Analysis of space-time fractional stochastic partial differential equations

Transcription

1 Analysis of space-time fractional stochastic partial differential equations Erkan Nane Department of Mathematics and Statistics Auburn University April 2, 2017 Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

2 Outline Fractional Diffusion Time Fractional SPDEs Intermittency Intermittency Fronts SPDEs in bounded domains SPDEs with space-colored noise Future work/directions Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

3 Acknowledgement Joint work with Jebessa Mijena, Georgia College and State University Mohammud Foondun, University of Strathclyde, Glasgow Wei Liu, Shanghai Normal University, China Current students: Sunday Asogwa, Ngartelbaye Guerngar, Auburn University Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

4 Fractional Diffusion Fractional time derivative: Two popular approaches! Riemann-Liouville fractional derivative of order 0 < β < 1; [ D β 1 t ] ds t g(t) = g(s) Γ(1 β) t (t s) β with Laplace transform s β g(s), g(s) = e st g(t)dt denotes 0 the usual Laplace transform of g. Caputo fractional derivative of order 0 < β < 1; D β t g(t) = 1 Γ(1 β) t 0 0 dg(s) ds ds (1) (t s) β was invented to properly handle initial values (Caputo 1967). Laplace transform of D β t g(t) is s β g(s) s β 1 g(0) incorporates the initial value in the same way as the first derivative. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

5 Fractional Diffusion examples D β t (t p ) = D β t (e λt ) = λ β e λt Γ(1 + p) Γ(p + 1 β) tp β t β Γ(1 β)? D β t (sin t) = sin(t + πβ 2 ) Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

6 Fractional Diffusion Space-Time fractional PDE The solution to the equation (Nigmatullin 1986; Zaslavsky 1994) β t u(t, x) = ν( ) α/2 u(t, x); u(0, x) = u 0 (x), (2) where β t is the Caputo fractional derivative of index β (0, 1) and α (0, 2] is given by u(t, x) = E x (u 0 (Y (E t ))) = P(s, x)f Et (s)ds 0 ( ) = p(s, x y)f Et (s)ds u 0 (y)dy R d 0 (3) where f Et (s) is the density of inverse stable subordinator of index β (0, 1), and Y is α-stable Lévy process. Here P(t, x) = E x (u 0 (Y (t))) is the semigroup of α-stable Lévy process. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

7 Time fractional SPDE s Time fractional spde model? We want to study the equations of the following type β t u(t, x) = ν( ) α/2 u(t, x) + λσ(u) W (t, x); u(0, x) = u 0 (x), (4) where W (t, x) is a space-time white noise with x R d. Assume that σ( ) satisfies the following global Lipschitz condition, i.e. there exists a generic positive constant Lip such that : σ(x) σ(y) Lip x y for all x, y R. (5) Assume also that the initial datum is L p (Ω) bounded (p 2), that is sup E u 0 (x) p <. (6) x R d Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

8 Time fractional SPDE s Time fractional Duhamel s principle Let G(t, x) be the fundamental solution of the time fractional PDE β u = L x u. The solution to the time-fractional PDE with force term f (t, x) β t u(t, x) = L x u(t, x) + f (t, x); u(0, x) = u 0 (x), (7) is given by Duhamel s principle (Umarov and Saydmatov, 2006), the influence of the external force f (t, x) to the output can be count as β t V (τ, t, x) = L x V (τ, t, x); V (τ, τ, x) = 1 β t f (t, x) t=τ, (8) which has solution V (t, τ, x) = G(t τ, x y) τ 1 β f (τ, x)dx R d Hence solution to (7) is given by t u(t, x) = G(t, x y)u 0 (y)dy+ R d G(t r, x y) r 1 β f (r, x)dxdr. R d 0 Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

9 Time fractional SPDE s Model? Hence if we use this approach we will get the solution of β t u(t, x) = L x u(t, x)+ W (t, x); u(0, x) = u 0 (x), (9) to be of the form (informally): u(t, x) = G(t, x y)u 0 (y)dy R d t + G(t r, x y) r 1 β [ W (r, y)]dydr.! 0 R d (10) here I am not sure what the fractional derivative in the Walsh-Dalang integral mean? Another point is that the stochastic integral maybe non-gaussian!? Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

10 Time fractional SPDE s Let γ > 0, define the fractional integral by I γ t f (t) := 1 Γ(γ) t 0 (t τ) γ 1 f (τ)dτ. For every β (0, 1), and g L (R + ) or g C(R + ) β t I β t g(t) = g(t). Therefore if we consider the time fractional PDE with a force given by f (t, x) = I 1 β t g(t, x), then by the Duhamel s principle the solution to (7) will be given by u(t, x) = G(t, x y)u 0 (y)dy R d t + G(t r, x y) r 1 β [Ir 1 β g(r, x)]dxdr 0 R d t = G(t, x y)u 0 (y)dy + G(t r, x y)g(r, x)dxdr. R d R d 0 (11) Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

11 Time fractional SPDE s Correct TFSPDE Model We should consider the following model problem (Here L x = ν( ) α/2 ): β t u(t, x) = L x u(t, x)+λi 1 β t [σ(u) W (t, x)]; u(0, x) = u 0 (x), (12) By the Duhamel s principle, mentioned above, (12) will have solution u(t, x) = G(t, x y)u 0 (y)dy R d t (13) + λ G(t r, x y)σ(u(r, y))w (dydr). 0 R d These space-time fractional SPDEs may arise naturally by considering the heat equation in a material with thermal memory, Chen et al. (2015). Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

12 Time fractional SPDE s The fractional integral above in equation (12) for functions φ L 2 (R d ) is defined as R d φ(x)i 1 β t [ W (t, x)]dx = 1 Γ(1 β) R d t 0 (t τ) β φ(x)w (dτdx), is well defined only when 0 < β < 1/2! An important reason to take the fractional integral of the noise in equation (12): Apply the fractional derivative of order 1 β to both sides of the equation (12) to see the forcing function, in the traditional units x/t ( B. Baeumer et al. (2005); K. Li et al. (2011) and (2013)). Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

13 Time fractional SPDE s G(t, x) is the density function of Y (Q t ), where Y is an isotropic α-stable Lévy process in R d and Q t is the first passage time of a β-stable subordinator D = {D r, r 0}. Let p Y (s) (x) and f Qt (s) be the density of Y (s) and Q t, respectively. Then the Fourier transform of p Y (s) (x) is given by p Y (s) (ξ) = e sν ξ α, f Qt (x) = tβ 1 x 1 1/β g β (tx 1/β ), (14) where g β ( ) is the density function of D 1. By conditioning, we have G(t, x) = Lemma 1. Let d < min{2, β 1 }α, then Rd G 2 (t, x)dx = t βd/α (ν) d/α 2π d/2 αγ( d 2 ) 1 (2π) d 0 p Y (s) (x)f Qt (s)ds. (15) 0 z d/α 1 (E β ( z)) 2 dz (16) Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

14 Time fractional SPDE s Theorem (Mijena and N., 2015) Let d < min{2, β 1 }α. If σ is Lipschitz continuous and u 0 is measurable and bounded, then there exists a continuous random field u γ>0 L γ,2 that solves (12) with initial function u 0. Moreover, u is a.s.-unique among all random fields that satisfy the following: There exists a positive and finite constant L-depending only on Lip, and sup z R d u 0 (z) - such that ) sup E ( u t (x) k L k exp(lk 1+α/(α βd) t) (17) x R d In contrast to SPDEs, TFSPDEs have random field (function) solutions for d < min{2, β 1 }α. We use Picard iteration and a stochastic Young inequality to prove this Theorem. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

15 Time fractional SPDE s Curse of Dimensionality for SPDEs Consider the SPDE, t Z(t, x) = ν Z(t, x)+ W (t, x) (t > 0, x R d, d 2) 2 subject to Z(0, x) = 0. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

16 Time fractional SPDE s Curse of Dimensionality for SPDEs Consider the SPDE, t Z(t, x) = ν Z(t, x)+ W (t, x) (t > 0, x R d, d 2) 2 subject to Z(0, x) = 0. The weak solution is Z(t, x) = (0,t) R p t s (y x)w (dsdy), where p t (x) = (2νπt) d/2 exp{ x 2 /(2νt)}. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

17 Time fractional SPDE s Curse of Dimensionality for SPDEs Consider the SPDE, t Z(t, x) = ν Z(t, x)+ W (t, x) (t > 0, x R d, d 2) 2 subject to Z(0, x) = 0. The weak solution is Z(t, x) = (0,t) R p t s (y x)w (dsdy), where p t (x) = (2νπt) d/2 exp{ x 2 /(2νt)}. Not a random function; If it were then it would be a GRF with t t E( Z(t, x) 2 ) = ds dy[p s (y)] 2 = p 2s (0)ds 0 R d 0 t (18) s d/2 ds =. 0 Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

18 Time fractional SPDE s Finite energy solution Random field u is a finite energy solution to the stochastic heat equation (12) when u γ>0 L γ,2 and there exists ρ > 0 such that If ρ (0, ), then 0 0 e ρ t E( u t (x) 2 )dt < for all x R d. e ρt E( u t (x) 2 )dt [N γ,2 (u)] 2 0 e (ρ 2γ)t dt. Therefore if ρ > 2γ and N γ,2 (u) = sup t>0, x R d e γt( E( u t (x) 2 ) ) 1/2 <, then the preceding integral is finite. When σ is Lipschitz-continuous function and u 0 is bounded and measurable, then there exists a finite energy solution to the time fractional stochastic type equation (12). Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

19 Time fractional SPDE s If we drop the assumption of linear growth for σ, then we have the next theorem that extends the result of Foondun and Parshad (2014). Theorem (Mijena and N., 2015) Suppose inf z R d u 0 (z) > 0 and inf y R d σ(y) / y 1+ɛ > 0. Then, there is no finite-energy solution to the time fractional stochastic heat equation (12). Hence there is no solution that satisfies sup E( u t (x) 2 ) L exp(lt) for all t > 0. x R d Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

20 Time fractional SPDE s Non-existence of global solutions Fujita (1966), considered t u t (x) = u t (x) + u t (x) 1+λ for t > 0 x R d, with initial condition u 0 and λ > 0. Set p c := 2/d, Fujita showed that for λ < p c, the only global solution is the trivial one. While for λ > p c, global solutions exist whenever u 0 is small enough. At first, this result might seem counterintuitive, but the right intuition is that for large λ, if the initial condition is small, then u 1+λ is even smaller and the dissipative effect of the Laplacian prevents the solution to grow too big for blow-up to happen. And when λ is close to zero, then irrespective of the size of the initial condition, the dissipative effect of the Laplacian cannot prevent blow up of the solution. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

21 The Erkan Nane above (Auburn result University) states that Space-time provided fractionalthat SPDEs the initial function April 2, 2017 is 19 / 51 Time fractional SPDE s In our case, we consider E u t (x) 2, there is still an interplay between the dissipative effect of the operator and the forcing term and we are able to shed light only on part of the true picture. We extend the results of Chow (2009; 2011) to space fractional SPDEs: Theorem (Foondun, N., Liu (2016)) Suppore that the function σ is a locally Lipschitz function satisfying the following growth condition. There exists a γ > 0 such that σ(x) x 1+γ for all x R. (19) Suppose also that inf x R d u 0 (x) := κ. Let u t be the solution to stochastic PDE (β = 1) and suppose that κ > 0. Then there exists a t 0 > 0 such that for all x R, E u t (x) 2 = whenever t t 0.

22 Intermittency Intermittency Definition: The random field u(t, x) is called weakly intermittent if inf z R d u 0 (z) > 0, and γ k (x)/k is strictly increasing for k 2 for all x R d, where γ k (x) := lim inf t 1 t log E( u(t, x) k ). Fact: If γ 2 (x) > 0 for all x R d, then γ k (x)/k is strictly increasing for k 2 for all x R d. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

23 Intermittency Theorem (Mijena and N., 2016) Let d < min{2, β 1 }α. If inf z R d u 0 (z) > 0, then inf γ 2 (x) [C (L σ ) 2 Γ(1 βd/α)] x R d 1 (1 βd/α) where L σ := inf z R d σ(z)/z. Therefore, the solution u(t, x) of (12) is weakly intermittent when inf z R d u 0 (z) > 0 and L σ > 0. This theorem extends the results of Foondun and Khoshnevisan (2009) to the time fractional stochastic heat type equations. Recall the constant C = const ν d/α. Hence Theorem above implies the so-called very fast dynamo property, lim ν inf x R d γ 2 (x) =. This property has been studied in fluid dynamics: Arponen and Horvai (2007), Baxendale and Rozovskii(1993), Galloway (2003). This theorem is proved by using an application of non-linear renewal theory. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

24 Intermittency Non-linear noise excitation Recall our equation β t u(t, x) = L x u(t, x) + λi 1 β t [σ(u) W (t, x)]; u(0, x) = u 0 (x). We set E t (λ) := E u t (x) 2 dx. R d and define the nonlinear excitation index by e(t) := lim λ log log E t (λ) log λ This was first studied by Khoshnevisan and Kim (2013). This measures in some sense the effect of non-linear noise on the solution of the SPDE. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

25 Intermittency The following theorem shows that as the value of λ increases, the solution rapidly develops tall peaks that are distributed over relatively small islands! Theorem (Foondun and N. 2017) Fix t > 0 and x R, we then have log log E u t (x) 2 lim λ log λ = 2α α βd. Moreover, if the energy of the solution exists, then the excitation index, e(t) is also equal to 2α. α βd Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

26 Intermittency fronts Intermittency fronts What if inf u 0 (x) = 0, say u 0 has compact support. We consider a non-random initial function u 0 : R d R that is measurable and bounded, has compact support, and is strictly positive on an open subinterval of (0, ) d. We consider α = 2. σ : R R Lipschitz and σ(0) = 0. A kind of weak intermittency occur. Roughly, tall peaks arise as t, but the farthest peaks move roughly linearly with time away from the origin intermittency fronts. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

27 Intermittency fronts Define, for all p 2 and for all θ 0, L p (θ) := lim sup t 1 t sup log E ( u t (x) p ). (20) x >θt We can think of θ Lp > 0 as an intermittency lower front if L p (θ) < 0 for all θ > θ Lp, and of θ Up > 0 as an intermittency upper front if L p (θ) > 0 whenever θ < θ Up. If there exists θ that is both a lower front and an upper front then θ is the intermittency front Phase transition. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

28 Intermittency fronts Theorem (Mijena, N. (2016), Asogwa, N. (2017)) Under the above conditions, the time fractional stochastic heat equation (12) has a positive intermittency lower front. In fact, L p (θ) < 0 if θ > p2 4 ( ) 1/β 4ν 2 β (Lip p σ c 0 ) 2( 2 βd ). (21) In addition, under the cone condition L σ = inf z R σ(z)/z > 0, there exists θ 0 > 0 such that L p (θ) > 0 if θ (0, θ 0 ). (22) That is, in this case, the stochastic heat equation has a finite intermittency upper front. This theorem in the case of the stochastic heat equation (for p = 2, and d = 1) was proved by Conus and Khoshnevisan (2012). Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

29 Intermittency fronts We use a version of stochastic Young inequality for the proof of the intermittency lower front. Let d = 1. When σ(x) = Ax. PAM. Our theorem implies that, if there were an intermittency front, then it would lie between θ 0 and 2 1/β (Ac 0 ) 4/(2 β) (Ac 0 ) 2β/(2 β). When β = 1. The existence of an intermittency front has been proved recently by Le Chen and Dalang (2012); in fact, they proved that the intermittency front is at A 2 /2. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

30 SPDEs in bounded domains TFSPDE in bounded domains with Dirichlet boundary conditions. How about the moment estimates and growth of the solution of the following TFSPDEs with Dirichlet boundary conditions? (t fixed large λ and λ fixed, large t.) β t u(t, x) = L x u(t, x) + λi 1 β t [σ(u) W (t, x)], u(0, x) = u 0 (x), x B(0, R); (23) Following Walsh (1986) and using the time fractional Duhamel s principle (Umarov 2012), (23) will have (mild/integral) solution u(t, x) = G B (t, x y)u 0 (y)dy B t (24) + λ G B (t r, x y)σ(u(s, y))w (dydr). 0 B Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

31 SPDEs in bounded domains Theorem (Foondun, Mijena, N. (2016)) Suppose that d < (2 β 1 )α. Then under Lipschitz condition on σ, there exists a unique random-field solution to (23) satisfying sup x B(0,R) E u t (x) 2 c 1 e c 2λ Here c 1 and c 2 are positive constants. α dβ 2α t for all t > 0. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

32 SPDEs in bounded domains Theorem (Foondun, Mijena, N. (2016)) Fix ɛ > 0 and let x B(0, R ɛ), then for any t > 0, log log E u t (x) 2 lim = 2α λ log λ α dβ, where u t is the mild solution to (23). The excitation index of the solution to (23), e(t) is equal to 2α. α dβ This result for β = 1, α (1, 2) is established by Foondun et al (2015), and for β = 1, α = 2 it is established by Khoshnevisan and Kim (2015) and Foondun and Joseph (2015). Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

33 SPDEs in bounded domains Large time fixed λ behavior of the solution Recently Foondun and Nualart (2015) studied the long time (t ) behavior of the the second moment of the solution to (23) when α = 2, β = 1 d = 1. We have extended their results to α (1, 2) Theorem (Foondun, Guerngar, and N. (2017)) There exists λ 0 > 0 such that for all λ < λ 0 and x B(0, R) < lim sup t 1 t log E u t(x) 2 < 0. On the other hand, for all ɛ > 0, there exists λ 1 > 0 such that for all λ > λ 1 and x B(0, R ɛ)], 0 < lim inf t 1 t log E u t(x) 2 <. Theorem says that the solution grows exponentially when λ is large. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

34 SPDEs in bounded domains When λ is small the solution decays exponentially. We define the energy of the solution u as We have also the following < lim sup t 0 < lim inf t E t (λ) := E u t 2. 1 t log E t(λ) < 0 for all λ < λ 0. 1 t log E t(λ) < for all λ > λ 1. Intuitively these results show that if the amount of noise is small, then the heat lost in the system modeled by the Dirichlet equation is not enough to increase the energy in the long run. On the other hand, if the amount of noise is large enough, then the energy will increase Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

35 SPDEs in bounded domains SPDEs with space-colored noise Look at the equation with colored noise. β t u(t, x) = L x u(t, x) + λi 1 β t [σ(u) F (t, x)]; u(0, x) = u 0 (x), (25) in (d + 1) dimensions, where ν > 0, β (0, 1), α (0, 2], F (t, x) is ( ) α/2 is the generator of an isotropic stable process, white noise in time and colored in space, and σ : R R is Lipschitz continuous, and satisfies L σ x σ(x) Lip σ x with L σ and Lip σ being positive constants. F denotes the Gaussian colored noise satisfying the following property, E[ F (t, x) F (s, y)] = δ 0 (t s)f (x, y). This can be interpreted more formally as ( ) Cov φdf, ψdf = ds dx dyφ s (x)ψ s (y)f (x y) 0 R d R d (26) where we use the notation φdf to denote the wiener integral of φ Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

36 SPDEs in bounded domains We will assume that the spatial correlation of the noise term is given by the following function for γ < d, f (x, y) := 1 x y γ. Following Walsh (1986), we define the mild solution of (25) as the predictable solution to the following integral equation t u t (x) = (Gu 0 ) t (x) + λ G t s (x, y)σ(u s (y))f (dsdy). (27) R d 0 where (Gu 0 ) t (x) := G t (x, y)u 0 (y)dy, R d and G t (x, y) is the space-time fractional heat kernel. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

37 SPDEs in bounded domains Future work/open problems 1 Large space behavior of STF-SPDEs 2 Employing the STF-SPDEs for modelling data! 3 Inverse problems for SPDEs 4 Existence-non-existence of solutions, blow up of solutions in finite time. 5 Comparison of solutions for different initial values and/or σs! 6 Approximations of SPDEs: what happens when you have time fractional derivatives? For regular diffusions and stable process, one can have random walk approximations and come up with a system of SDEs. Can one follow the same strategy here? We can approximate time fractional diffusions by CTRWs.! Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

38 SPDEs in bounded domains Thank You! Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

39 SPDEs in bounded domains Some References S. Asogwa and E. Nane. Intermittency fronts for space-time fractional stochastic partial differential equations in (d+1) dimensions. Stochastic Process. Appl. 127 (2017), no. 4, Zhen-Qing Chen, Kyeong-Hun Kim, Panki Kim. Fractional time stochastic partial differential equations. Stochastic Process Appl. 125 (2015), L. Chen, G. Hu, Y. Hu and J. Huang. Space-time fractional diffusions in Gaussian noisy environment. Stochastics, L. Chen, Y. Hu and D. Nualart. Nonlinear stochastic time-fractional slow and fast diffusion equations on R d. Preprint, M. Foodun, N. Guerngar and E. Nane. Some properties of non-linear fractional stochastic heat equations on bounded domains. Chaos, Solitons & Fractals (To appear 2017). Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

40 SPDEs in bounded domains M. Foondun and D. Khoshnevisan, Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab. 14 (2009), no. 21. M. Foondun, W. Liu, M. Omaba, Moment bounds for a class of fractional stochastic heat equations M. Foodun, J.B. Mijena and E. Nane. Non-linear noise excitation for some space-time fractional stochastic equations in bounded domains. Fract. Calc. Appl. Anal. Vol. 19, No 6 (2016), pp M Foondun and E. Nane. Moment bounds for a class of space-time fractional stochastic equations. Mathematische Zeitschrift (2017), pp M. Foondun,and E. Nualart. On the behaviour of stochastic heat equations on bounded domains. ALEA Lat. Am. J. Probab. Math. Stat. 12 (2015), no. 2, Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

41 SPDEs in bounded domains J. B. Mijena, and E.Nane. Intermittence and time fractional partial differential equations. Potential Anal. Vol. 44 (2016) J.B. Mijena and E. Nane. Space-time fractional stochastic partial differential equations. Stochastic Process. Appl. 125, (2015) Umarov, Sabir; Saydamatov, Erkin A fractional analog of the Duhamel principle. Fract. Calc. Appl. Anal. 9 (2006), no. 1, S. Umarov, and E. Saydamatov. A fractional analog of the Duhamel principle. Fract. Calc. Appl. Anal. 9 (2006), no. 1, J.B. Walsh, An introduction to stochastic partial differential equations. Ecole d?eté de Probabilités de Saint-Flour XIV , Lect. Notes Math (1986) Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

42 SPDEs in bounded domains Physical explanation/motivation of the model! The following discussion is adapted from Chen et al.(2015). Let u(t, x), e(t, x) and F (t, x) denote the body temparature, internal energy and flux density, reps. the the relations t e(t, x) = div F (t, x) e(t, x) = βu(t, x), F (t, x) = λ u(t, x) (28) yields the classical heat equation β t u = λ u. According to the law of classical heat equation, the speed of heat flow is infinite. However in real modeling, the propagation speed can be finite because the heat flow can be disrupted by the response of the material. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

43 SPDEs in bounded domains In a material with thermal memory e(t, x) = βu(t, x) + t 0 n(t s)u(s, x)ds holds with some appropriate constant β and kernel n. Typically n(t) = Γ(1 β) 1 t β 1. The convolution implies that the nearer past affects the present more! If in addition the internal energy also depends on past random effects, then t t e(t, x) = βu(t, x)+ n(t s)u(s, x)ds+ l(t s)h(s, u) W (s, x)ds 0 0 (29) Where W is the space time white noise, modeling the random effects. Take β = 0, l(t) = Γ(2 β 2 ) 1 t 1 β 2, then after differentiation (29) gives β 1 t u = div F + 1 Γ(1 β 2 ) t 0 (t s) β 2 h(s, u(s, x)) W (s, x)ds Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

44 SPDEs in bounded domains Walsh-Dalang Integral Need to make sense of the stochastic integral in the mild solution (13). We use the Brownian Filtration {F t } and the Walsh-Dalang integrals: (t, x) Φ t (x) is an elementary random field when 0 a < b and an F a -meas. X L 2 (Ω) and φ L 2 (R d ) such that Φ t (x) = X 1 [a,b] (t)φ(x) (t > 0, x R d ). If h = h t (x) is non-random and Φ is elementary, then hφdw := X h t (x)φ(x)w (dtdx). (a,b) R d The stochastic integral is Wiener s; well defined iff h t (x)φ(x) L 2 ([a, b] R d ). We have Walsh isometry, ( 2) E hφdw = ds dy[h s (y)] 2 E( Φ s (y) 2 ) 0 R d Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

45 SPDEs in bounded domains Picard iteration Define u (0) t (x) := u 0 (x), and iteratively define u (n+1) t follows: from u (n) t as u (n+1) t (x) := (G t u 0 )(x) + λ G(t r, x y)σ(u (n) (r, y))w (drdy) (0,t) R d (30) for all n 0, t > 0, and x R d. Moreover, we set u (k) 0 (x) := u 0(x) for every k 1 and x R d. Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

46 SPDEs in bounded domains Proposition (A stochastic Young inequality) For all γ > 0, k [2, ), d < min{2, β 1 }α, and Φ L γ,2, N γ,k (G Φ) c 0 k 1/2 N γ,k (Φ). t (G Φ) t (x) 2 k 4k ds dy[g(t s, y x)] 2 Φ(y) 2 k 0 R d t 4k[N γ,k (Φ)] 2 e 2γs ds [G(t s, y x)] 2 dy 0 R d t = 4kC [N γ,k (Φ)] 2 e 2γs (t s) βd/α ds t = 4kC [N γ,k (Φ)] 2 e 2γt e 2γu u βd/α du kc α,β,d [N γ,k (Φ)] 2 e 2γt (γ) (1 βd/α). 0 0 Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

47 SPDEs in bounded domains Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

48 SPDEs in bounded domains Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

49 SPDEs in bounded domains Continuous time random walks (CTRW) Particle jump random walk has scaling limit c 1/2 S([ct]) = W (t). Number of jumps has scaling limit c β N(ct) = Q(t). CTRW is a random walk subordinated to (a renewal process) N(t) S(N(t)) = X 1 + X X N(t) CTRW scaling limit is a subordinated process: c β/2 S(N(ct)) = (c β ) 1/2 S(c β c β N(ct)) (c β ) 1/2 S(c β Q(t)) = W (Q(t)). Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

50 SPDEs in bounded domains Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

51 SPDEs in bounded domains Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

52 SPDEs in bounded domains Figure: Typical sample path of the iterated process W (Q(t)). Here W (t) is a Brownian motion and Q(t) is the inverse of a β = 0.8-stable subordinator. Graph has dimension 1 + β/2 = The limit process retains long resting times Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51

53 SPDEs in bounded domains Power law waiting times Wait between solar flares 1 < β < 2 Wait between raindrops β = 0.68 Wait between money transactions β = 0.6 Wait between s β 1.0 Wait between doctor visits β 1.4 Wait between earthquakes β = 1.6 Wait between trades of German bond futures β 0.95 Wait between Irish stock trades β = 0.4 (truncated) Erkan Nane (Auburn University) Space-time fractional SPDEs April 2, / 51