X t = x + W t + { γk+1 γ k 2 k Z l l k t + γ } k+1 γ k l r k 2 t + γ γ l 0 t 2 G. Trutnau (SNU) CSBM 14 / 25

15 Proof. Proof Ibp gives E(f, g) = g(x) 1 R 2 f (x)ρ(x)dx + g(x)f (x) ( γk+1 γ k δ lk (dx) + γ ) k+1 γ k δ rk (dx) R 2 2 k Z + g(x)f (x) lim k γ k lim k γ k δ 0 (dx) R 2 = g(x) 1 2 f (x)ρ(x)dx + g(x)dν f. R Revuz correspondence: pos. Radon measures pos.increasing cont. processes signed Radon measures cont. processes in BV loc ([0, )) ν id signed Radon ( ). R G. Trutnau (SNU) CSBM 15 / 25

16 Again, one can show l l k t = 2 γ k+1 + γ k l l k t (X ), l r k t = 2 γ k+1 + γ k l r k t (X ), l 0 t = 2 γ + γ l0 t (X ) and so ((X t ) t 0, P x ) solves X t = x + W t + k Z where ( γ k+1 γ k γ k+1 + γ }{{ k } 2α k 1 l l k t (X ) + γ k+1 γ k γ k+1 + γ k } {{ } =2α k 1 l r k t (X ) ) + γ γ γ + γ }{{} =2α 1 alpha k = γ k+1 γ k+1 + γ k, α k = γ k+1 γ k+1 + γ k, k Z, α = γ γ + γ l 0 t (X ) G. Trutnau (SNU) CSBM 16 / 25

17 Example: Let ρ be the lower Riemann step function of x δ 1 (= density of reversible measure for Bessel processes of dimension δ) according to the partition l k = r k = 1 k in a neighborhood of zero. Then (1) For δ (0, 1), ((X t ) t 0, P x ) is not a semimartingale (2) For δ (1, 2), ((X t ) t 0, P x ) is a semimartingale, but 2α k 1 + 2α k 1 = γ k+1 γ k γ k+1 + γ k + γ k+1 γ k γ k 0 k 0 k 0 k 0 k+1 + γ k = G. Trutnau (SNU) CSBM 17 / 25

18 Strong uniqueness and conservativeness of CSBM Assume ((X t ) t 0, P x ) is a semimartingale. Find h : R R, h, continuous and piecewise linear, h(0) = 0, such that dh(x t ) = h h 1 (h(x t ))dw t For this, one needs that h eliminates local times, works with αγ γ k+1 on (l k, l k+1 ) h = (1 α)γ γ on (r k, r k+1 ) k+1 In order to apply the symmetric Tanaka fromula, one needs h difference of two convex functions h BV loc (R) 1 1 γ k γ k γ k 0 k 0 k γ < ( ) k+1 G. Trutnau (SNU) CSBM 18 / 25

19 In this case, we may call h a canonical scale and h strong uniqueness holds provided conservativeness holds. We assumed conservativeness but in case of canonical scale there is a sharp criterion, called Feller s test for non-explosion that is also applicable in our case, namely ± z (X t ) t 0 is conservative h 1 (z) 0 0 h dy dz = ± (y) r l+1 r l lim γ n γ k+1 (r k+1 r k ) = and 1 l n l+1 1 k l l m+1 l m lim n γ m+1 γ k+1 (l k+1 l k ) = ( ) n m 1 m k 1 G. Trutnau (SNU) CSBM 19 / 25

20 It also holds Theorem (ORT 2015) (X ) t 0 is conservative (i.e. ζ = + a.s.), iff there exist u n D(E), n 1, 0 u n 1 dx-a.e. as n such that lim E(u n, G 1 w) 0 n for some w L 2 (R, ρdx) L 1 (R, ρdx) such that w > 0 a.e. Example for explosion: Let and r k := k l=1 γ k+1 = C k (k + 1), k 1, 1 l, k 1 (C > 1 is some constant) (remaining r k, γ k+1, l k, γ k+1 arbirary, just as at beginning). Then α k > ε for k N 0 for some N 0 N. C G. Trutnau (SNU) CSBM 20 / 25

21 Theorem (ORT 2015) Let (α k ) k Z, (α k ) k Z (0, 1). Suppose and that ( ) holds for γ k = 1 j=k 2α k 1 + 2α k 1 < k 0 k 0 k 1 1 α j, k 1, γ α k = j j=0 α j 1 α j, k 1 and some partition (l k ) k Z, (r k ) k Z of R as described at the beginning. Then for any α (0, 1) there exists a unique strong solution to X t = X 0 +W t + ( ) (2α k 1)l l k t (X ) + (2α k 1)l r k t (X ) +(2α 1)l 0 t (X ) k Z G. Trutnau (SNU) CSBM 21 / 25

22 Recurrence and positive recurrence of CSBM Let (l k ) k Z, (r k ) k Z, (γ k ) k Z, (γ k ) k Z as at beginning and satisfy ( ) and ( ). Let D y := inf{t 0 X t = y}, y R. Then: Theorem (ORT 2015) The following are equivalent: (i) (X ) t 0 is recurrent, i.e. P x (D y < ) = 1 x, y R. (ii) h( ) = and h( ) =. (iii) k Z (iv) 0 l k+1 l k γ k+1 = and k Z 1 ρ(x) dx = and 0 r k+1 r k γ k+1 =. 1 ρ(x) dx =. (v) There exist u n D(E), n 1, 0 u n 1 dx-a.e. as n such that E(u n, u n ) 0 as n. G. Trutnau (SNU) CSBM 22 / 25

23 Remark The Dirichlet form (E, D(E)) is irreducible., then it is either recurrent or transient. Thus the preceding Theorem provides also sharp conditions about transience. Lemma Let (X ) t 0 be recurrent. Let (θ t ) t 0 be the shift operator of (X ) t 0. Then for any x, y R lim t sup P x θt 1 (A) P y θt 1 (A) = 0. A F G. Trutnau (SNU) CSBM 23 / 25

24 Suppose (X t ) t 0 is recurrent. Then (X t ) t 0 is called null-recurrent if lim p t1 K (x) = lim P x (X t K) = 0 t t for any x R and any compact set K with non-empty interior. Otherwise it is called positive recurrent. Theorem (ORT 2015) Suppose (X t ) t 0 is recurrent. Then the following are equivalent: (i) (X t ) t 0 is positive recurrent. (ii) 1 h (x) dx <. (iii) The invariant measure ρdx is finite, i.e. k Z {γ k+1(l k+1 l k ) + γ k+1 (r k+1 r k )} <. (iv) p t (x, dy) = P x (X t ) converges weakly to the invariant distribution ρdx as t for any x R. R ρ(x)dx (v) E x [D y ] < x, y R. G. Trutnau (SNU) CSBM 24 / 25

25 Corollary Assume (X t ) t 0 is positive recurrent. Then invariant distribution. R ρdx ρ(x)dx is the unique G. Trutnau (SNU) CSBM 25 / 25