On time regularity of generalized Problems Main Results - PowerPoint PPT Presentation

On time regularity of generalized Problems Main Results Ornstein-Uhlenbeck processes with Proofs Some Discussions cylindrical stable noise ➊ ➥ ❒ ➄ ■ ❑ ➄ Yong LIU , Jianliang ZHAI ◭◭ ◮◮ School of Mathematical Sciences, Peking University ◭ ◮ ✶ 1 ➄ ✁ 30 ➄ ❼ ↔ 2011 SALSIS Dec. 5th 2011, Kochi University, Japan ✜ ➯ ✇ ➠ liuyong@math.pku.edu.cn ✬ ✹ ò Ñ

Problems Main Results Proofs Some Discussions Outline • Problems ➊ ➥ ❒ ➄ • Main Results ■ ❑ ➄ ◭◭ ◮◮ • Proofs ◭ ◮ • Some Discussions ✶ 2 ➄ ✁ 30 ➄ ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ

1 Problems Problems Main Results Proofs Some Discussions dX ( t ) = AX ( t ) dt + dL ( t ) , t ≥ 0 . (1) H , a separable Hilbert space , �· , ·� H . ➊ ➥ ❒ ➄ generator of a C 0 -semigroup on H , A ∗ the adjoint operator of A . ■ ❑ ➄ A , evy process, L = � ∞ ◭◭ ◮◮ n =1 β n L n ( t ) e n , L , L´ ◭ ◮ L n i.i.d., c ` a dl ` a g real-valued L ´ e vy processes. ✶ 3 ➄ ✁ 30 ➄ { e n } n ∈ N fixed reference orthonormal basis in H . ❼ ↔ β n a sequence of positive numbers. ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ

Problems Main Results Proofs Some Discussions Problem ➭ ➭ ➭ If the solution of Eq. (1) ( X ( t )) t ≥ 0 takes value in H for any t , is there a H - ➊ ➥ ❒ ➄ a g ( ˜ valued c ` a dl ` a g modification of X ? i.e. ∃ ? a H -valued c ` a dl ` X t ) t ≥ 0 such ■ ❑ ➄ that, P ( X t = ˜ ◭◭ ◮◮ X t ) = 1 , for any t. (2) ◭ ◮ ✶ 4 ➄ ✁ 30 ➄ ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ

Problems Assume that { e n } n ∈ N ⊂ D ( A ∗ ) , the weak solution of Eq. (1), Main Results Proofs Some Discussions dX ( t ) = AX ( t ) dt + dL ( t ) , t ≥ 0 . can be represented by for any n ∈ N , ➊ ➥ ❒ ➄ d � X ( t ) , e n � H = � X ( t ) , A ∗ e n � H dt + β n dL n ( t ) . ■ ❑ ➄ (3) ◭◭ ◮◮ � X ( t ) , e n � H ≡ X n ( t ) . ◭ ◮ ✶ 5 ➄ ✁ 30 ➄ L n , α − stable processes, α ∈ (0 , 2) . ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ

1.1. Property of Sample Paths Problems Main Results Proofs Kolmogorov’s Extension Theorem: Some Discussions S : State space. construct distribution on S [0 , ∞ ) . ➊ ➥ ❒ ➄ However, this theorem does not describe the properties of sample paths. ■ ❑ ➄ Continuous or c` adl` ag modification of sample path is a fundamental property in ◭◭ ◮◮ Theory of Stochastic Processes, such as Martingale Theory, Markov Processes ◭ ◮ and Probabilistic Potential Theory and SDE. ✶ 6 ➄ ✁ 30 ➄ ❼ ↔ [1] Doob, J.L. Stochastic Processes. John Wiley & Sons Inc., New York 1953 ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ

1.2. Generalized Ornstein-Uhlenbeck Processes Problems Main Results Proofs Some Discussions dX ( t ) = AX ( t ) dt + dL ( t ) . n =1 β n L n ( t ) e n , L n i.i.d., c` ➊ ➥ ❒ ➄ L = � ∞ ag α -stable processes. adl` ■ ❑ ➄ Modeling some heavy tail phenomenon. ◭◭ ◮◮ ◭ ◮ The time regularity of the process X is of prime interest in the study of ✶ 7 ➄ ✁ 30 ➄ non-linear Stochastic PDEs. And these studies of generalized O-U processes is a beginning point . ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ

l 2 -valued O-U processes driven by Brownian motion 1.3. • l 2 -valued O-U processes driven by Brownian motion [2] Iscoe, Marcus, McDonald, Talagrand, Zinn, (1990) Ann. Proba. Problems √ Main Results dx k ( t ) = − λ k x k ( t ) dt + 2 a k dB k , k = 1 , 2 , · · · . Proofs Some Discussions They gave a simple but quite sharp criterion for continuity of X t in l 2 . Theorem 1 in [2] f ( x ) positive function on [0 , ∞ ) such that f ( x ) nonde- x ➊ ➥ ❒ ➄ creasing for x ≥ x 1 > 0 and ■ ❑ ➄ � ∞ dx a k f ( a k ) ∨ x 1 � ◭◭ ◮◮ f ( x ) < ∞ , < ∞ , sup < ∞ . (4) λ k λ k ∨ 1 k x 1 k ◭ ◮ Then, x t is continuous in l 2 a.s. Moreover, this result is best possible in ✶ 8 ➄ ✁ 30 ➄ the sense that it is false for any function f ( x ) , which satisfies all the above ❼ ↔ � ∞ dx f ( x ) = ∞ . hypotheses with the exception that ✜ ➯ ✇ ➠ x 1 ✬ ✹ • H or B -valued O-U processes ò Ñ [3] Millet, Smolenski (1992) Prob. Theory Related Fields.

1.3.1. O-U Eq. with L´ evy noise [4 ] Fuhrman, R¨ ockner (2000) Generalized Mahler semigroups: the non Gaussian case, Problems Potential Anal., 12(2000), 1-47. Main Results Proofs • There is an enlarged space E , H ⊂ HS E , such that ( X ( t )) t ≥ 0 has a c` adl` ag Some Discussions path in E . [5 ] Priola, E., Zabczyk, J. On linear evolution with cylindrical L´ evy noise, in: SPDE and ➊ ➥ ❒ ➄ Applications VIII, Proceedings of the Levico 2008 Conference. ■ ❑ ➄ • L ( t ) symmetric, and L ( t ) ∈ U ⊃ H , they give a necessary and sufficient ◭◭ ◮◮ condition of X t ∈ H , for any t > 0 . ◭ ◮ [6 ] Brze´ ✶ 9 ➄ ✁ 30 ➄ zniak, Z., Zabczyk, J. Regularity of Ornstein-Uhlenbeck processes driven by L´ evy white noise, Potential Anal. 32(2010)153-188. ❼ ↔ • L ( t ) , L´ evy white noise obtained by subordination of a Gaussian white ✜ ➯ ✇ ➠ noise. L t = W ( Z ( t )) , Spatial continuity, Time irregularity. ✬ ✹ ò Ñ

Problems Main Results Proofs [7 ] Priola, E., Zabczyk, J. Structural properties of semilinear SPDEs driven by cylindri- Some Discussions cal stable process, Probab. Theory Related Fields, 149(2011), 97-137 [PZ11] • They conjectured in Section 4 in [7], If L n are symmetric α -stable pro- ➊ ➥ ❒ ➄ cesses, α ∈ (0 , 2) , the H -c ` a dl ` a g property of Eq. (1) holds under much ■ ❑ ➄ weaker conditions than � ∞ n =1 β α n < ∞ . ◭◭ ◮◮ Remark 1. � ∞ n < ∞ ⇔ L ( t ) = � ∞ n =1 β α n =0 β n L n ( t ) e n has H -c ` a dl ` a g property. ◭ ◮ Remark 2. In general, L ∈ H ⇒ X has H -c ` a dl ` a g path. ✶ 10 ➄ ✁ 30 ➄ ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ

[8 ] Brze´ zniak, Z., Goldys, B., Imkeller, P., Peszat, S., Priola, E., Zabczyk, J. Time ir- regularity of generalized Ornstein-Uhlenbeck processes, C. R. Acad. Sci. Paris, Ser. I Problems Main Results 348(2010), 273-276. [BGIPPZ10] Proofs Some Discussions dX ( t ) = AX ( t ) dt + dL ( t ) , t ≥ 0 . ➊ ➥ ❒ ➄ ■ ❑ ➄ d � X ( t ) , e n � H = � X ( t ) , A ∗ e n � H dt + β n dL n ( t ) , n ∈ N . (5) ◭◭ ◮◮ � X ( t ) , e n � H ≡ X n ( t ) . ◭ ◮ • Theorem 2.1 [8] X , H -valued process ( e n ) ∈ D ( A ∗ ) , β n � 0 , then X ✶ 11 ➄ ✁ 30 ➄ has no H -c ` a dl ` a g modification with probability 1. ❼ ↔ • Question 1,2,3,4 ... ... ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ

Problems Main Results Proofs Some Discussions [9 ] Brze´ zniak, Z., Otobe, Y. and Xie B. Regularity for SPDE driven by α -stable cylindri- ➊ ➥ ❒ ➄ cal noise. 2011, preprint ■ ❑ ➄ • They obtained detailed results of spatial regularity and temporal integra- bility. ◭◭ ◮◮ ◭ ◮ ✶ 12 ➄ ✁ 30 ➄ ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ

2 Main Results Problems Main Results Proofs n =1 β n L n ( t ) e n , L n i.i.d. real-valued L ´ L = � ∞ e vy processes, L´ evy characteristic Some Discussions measure ν . { e n } n ∈ N ⊂ D ( A ∗ ) , ➊ ➥ ❒ ➄ d � X ( t ) , e n � H = � X ( t ) , A ∗ e n � H dt + β n dL n ( t ) . ■ ❑ ➄ Theorem 1 Assume that the process X in Eq. (1) has H -c` adl` ag modification, ◭◭ ◮◮ then for any ǫ > 0 , � ∞ n =1 ν ( | y | ≥ ǫ/β n ) < ∞ . ◭ ◮ Remark 3. This theorem implies Theorem 2.1 in [BGIPPZ10] ✶ 13 ➄ ✁ 30 ➄ ❼ ↔ β n � 0 ⇒ no H -c` adl` ag modification ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ

� c 1 y − 1 − α dy, y > 0 , L n , i.i.d. α -stable process. ν ( dy ) = Problems c 2 | y | − 1 − α dy, y < 0 . Main Results Proofs Theorem 2 Assume ( L n , n = 1 , 2 , · · · ) are i.i.d., non-trivial α -stable processes, Some Discussions α ∈ (0 , 2) , and S ( t ) = e At satisfying � S ( t ) � L ( H,H ) ≤ e βt , β ≥ 0 , (generalized contraction principle ), the following three assertions are equivalent: ➊ ➥ ❒ ➄ (1) the process ( X ( t ) , t ≥ 0) in Eq. (1) has H -c ` a dl ` a g modification; ■ ❑ ➄ n =1 | β n | α < ∞ ; (2) � ∞ ◭◭ ◮◮ (3) the process L is a L´ evy process on H . ◭ ◮ Remark 4. This result denies the conjecture in [PZ11]. And more, Theorem 2 ✶ 14 ➄ ✁ 30 ➄ does not need the assumption of symmetry of L n . ❼ ↔ n =1 | β n | α < ∞ . much weaker than � ∞ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ

Problems Main Results Proofs Some Discussions Remark 5. In [BGIPPZ10], Question 3: Is the requirement of the process L evolves in H also necessary ➊ ➥ ❒ ➄ for the existence of H -c` ag modification of X ? adl` ■ ❑ ➄ Theorem 2 partly answers Question 3, i.e. at least if L n , i.i.d. α -stable processes, ◭◭ ◮◮ L evolving in H is a necessary condition of X having H - c` adl` ag modification. ◭ ◮ ✶ 15 ➄ ✁ 30 ➄ ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ