The pathwise solution of an SPDE with fractal noise Elena Issoglio - PowerPoint PPT Presentation

Introduction of the problem Interpretation of the involved objects The theorem of existence and uniqueness The pathwise solution of an SPDE with fractal noise Elena Issoglio Friedrich-Schiller Universit¨ at, Jena March 15, 2010 This work has been financially supported by Marie Curie Initial Training Network (ITN), FP7-PEOPLE-2007-1-1-ITN, no. 213841-2, “Deterministic and Stochastic Controlled Systems and Applications” Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem Interpretation of the involved objects The theorem of existence and uniqueness Outline Introduction of the problem The Cauchy problem with Dirichlet conditions The abstract Cauchy problem Interpretation of the involved objects The Dirichlet Laplacian Fractional Sobolev Spaces The noise ∇ B H The theorem of existence and uniqueness Definition of the mild solution The integral operator The main result Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem The Cauchy problem with Dirichlet conditions Interpretation of the involved objects The abstract Cauchy problem The theorem of existence and uniqueness Introduction of the problem The Cauchy problem with Dirichlet conditions The abstract Cauchy problem Interpretation of the involved objects The Dirichlet Laplacian Fractional Sobolev Spaces The noise ∇ B H The theorem of existence and uniqueness Definition of the mild solution The integral operator The main result Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem The Cauchy problem with Dirichlet conditions Interpretation of the involved objects The abstract Cauchy problem The theorem of existence and uniqueness Stochastic transport equation  ∂ u ∂ t ( t , x ) = ∆ u ( t , x ) + ∇ B H ( x ) · ∇ u ( t , x ) , t ∈ (0 , T ] , x ∈ D  u (0 , x ) = u 0 ( x ) , x ∈ D u ( t , x ) = 0 , t ∈ (0 , T ] , x ∈ ∂ D  ◮ u ( t , x ): unknown concentration of the substance at time t and position x ◮ D ⊂ R d : bounded domain with smooth boundary ◮ B H ( x ) = B H ( x , ω ): suitable stochastic noise — > In this session B H ( x ) will be a fractional Brownian field with Hurs index 0 < H < 1. Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem The Cauchy problem with Dirichlet conditions Interpretation of the involved objects The abstract Cauchy problem The theorem of existence and uniqueness Fractional Brownian motion ( d = 1) { B H ( x ) , x ∈ R + } is a fractional Brownian motion with Hurst parameter H ∈ (0 , 1) if it is a centred Gaussian process with covariance function given by y ) = 1 � x 2 H + y 2 H − | x − y | 2 H � E ( B H x B H 2 ◮ homogeneous increments but not indipendent (negatively correlated if H < 1 / 2, positively if H > 1 / 2) ◮ there exists a version of B H with α -H¨ older continuous trajectories, for α < H ◮ if H � = 1 / 2 then B H is not a semimartingale: Itˆ o-type theory can not be used Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem The Cauchy problem with Dirichlet conditions Interpretation of the involved objects The abstract Cauchy problem The theorem of existence and uniqueness The abstract Cauchy problem ◮ X Banach space ◮ A linear operator on X ◮ A generates a semigroup ( T ( t ) , t ≥ 0) ◮ f : [0 , T ) → X given function The abstract Cauchy problem is � d u ( t ) = Au ( t ) + f ( t ) , t > 0 (1) d t u (0) = h where u is a X -valued function. We define the mild solution as the function � t u ( t ) = T ( t ) h + T ( t − s ) f ( s ) d s . 0 Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem The Cauchy problem with Dirichlet conditions Interpretation of the involved objects The abstract Cauchy problem The theorem of existence and uniqueness The stochastic transport equation as abstract Cauchy problem ◮ X infinite dimensional Banach space ◮ h ∈ X function depending on x ∈ D ⊂ R d : h ( x ) The function u ( t , x ) is now interpreted only as function of time and takes values in X . u : [0 , T ] → X �→ u ( t ) t where u ( t ) is a function of x defined by u ( t ) : D → R x �→ u ( t )( x ) where u ( t )( x ) := u ( t , x ). Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem The Cauchy problem with Dirichlet conditions Interpretation of the involved objects The abstract Cauchy problem The theorem of existence and uniqueness The Cauchy problem with Dirichlet conditions is now rewritten as � u t = ∆ D u + ∇ B H · ∇ u , t ∈ (0 , T ] u (0) = u 0 where ◮ u t indicates the derivative of u with respect to time ◮ u 0 ( x ) := u (0 , x ) = u 0 ( x ) ◮ ∆ D is the Dirichlet laplacian on D : it encodes the condition u ( t )( x ) ≡ 0 for x ∈ ∂ D ◮ ∇ B H · ∇ u has still to be defined since ∇ B H is a distribution ◮ pathwise interpretation : fix ω ∈ Ω and study the equation for almost every ω Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem The Dirichlet Laplacian Interpretation of the involved objects Fractional Sobolev Spaces The noise ∇ B H The theorem of existence and uniqueness Introduction of the problem The Cauchy problem with Dirichlet conditions The abstract Cauchy problem Interpretation of the involved objects The Dirichlet Laplacian Fractional Sobolev Spaces The noise ∇ B H The theorem of existence and uniqueness Definition of the mild solution The integral operator The main result Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem The Dirichlet Laplacian Interpretation of the involved objects Fractional Sobolev Spaces The noise ∇ B H The theorem of existence and uniqueness ∆ e ∆ D : probabilistic interpretation ◮ Laplacian ∆ on R d . generates a semigroup { T t } t ≥ 0 � T t u ( x ) = R d p ( t , x , y ) u ( y ) d y where p ( t , x , y ) is the heat kernel −| x − y | 2 1 � � p ( t , x , y ) = (2 π t ) d / 2 exp 2 t < — > Brownian motion on R d where p ( t , x , y ) = P x ( B t ∈ d y ) is the transition probability density function of a Brownian motion { B t } t ≥ 0 . Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem The Dirichlet Laplacian Interpretation of the involved objects Fractional Sobolev Spaces The noise ∇ B H The theorem of existence and uniqueness ◮ Laplacian ∆ D . generates a semigroup { P t } t ≥ 0 � P t u ( x ) = p D ( t , x , y ) u ( y ) d y D where p D ( t , x , y ) = p ( t , x , y ) − r ( t , x , y ) with r ( t , x , y ) = E x [ p ( t − τ D , B τ D , y ); τ D < t ] and τ D is the first exit time from D . < — > killed Brownian motion (killed at exiting D ) � B t if t < τ D ¯ B t = ζ if t > τ D . Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem The Dirichlet Laplacian Interpretation of the involved objects Fractional Sobolev Spaces The noise ∇ B H The theorem of existence and uniqueness ∆ e ∆ D : analytical interpretation ◮ The semigroup T t acts (for instance) on L 2 ( R d ). In this case we have dom(∆) = W 2 ( R d ) ⊂ W 0 ( R d ) = L 2 ( R d ). Property : λ − ∆ : H γ ( R d ) → H γ − 2 ( R d ), for every γ ∈ R , λ > 0. ◮ The semigroup P t and its generator act on a space restricted to D which contains information on ∂ D . — > fractional Sobolev spaces on D . Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem The Dirichlet Laplacian Interpretation of the involved objects Fractional Sobolev Spaces The noise ∇ B H The theorem of existence and uniqueness Fractional Sobolev spaces on R d ◮ Sobolev spaces. Let m ∈ N � � W m p ( R d ) := f ∈ S ′ ( R d ) : ∂ γ f ∈ L p ( R d ) for every | γ | ≤ m | γ |≤ m � ∂ γ f | L p � p � 1 / p �� endowed with the norm � f | W m p � := ◮ Fractional Sobolev Spaces. Let α ∈ R � f ) ∨ ∈ L p ( R d ) � H α f ∈ S ′ ( R d ) : ((1 + | ξ | 2 ) α/ 2 ˆ p ( R d ) := p ( R d ) � = � ((1 + | ξ | 2 ) α/ 2 ˆ endowed with the norm � f | H α f ) ∨ � L p Property: if α = m ∈ N then H m p ( R d ) = W m p ( R d ). Elena Issoglio The pathwise solution of an SPDE with fractal noise

Introduction of the problem The Dirichlet Laplacian Interpretation of the involved objects Fractional Sobolev Spaces The noise ∇ B H The theorem of existence and uniqueness Fractional Sobolev spaces on D Let α ∈ R ◮ define � � H α f ∈ S ′ ( D ) : ∃ g ∈ H α p ( R d ) s.t. g | D = f p ( D ) := endowed with the norm � � � f | H α � g | H α p ( R d ) � s.t. g ∈ H α p ( R d ) and g | D = f p ( D ) � = inf ◮ define ˜ p ( R d ) : supp( f ) ⊂ ¯ H α � f ∈ H α � p ( D ) := D endowed with the norm � · | H α p ( R d ) � Elena Issoglio The pathwise solution of an SPDE with fractal noise