An Introduction to Stochastic Partial Dierential Equations

Transcription

1 An Introduction to Stochastic Partial Dierential Equations Herry Pribawanto Suryawan Dept. of Mathematics, Sanata Dharma University, Yogyakarta 29. August 2014 Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

2 Outline SPDE Existence and Uniqueness of Solution A Stochastic Heat Equation Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

3 SPDE Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

4 SPDE is an interdisciplinary area at the crossroads of stochastic processes and partial dierential equations. Wave equation 2 u(t, x) t 2 = κ 2 u(t, x) x 2 + F (t, x), t 0, 0 x L (1) If F is a random noise, (1) can be interpreted as a guitar in the desert. For example, F (t, x) = W (t, x) is the space-time white noise. Heuristically, W (t, x) is a (Gaussian) random eld such that E (W (t, x)w (s, y)) = δ(t s)δ(x y). In this case, (1) does not have classical meaning and must be interpreted as an innite dimensional integral equation. Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

5 Our topic: Semilinear parabolic problems driven by additive Brownian noise. Our approach: Hilbert space approach based on the theory of operator semigroup. See [DaPrato-Zabczyk] and [Prevot-Röckner]. This is an (elementary) introduction since we do not consider 1 Equations with multiplicative noise 2 Equations driven by fractional Brownian noise (Gaussian, non-markov, non-semimartingale) 3 Equations driven by non-gaussian noise (e.g. Levy noise, alpha-stable noise) 4 Equations with rough (non-lipschitz) nonlinearities 5 Variational solution in Gelfand triplets 6 Hyperbolic and Elliptic problems 7 Malliavian calculus approach and densities of solution 8 Hida calculus approach (Wick type equations) 9 Solutions via Dirichlet Forms 10 Numerical methods 11 etc Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

6 Our approach: An SPDE is translated into a stochastic evolution equation (Cauchy problem) in some innite dimensional Banach space. There are some crucial problems due to -dimension! Innite dimensional Lebesgue measure does not exist! Escape from problem: Gaussian measure! Theorem (Minlos-Sazanov) Let H be a separable Hilbert space. Let Q be a positive denite, symmetric, trace-class operator in H and let m H. Then there exists a Gaussian measure µ = N (m, Q) on (H, B(H)) given via ˆµ(h) := e i h,u µ(du) = e i m,h 1 2 Qh,h, h H. H Importance: to dene Hilbert-space valued Brownian motion B and, hence, to construct Hilbert-space valued Itô integral with respect to B Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

7 Existence result is a priori not clear! Theorem (Peano) For each continuous function f : R B B dened on some open set V R B and for each point (t 0, x 0 ) V the Cauchy problem x (t) = f (t, x(t)), x(t 0 ) = x 0 has a solution which is dened on some neighborhood of t 0. Theorem (Godunov) Each Banach space in which Peano's theorem is true is nite dimensional. Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

8 The Stochastic Evolution Equation Setting: H and U are two separable Hilbert spaces (Ω, F, P) is a complete probability space B : [0, T ] Ω U is a trace-class Wiener process on U adapted to a normal ltration (F t ) t [0,T ] A : dom(a) H H is a densely dened, self-adjoint and positive denite linear operator with compact inverse. Aim: existence and uniqueness of a predictable stochastic process X : [0, T ] Ω H which solves the semilinear stochastic evolution equation driven by the Wiener process B dx (t) + (AX (t) + f (t, X (t))) dt = g(t, X (t)) db(t), X (0) = X 0, 0 t T for some nice functions f and g. Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

9 Existence and Uniqueness of Solution Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

10 Stochastic evolution equation (SEE): dx (t) + (AX (t) + f (t, X (t))) dt = g(t, X (t)) db(t), X (0) = X 0, 0 t T Denition (of mild solution of SEE) Let p 2. A predictable stochastic process X : [0, T ] Ω H is called a p-fold integrable mild solution of SEE if and, for all t [0, T ], it holds P-a.s. sup X (t) Lp (Ω;H) < t [0,T ] t t X (t) = E(t)X 0 E(t s)f (s, X (s)) ds + E(t s)g(s, X (s)) db(s), 0 0 where (E(t)) t [0, ) is the analytic semigroup on H generated by A, the rst integral is a Bochner integral and the second integral is the Hilbert-space valued Itô integral. Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

11 Existence-Uniqueness of Mild Solution Theorem (DaPrato-Zabczyk) Under some measurability, L p -regularity and linear growth conditions on X 0, f and g, there exists a unique p-fold integrable mild solution X : [0, T ] Ω H to SEE such ( that for every t [0, T ] and every s [0, 1) it holds that P X (t) H ) s = 1 with sup X (t) Lp (Ω;Ḣ s ) <, t [0,T ] where H s is the Hilbert space given by dom(a s/2 ). Furthermore, for every δ (0, 1 ) there exists a constant CY > 0 with 2 for all t 1, t 2 [0, T ]. X (t 1 ) X (t 2 ) L p (Ω;H) C t 1 t 2 δ Uniqueness here is in the sense of modication of stochastic processes. Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

12 A Stochastic Heat Equation Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

13 Stochastic heat equation with additive noise on the unit interval Setting: H = L 2 ([0, 1], B([0, 1]), dx; R) B is a trace-class Wiener process on H. Problem: Find a measurable mapping X : [0, T ] Ω R such that dx (t, x) = 2 X (t, x) dt + db(t, x) for all t (0, T ], x [0, 1] x 2 X (t, 0) = X (t, 1) = 0 for all t (0, T ] X (0, x) = X 0 (x) for all x [0, 1], where X 0 : Ω [0, 1] R is such that for almost all ω Ω, X 0 (ω, ) is a suciently smooth function which also satises the boundary conditions. Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

14 We translate into an abstract stochastic evolution equation on the Hilbert space H: where dx (t) + AX (t) dt = db(t), for all t [0, T ], X (0) = X 0, with dom(a) = H 1 0 (0, 1) H 2 (0, 1). A := 2 x 2 The mild solution is then given by the stochastic process X : [0, T ] Ω H with t X (t) = E(t)X 0 + E(t s) db(s) 0 Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

15 References R. Dalang et al. A Minicourse on Stochastic Partial Dierential Equations, Springer, L. Gawarecki and V. Mandrekar. Stochastic Dierential Equations in Innite Dimensions with Applications to Stochastic Partial Dierential Equations, Springer, G. da Prato and J. Zabczyk. Stochastic Equations in Innite Dimensions, Cambridge University Press, C. Prevot and M. Röckner. A Concise Course on Stochastic Partial Dierential Equations, Springer, S. Tappe. Foundation of the theory of semilinear stochastic partial dierential equations, Int. J. Stoch. Anal. Volume Article ID , Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16

16 Thank You!! Herry Pribawanto Suryawan (Math USD) SNAMA 2014 ITB 29. August / 16