Markov Chain Approximation of Pure Jump Processes. Nikola Sandrić (joint work with Ante Mimica and René L. Schilling) University of Zagreb

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1 Markov Chain Approximation of Pure Jump Processes Nikola Sandrić (joint work with Ante Mimica and René L. Schilling) University of Zagreb 8 th International Conference on Lévy processes Angers, July 25-29, 2016 Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

2 Outline 1 Problem 2 Related results 3 Semimartingale approach 4 Dirichlet form approach Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

3 Problem Let {X n } n N be a sequence of Markov chains on Z d n := n 1 Z d, and let X be a Markov process on R d. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

4 Problem Let {X n } n N be a sequence of Markov chains on Z d n := n 1 Z d, and let X be a Markov process on R d. The following two questions naturally arise: When does {X n } n N converge weakly to a Markov process? Can X be approximated (in the sense of weak convergence) by a sequence of Markov chains? Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

5 Related results Stroock-Varadhan, Multidimensional diffusion processes 1979: X is a diffusion process determined by a generator in non-divergence form. Stroock-Zheng, AIHP 1997: X is a symmetric diffusion process determined by a generator in divergence form. Bass-Kumagai, TAMS 2008: X is a symmetric diffusion process determined by a generator in divergence form. Deuschel-Kumagai, CPAM 2013: X is a non-symmetric diffusion process determined by a generator in divergence form. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

6 Related results Husseini-Kassmann, PA 2007: X is a symmetric pure jump process whose corresponding jump kernel is comparable to the jump kernel of a symmetric stable Lévy process. Bass-Kassmann-Kumagai, AIHP 2010: X is a symmetric pure jump process with stable-like kernel. Bass-Kumagai-Uemura, PTRF 2010: X is a symmetric process which admits continuous and jump part. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

7 Related results Husseini-Kassmann, PA 2007: X is a symmetric pure jump process whose corresponding jump kernel is comparable to the jump kernel of a symmetric stable Lévy process. Bass-Kassmann-Kumagai, AIHP 2010: X is a symmetric pure jump process with stable-like kernel. Bass-Kumagai-Uemura, PTRF 2010: X is a symmetric process which admits continuous and jump part. The main step in the proofs of all above mentioned results is to obtain a prior heat kernel estimates of the chains {X n } n N. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

8 Related results By using completely different approach, Chen-Kim-Kumagai, PTRF 2013, consider the situation when X is a symmetric pure jump process (on a metric measure space). Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

9 Related results By using completely different approach, Chen-Kim-Kumagai, PTRF 2013, consider the situation when X is a symmetric pure jump process (on a metric measure space). Their approach consists of two steps: to conclude tightness of {X n } n N they use the Lyons-Zhang decomposition, Lyons-Zhang, AP 1994; to prove convergence of finite-dimensional distributions of {X n } n N to finite-dimensional distributions of X they apply the Mosco convergence of symmetric Dirichlet forms, obtained by Mosco, JFA 1994, and generalized by Kim, SPA Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

10 Goal Discuss the questions of convergence and approximation in the case when X is a non-symmetric pure jump process. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

11 Goal Discuss the questions of convergence and approximation in the case when X is a non-symmetric pure jump process. The approach consists of two steps: to conclude tightness of {X n } n N we use stochastic analysis tools (characteristics of semimartingales) discussed in Jacod-Shiryaev, Limit theorems for stochastic processes, 2003; to prove convergence of finite-dimensional distributions of {X n } n N to finite-dimensional distributions of X we apply the Mosco convergence of non-symmetric Dirichlet forms, obtained by Hino, JMKU 1998, and generalized by Tölle, Master s thesis Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

12 Semimartingale approach Let {S t } t 0 a semimatingale and let h : R d R d be a truncation function. Define, S(h) t := s t ( S s h( S s )) and S(h) t := S t S(h) t. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

13 Semimartingale approach Let {S t } t 0 a semimatingale and let h : R d R d be a truncation function. Define, S(h) t := s t ( S s h( S s )) and S(h) t := S t S(h) t. The process {S(h) t } t 0 is a special semimartingale, that is, it admits a unique decomposition S(h) t = S 0 + M(h) t + B(h) t, where {M(h) t } t 0 is a local martingale and {B(h) t } t 0 is a predictable process of bounded variation. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

14 Semimartingale approach Further, let N(ω, ds, dy) be the compensator of the jump measure µ(ω, ds, dy) := s: S s(ω) 0 δ (s, Ss(ω))(ds, dy) of {S t } t 0, and let {A t } t 0 = {(A ij t ) 1 i,j d)} t 0 be the quadratic co-variation process for {S c t } t 0. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

15 Semimartingale approach Further, let N(ω, ds, dy) be the compensator of the jump measure µ(ω, ds, dy) := s: S s(ω) 0 δ (s, Ss(ω))(ds, dy) of {S t } t 0, and let {A t } t 0 = {(A ij t ) 1 i,j d)} t 0 be the quadratic co-variation process for {S c t } t 0. The triplet (B, A, N) is called the characteristics of {S t } t 0 (relative to h(x)). Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

16 Semimartingale approach Further, let N(ω, ds, dy) be the compensator of the jump measure µ(ω, ds, dy) := s: S s(ω) 0 δ (s, Ss(ω))(ds, dy) of {S t } t 0, and let {A t } t 0 = {(A ij t ) 1 i,j d)} t 0 be the quadratic co-variation process for {S c t } t 0. The triplet (B, A, N) is called the characteristics of {S t } t 0 (relative to h(x)). In addition, by defining Ã(h)ij t := M(h) i t, M(h)j t, the triplet (B, Ã, N) is called the modified characteristics of {S t } t 0 (relative to h(x)). Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

17 Semimartingale approach Further, let N(ω, ds, dy) be the compensator of the jump measure µ(ω, ds, dy) := s: S s(ω) 0 δ (s, Ss(ω))(ds, dy) of {S t } t 0, and let {A t } t 0 = {(A ij t ) 1 i,j d)} t 0 be the quadratic co-variation process for {S c t } t 0. The triplet (B, A, N) is called the characteristics of {S t } t 0 (relative to h(x)). In addition, by defining Ã(h)ij t := M(h) i t, M(h)j t, the triplet (B, Ã, N) is called the modified characteristics of {S t } t 0 (relative to h(x)). Jacod-Shiryaev, Limit theorems for stochastic processes, 2003: Problem of weak convergence of a sequence of semimartingales to a semimartingale translate in terms of convergence of the corresponding characteristics. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

18 Semimartingale approach Let C n : Z d n Z d n [0, ), n N, be a family of functions satisfying (S1) C n (a, a) = 0 for all a Z d n and all n N; (S2) sup a Z d n C n (a, b) < for all n N. b Z d n Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

19 Semimartingale approach Let C n : Z d n Z d n [0, ), n N, be a family of functions satisfying (S1) C n (a, a) = 0 for all a Z d n and all n N; (S2) sup a Z d n C n (a, b) < for all n N. b Z d n Then, under (S1) and (S2), C n, n N, define a family of regular continuous-time Markov chains {Xt n} t 0 on Z d n determined by infinitesimal generator of the form A n f (a) = b Z d n (f (b) f (a))c n (a, b). Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

20 Semimartingale approach Clearly, the processes {X n t } t 0, n N, are semimartingales and the corresponding (modified) characteristics are given by: B n (h) t = t 0 A n t = 0, Ã n (h) ij t = t 0 h(b)c n (Xs n, Xs n + b)ds, b Z d n h i (b)h j (b)c n (Xs n, Xs n + b)ds, b Z d n N n (ds, b) = C n (X n s, X n s + b)ds. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

21 Semimartingale approach Pure jump homogeneous diffusion with jumps is a semimartingale {X t } t 0 determined with (modified) characteristics of the form B t := A n,ij t 0 b(x s )ds, t := 0, i, j = 1,..., d, t à n,ij t := h i (y)h j (y)ν(x s, dy)ds, i, j = 1,..., d, 0 R d N(ds, dy) := ν(x s, dy)ds, where b : R d R d and ν : R d B(R d ) [0, ] are, respectively, Borel function and Borel kernel satisfying ν(x, {0}) = 0 and R d (1 y 2 )ν(x, dy) <. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

22 Semimartingale approach For a = (a 1,..., a d ) Z d n set ā := [a 1 1/2n, a 1 + 1/2n) [a d 1/2n, a d + 1/2n), and for x = (x 1,..., x d ) R d define [x] n := ([nx 1 + 1/2n] /n,..., [nx d + 1/2n] /n). Note that for a Z d n, [x] n = a for all x ā. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

23 Semimartingale approach For a = (a 1,..., a d ) Z d n set ā := [a 1 1/2n, a 1 + 1/2n) [a d 1/2n, a d + 1/2n), and for x = (x 1,..., x d ) R d define [x] n := ([nx 1 + 1/2n] /n,..., [nx d + 1/2n] /n). Note that for a Z d n, [x] n = a for all x ā. Theorem Under (S1), (S2), (S3) the functions b(x), x R d h i (y)h j (y)ν(x, dy) and x R d g(y)ν(x, dy) are continuous for any bounded and continuous function g : R d R vanishing in a neighborhood of the origin; (S4) for all R > 0, lim sup r x B R (0) ν(x, B c r (0)) = 0; Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

24 Semimartingale approach Theorem (continued) (S5) for all R > 0, lim sup n h i (b)c n ([x] n, [x] n + b) b i (x) x B R (0) = 0; b Z d n (S6) for all R > 0, lim sup n x B R (0) h i (b)h j (b)c n ([x] n, [x] n + b) b Z d n h i (y)h j (y)ν(x, dy) = 0; R d Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

25 Semimartingale approach Theorem (continued) (S7) for all R > 0 and all bounded and continuous functions g : R d R vanishing in a neighbourhood of the origin, lim sup n g(b)c n ([x] n, [x] n + b) g(y)ν(x, dy) x B R (0) R d = 0, b Z d n {Xt n d } t 0 {X t} t 0. n Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

26 Semimartingale approach Assume sup ν(x, Bρ(0)) c <, ρ > 0. x R d For 0 < p 1, define C n,p : Z d n Z d n [0, ) by { C n,p ν(a, (a, b) := b a), a b > 0, a b d n p d n. p Observe that C n,p, n N, automatically satisfy (S1) and (S2). Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

27 Semimartingale approach Assume sup ν(x, Bρ(0)) c <, ρ > 0. x R d For 0 < p 1, define C n,p : Z d n Z d n [0, ) by { C n,p ν(a, (a, b) := b a), a b > 0, a b d n p d n. p Observe that C n,p, n N, automatically satisfy (S1) and (S2). Theorem The conditions in (S5)-(S7) will be satisfied if for all ρ > 0 and R > 0, lim ε sup ε 0 x B R (0) lim ε 0 εp ν(x, B ρ (0) \ B ε p(0)) = 0, sup ν(x, B dε p +( d/2)ε (0) \ B dε p ( d/2)ε (0)) = 0, x B R (0) Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

28 Semimartingale approach Theorem (continued) lim sup y 2 ν(x, dy) = 0, ε 0 x B R (0) lim sup n x B R (0) lim sup n x B R (0) lim sup n x B R (0) lim sup ε 0 x B R (0) B ε(0) B ρ(0) y 2 ν([x] n, dy) ν(x, dy) TV = 0, B ρ(0)\b d/n p d/2n (0) y ν([x] n, dy) ν(x, dy) TV = 0, ν([x] n, B c ε (0)) ν(x, B c ε (0)) TV = 0, ε > 0, B c ε(0) h i (y)ν(x, dy) b i (x) = 0. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

29 Semimartingale approach Examples Pure jump Lévy processes. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

30 Semimartingale approach Examples Pure jump Lévy processes. Stable-like processes: Let α : R d (0, 2) be bounded and continuously differentiable function with bounded derivatives such that 0 < α = α(x) = α < 2. Under this assumptions, Bass, PTRF 1988, Schilling, PTRF 1998, and Schilling-Wang, TAMS 2013, have shown that there exists a unique Feller semimartingale {X t } t 0, called a stable-like process, determined by (modified) characteristics (with respect to an odd truncation function h(x)) of the form B(h) t = 0, t à i,j dy t = h i (y)h j (y) 0 R d y N(ds, dy) = dyds y d+α(xs). d+α(xs) ds, Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

31 Semimartingale approach Examples Lévy-driven SDEs: Let {L t } t 0 be an n-dimensional Lévy process and let Φ : R d R d n be bounded and locally Lipschitz continuous. Then, Schilling-Schnurr, EJP 2010, have shown that the SDE dx t = Φ(X t )dl t, X 0 = x R d, admits a unique strong solution which is a Feller semimartingale. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

32 Semimartingale approach Examples Lévy-driven SDEs: Let {L t } t 0 be an n-dimensional Lévy process and let Φ : R d R d n be bounded and locally Lipschitz continuous. Then, Schilling-Schnurr, EJP 2010, have shown that the SDE dx t = Φ(X t )dl t, X 0 = x R d, admits a unique strong solution which is a Feller semimartingale. In particular, if Lt = (l t, t), where {l t } t 0 is a d-dimensional Lévy process determined by Lévy triplet (0, 0, ν(dy)) such that ν(dy) is symmetric; Φ(x) = (φ(x)i, 0), where φ : R d R is locally Lipschitz continuous and 0 < inf x R d φ(x) sup x R d φ(x) <, Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

33 Semimartingale approach Examples then {X t } t 0 is determined by (modified) characteristics (with respect to an odd truncation function h(x)) of the form B(h) t = 0, t à i,j t = h i (y)h j (y)ν (dy/ φ(x s ) ) ds, 0 R d N(ds, dy) = ν (dy/ φ(x s ) ). Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

34 Dirichlet form approach Recall, functions C n : Z d n Z d n [0, ), n N, satisfying (T1) C n (a, a) = 0 for all a Z d n and all n N; (T2) sup C n (a, b) < for all n N, a Z d n b Z d n define a family of regular continuous-time Markov chains {Xt n} t 0 with (modified) characteristics: B n (h) t = t 0 A n t = 0, Ã n (h) ij t = t 0 h(b)c n (Xs n, Xs n + b)ds, b Z d n h i (b)h j (b)c n (Xs n, Xs n + b)ds, b Z d n N n (ds, b) = C n (X n s, X n s + b)ds. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

35 Dirichlet form approach Theorem (tightness) The family {Xt n} t 0 will be tight if (T1), (T2) and (T3) lim sup sup C n (a, a + b) <, ρ > 0, n a Z d n b >ρ lim lim sup sup C n (a, a + b) = 0; r n a Z d n b >r (T4) there exists ρ > 0 such that lim sup sup b i C n (a, a + b) n < hold true. lim sup n a Z d n b <ρ sup b i b j C n (a, a + b) < a Z d n b <ρ Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

36 Dirichlet form approach Let k : R d R d \ diag [0, ) be a Borel measurable function. Denote k s (x, y) := 1 (k(x, y) + k(y, x)) 2 k a (x, y) := 1 (k(x, y) k(y, x)). 2 Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

37 Dirichlet form approach Let k : R d R d \ diag [0, ) be a Borel measurable function. Denote k s (x, y) := 1 (k(x, y) + k(y, x)) 2 k a (x, y) := 1 (k(x, y) k(y, x)). 2 Under assumption (C1) x (1 y 2 )k s (x, x + y)dy L 1 loc (Rd, dx) R d α 0 := sup x R d {y R d : k s(x,y) 0} k a (x, y) 2 dy <, k s (x, y) Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

38 Dirichlet form approach k(x, y) defines a regular symmetric Dirichlet form (E, F) on L 2 (R d, dx), where E(f, g) := (f (y) f (x))(g(y) g(x))k s (x, y)dxdy, f, g F, R d R d \diag F := {f L 2 (R d, dx) : E(f, f ) < } and F is the E 1/2 1 -closure of Cc Lip (R d ) in F. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

39 Dirichlet form approach k(x, y) defines a regular symmetric Dirichlet form (E, F) on L 2 (R d, dx), where E(f, g) := (f (y) f (x))(g(y) g(x))k s (x, y)dxdy, f, g F, R d R d \diag F := {f L 2 (R d, dx) : E(f, f ) < } and F is the E 1/2 1 -closure of Cc Lip (R d ) in F. Further, Fukushima-Uemura, AP 2012, and Schilling-Wang, FMF 2015, have shown that the (non-symmetric) form H(f, g) := lim (f (y) f (x))k(x, y)dy g(x)dx, f, g Cc Lip (R d ), ε 0 R d B c ε(x) Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

40 Dirichlet form approach is well defined, has a representation H(f, g) = 1 2 E(f, g) (f (y) f (x))g(y)k a (x, y)dxdy, R d R d \diag extends to F F such that (H, F) defines a regular lower bounded coercive semi-dirichlet form on L 2 (R d, dx) (and hence a Hunt process ({X t } t 0, {P x } x R d ) defined on the complement of an exceptional set). Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

41 Dirichlet form approach is well defined, has a representation H(f, g) = 1 2 E(f, g) (f (y) f (x))g(y)k a (x, y)dxdy, R d R d \diag extends to F F such that (H, F) defines a regular lower bounded coercive semi-dirichlet form on L 2 (R d, dx) (and hence a Hunt process ({X t } t 0, {P x } x R d ) defined on the complement of an exceptional set). Moreover, it satisfies and 1 4 (1 α 0)E 1 (f ) H α0 (f ) (1 α 0 ) E 1 (f ), f F, 2 (1 α 0 )H 1 (f ) H α0 (f ) (1 α 0 ) H 1 (f ), f F. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

42 Dirichlet form approach Denote by L 2 (Z d n) the standard Hilbert space on Z d n with scalar product f, g n := n d a Z d n f (a)g(a), f, g L 2 (Z d n). Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

43 Dirichlet form approach Denote by L 2 (Z d n) the standard Hilbert space on Z d n with scalar product f, g n := n d a Z d n f (a)g(a), f, g L 2 (Z d n). Proposition Assume (T1), (T2) and (C1). Then, for each n N, the following operator is well defined (non-symmetric) bilinear form on F n := {f L 2 (Z d n) : E n (f, f ) < }, H n (f, g) = 1 2 En (f, g) n d a Z d n (f (b) f (a))g(b)ca(a, n b); b Z d n Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

44 Dirichlet form approach Proposition (continued) (H n, F n ) is a regular lower bounded coercive semi-dirichlet form; Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

45 Dirichlet form approach Proposition (continued) (H n, F n ) is a regular lower bounded coercive semi-dirichlet form; for any f C c (Z d n) and g F n, it holds f D A n, A n f L 2 (Z d n) and H n (f, g) = A n f, g n ; Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

46 Dirichlet form approach Proposition (continued) (H n, F n ) is a regular lower bounded coercive semi-dirichlet form; for any f C c (Z d n) and g F n, it holds f D A n, A n f L 2 (Z d n) and H n (f, g) = A n f, g n ; for all f F n, and 1 4 (1 αn 0 )En 1 (f ) Hn α 0 (f ) (1 α0 n 2 ) En 1 (f ) (1 α0 n )Hn 1 (f ) Hn α 0 (f ) (1 α0 n ) Hn 1 (f ). Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

47 Dirichlet form approach Let r n : L 2 (R d, dx) L 2 (Z d n) and e n : L 2 (Z d n) L 2 (R d, dx), n N, denote the restriction and extension operators, respectively, defined as follows r n f (a) = n d f (x)dx, a Z d n e n f (x) = f (a), x ā. ā Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

48 Dirichlet form approach Let r n : L 2 (R d, dx) L 2 (Z d n) and e n : L 2 (Z d n) L 2 (R d, dx), n N, denote the restriction and extension operators, respectively, defined as follows r n f (a) = n d f (x)dx, a Z d n e n f (x) = f (a), x ā. ā We say that f n L 2 (Z d n), n N, converge strongly to f L 2 (R d, dx) if and they converge weakly if lim e nf n f L 2 = 0, n lim e nf n, g = f, g, n g L 2 (R d, dx). Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

49 Dirichlet form approach Theorem Assume (T1)-(T4), (C1) and that {P n t r nf } n 1 converges strongly to P t f for all t 0 and all f L 2 (R d, dx). Then, there exists a Lebesgue measure zero set, say B, such that for any initial distribution µ(dx) of {X t } t 0 with µ(b) = 0 and any sequence of initial distributions of {X n t } t 0, n N, converging weakly to µ(dx), {Xt n d } t 0 {X t} t 0. n Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

50 Dirichlet form approach Definition Let C be dense in (F, H 1/2 1 ). Assume the following (i) for every sequence {f n } n 1, f n F n, converging weakly to some f L 2 (R d, dx) and satisfying lim inf n Hn 1 (f n) <, we have that f F; (ii) for any g C, any f F and any sequence {f n } n 1, f n F n, converging weakly to f, there exists a sequence g n F n converging strongly to g and lim n Hn (g n, f n ) = H(g, f ). Then, we say that the forms H n, n N, converge in generalized (Mosco) sense to H. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

51 Dirichlet form approach If (C2) x B 1 (0) y 2 k s (x, x + y) dy L 2 loc (Rd, dx), x k s (x, x + y) dy L 2 (R d, dx) L (R d, dx), B1 c(0) x B 1 (0) y ( k(x, x + y) k(x, x y) + k(x + y, x) k(x y, x) ) dy L 2 loc (Rd, dx), then (under (C1)) for the generator (A, D A ) of {P t } t 0 (or, equivalently, of (H, F)) it holds that Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

52 Dirichlet form approach Cc (R d ) D A ; for every g Cc (R d ), Ag(x) = (g(x + y) g(x) g(x), y 1 B1 (0)(y))k(x, x + y) dy R d + 1 g(x), y (k(x, x + y) k(x, x y)) dy; 2 B 1 (0) for all g C c (R d ) and all f F, H(g, f ) = Ag, f. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

53 Dirichlet form approach Theorem Assume (T1), (T2), (C1), (C2), (C3) 0 < lim inf n αn 0 (C4) for every ρ > 0, lim sup α0 n < ; n sup x B ρ(0) R d (1 y 2 )k s (x, x + y)dy < ; (C5) for every ρ > 0, lim sup n sup a B ρ(0) b Z d n (1 b 2 )Cs n (a, a + b) < ; Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

54 Dirichlet form approach Theorem (continued) (C6) for every ε > 0 there is n 0 N such that for all n 0 m n and all f L 2 (Z d m), E n (r n e m f, r n e m f ) 1/2 E m (f, f ) 1/2 + ε; (C7) for any sufficiently small ε > 0 and large m N, lim Ēm,ε(f n, f ) = E m,ε (f, f ), n f C Lip c (R d ), where Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

55 Dirichlet form approach Theorem (continued) (C7) E m,ε (f, f ) := 1 2 {(x,y) B m(0) B m(0): x y >ε} (f (y) f (x)) 2 k s (x, y)dxdy, Ē n m,ε(f, f ) := nd 2 and {(x,y) B m(0) B m(0): x y >ε} C n s (x, y) := { C n s (a, b), x ā and y b 0, x / ā or y / b; (f (y) f (x)) 2 Cn s (x, y)dxdy, Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

56 Dirichlet form approach Theorem (continued) (C8) y 2 k(x, x + y) n d C n ([x] n, [x] n + [y] n ) dy L2 loc (Rd,dx) 0; n B 1 (0) (C9) Bc1(0) k(x, x + y) n d C n ([x] n, [x] n + [y] n ) dy L2 loc (Rd,dx) n 0; (C10) for all R > 0 large enough, B c 2R (0) ( 2 k(x, x + y)dy) dx < ; B R ( x) Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

57 Dirichlet form approach Theorem (continued) (C11) for all R > 0 large enough, B c 2R (0) n 0; ( 2 k(x, x + y) n d C n ([x] n, [x] n + [y] n ) dy) dx B R ( x) (C12) B 1 (0) y k(x, x + y) k(x, x y) n d C n ([x] n, [x] n + [y] n ) +n d C n ([x] n, [x] n [y] n ) dy L2 loc (Rd,dx) n 0. The the forms H n, n N, converge to H in Mosco sense. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

58 Dirichlet form approach For 0 < p 1 define C n,p n d (a, b) := ā b k(x, y)dxdy, a b > 2 d n p 0, a b 2 d n p. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

59 Dirichlet form approach For 0 < p 1 define C n,p n d (a, b) := ā b k(x, y)dxdy, a b > 2 d n p 0, a b 2 d n p. The conditions in (T2)-(T4) will be satisfied if for every ρ > 0, lim sup k(x, y)dy = 0; r x R d Br c (x) sup k(x, y)dy < ; x R d Bρ(x) c Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

60 Dirichlet form approach there exists ρ > 0 such that lim sup sup (y ε 0 i x i )k(x, y)dy < lim sup ε 0 x R d B ρ(x)\b ε(x) sup y i x i k(x, y)dx < B dε p \B dε p (x) ( d/2)ε (x) x R d lim sup ε sup k(x, y)dy <, ε 0 x R d B ρ(x)\b ε p (x) Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

61 Dirichlet form approach there exists ρ > 0 such that lim sup sup (y ε 0 i x i )(y j x j )k(x, y)dy < lim sup ε 0 x R d B ρ(x)\b ε(x) sup y i x i y j x j k(x, y)dx < B dε p (x)\b dε p ( d/2)ε (x) x R d lim sup ε sup y i x i k(x, y)dy < ε 0 x R d B ρ(x)\b ε p (x) Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

62 Dirichlet form approach Examples Symmetric processes. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

63 Dirichlet form approach Examples Symmetric processes. Non-symmetric Lévy processes: Let B R d be Borel and let ν 1 (dy) = n 1 (y)dy and ν 2 (dy) = n 2 (y)dy be Lévy measures. Define { ν1 (dy), y B ν(dy) := ν 2 (dy), y B c. Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

64 Dirichlet form approach Examples Symmetric processes. Non-symmetric Lévy processes: Let B R d be Borel and let ν 1 (dy) = n 1 (y)dy and ν 2 (dy) = n 2 (y)dy be Lévy measures. Define { ν1 (dy), y B ν(dy) := ν 2 (dy), y B c. Stable-like processes: Let α : R d (0, 2) be Borel measurable such that 0 < α α(x) α < 2 and 1 0 (β(u) log u ) 2 u 1+α du <, where β(u) := sup x y u α(x) α(y). Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

65 Dirichlet form approach Examples Then, the (non-symmetric) kernel k(x, y) := γ(x) y x α(x) d defines a regular lower bounded semi-dirichlet form whose corresponding Hunt process is called stable-like process. Here α(x) 1 Γ(α(x)/2 + d/2) γ(x) := α(x)2 π d/2 Γ(1 α(x)/2). Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43

66 T h a n k y o u f o r y o u r a t t e n t i o n! Nikola Sandrić (University of Zagreb) Lévy 2016 July 25-29, / 43