The stochastic heat equation driven by a Gaussian noise: Markov property Doyoon Kim 1 , 2 Raluca Balan 1 1 University of Ottawa 2 University of Southern California (after August 30, 2007) June 4-8, 2007 R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 1 / 23
Outline History of the problem 1 The Framework 2 The noise The stochastic integral The equation and its solution RKHS 3 General characterization Bessel kernel Riesz kernel Germ Markov property 4 The necessary and sufficient condition Main result R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 2 / 23
History of the problem Germ Markov property The germ σ -field Let S be a subset in [ 0 , T ] × R d . F S : the σ -field generated by { u ( t , x ) : ( t , x ) ∈ S } � G S = F S . O open : O ⊃ S Definition The process { u ( t , x ) : ( t , x ) ∈ [ 0 , T ] × R d } is germ Markov if for every precompact open set A ⊂ [ 0 , T ] × R d , A ⊥ G A c | G ∂ A , G ¯ where ∂ A = ¯ A ∩ A c . R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 3 / 23
History of the problem Some cases investigated Donati-Martin and Nualart, 1994 � − ∆ u + f ( u ) = ˙ W , x ∈ D , u | ∂ D = 0 where D is a bounded domain in R d , d = 1 , 2 , 3. f is an affine function. Nualart and Pardoux, 1994 � u t = u xx + f ( u ) + ˙ ( t , x ) ∈ [ 0 , 1 ] 2 W , 0 ≤ x ≤ 1 ; u ( t , 0 ) = u ( t , 1 ) = 0 , 0 ≤ t ≤ 1 . u ( 0 , x ) = u 0 ( x ) , Dalang and Hou, 1997 u tt = ∆ u + ˙ L , where L is locally finite Lévy process. R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 4 / 23
History of the problem Some cases investigated Donati-Martin and Nualart, 1994 � − ∆ u + f ( u ) = ˙ W , x ∈ D , u | ∂ D = 0 where D is a bounded domain in R d , d = 1 , 2 , 3. f is an affine function. Nualart and Pardoux, 1994 � u t = u xx + f ( u ) + ˙ ( t , x ) ∈ [ 0 , 1 ] 2 W , 0 ≤ x ≤ 1 ; u ( t , 0 ) = u ( t , 1 ) = 0 , 0 ≤ t ≤ 1 . u ( 0 , x ) = u 0 ( x ) , Dalang and Hou, 1997 u tt = ∆ u + ˙ L , where L is locally finite Lévy process. R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 4 / 23
History of the problem Some cases investigated Donati-Martin and Nualart, 1994 � − ∆ u + f ( u ) = ˙ W , x ∈ D , u | ∂ D = 0 where D is a bounded domain in R d , d = 1 , 2 , 3. f is an affine function. Nualart and Pardoux, 1994 � u t = u xx + f ( u ) + ˙ ( t , x ) ∈ [ 0 , 1 ] 2 W , 0 ≤ x ≤ 1 ; u ( t , 0 ) = u ( t , 1 ) = 0 , 0 ≤ t ≤ 1 . u ( 0 , x ) = u 0 ( x ) , Dalang and Hou, 1997 u tt = ∆ u + ˙ L , where L is locally finite Lévy process. R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 4 / 23
The Framework The noise Outline History of the problem 1 The Framework 2 The noise The stochastic integral The equation and its solution RKHS 3 General characterization Bessel kernel Riesz kernel Germ Markov property 4 The necessary and sufficient condition Main result R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 5 / 23
The Framework The noise The noise Gaussian noise with spatial correlation (Dalang, 1999) M= { M ( ϕ ) , ϕ ∈ D (( 0 , T ) × R d ) } Gaussian process with covariance � ∞ � � E ( M ( ϕ ) M ( ψ )) = R d ϕ ( t , x ) f ( x − y ) ψ ( t , y ) dx dy dt R d 0 � T � = R d F ϕ ( t , ξ ) F ψ ( t , ξ ) µ ( d ξ ) dt := � ϕ, ψ � 0 0 Here f = F µ , where µ is a tempered measure on R d Riesz kernel f ( x ) = c α, d | x | − α � ∞ 0 s ( α − d ) / 2 − 1 e − s −| x | 2 / ( 4 s ) ds Bessel kernel f ( x ) = c α Heat kernel f ( x ) = c α, d e −| x | 2 / ( 4 α ) Poisson kernel f ( x ) = c α, d ( | x | 2 + α 2 ) − ( d + 1 ) / 2 R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 6 / 23
The Framework The noise The noise Gaussian noise with spatial correlation (Dalang, 1999) M= { M ( ϕ ) , ϕ ∈ D (( 0 , T ) × R d ) } Gaussian process with covariance � ∞ � � E ( M ( ϕ ) M ( ψ )) = R d ϕ ( t , x ) f ( x − y ) ψ ( t , y ) dx dy dt R d 0 � T � = R d F ϕ ( t , ξ ) F ψ ( t , ξ ) µ ( d ξ ) dt := � ϕ, ψ � 0 0 Here f = F µ , where µ is a tempered measure on R d Riesz kernel f ( x ) = c α, d | x | − α � ∞ 0 s ( α − d ) / 2 − 1 e − s −| x | 2 / ( 4 s ) ds Bessel kernel f ( x ) = c α Heat kernel f ( x ) = c α, d e −| x | 2 / ( 4 α ) Poisson kernel f ( x ) = c α, d ( | x | 2 + α 2 ) − ( d + 1 ) / 2 R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 6 / 23
The Framework The stochastic integral Outline History of the problem 1 The Framework 2 The noise The stochastic integral The equation and its solution RKHS 3 General characterization Bessel kernel Riesz kernel Germ Markov property 4 The necessary and sufficient condition Main result R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 7 / 23
The Framework The stochastic integral Stochastic integral with respect to M Space of deterministic integrands P ( d ) is the completion of D (( 0 , T ) × R d ) with respect to �· , ·� 0 0 (This is a space of distributions in x !) Stochastic integral � T � M ( ϕ ) = R d ϕ ( s , x ) M ( ds , dx ) 0 is defined an an isometry ϕ �→ M ( ϕ ) between P ( d ) and the Gaussian 0 space H M : ∀ ϕ, ψ ∈ P ( d ) E M ( ϕ ) M ( ψ ) = � ϕ, ψ � 0 , 0 R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 8 / 23
The Framework The equation and its solution Outline History of the problem 1 The Framework 2 The noise The stochastic integral The equation and its solution RKHS 3 General characterization Bessel kernel Riesz kernel Germ Markov property 4 The necessary and sufficient condition Main result R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 9 / 23
The Framework The equation and its solution Stochastic heat equation Stochastic heat equation driven by ˙ M � u t = ∆ u + ˙ [ 0 , T ] × R d M in . u ( 0 , x ) = 0 Mild solution � t � R d G ( t − s , x − y ) M ( ds , dy ) , u ( t , x ) = 0 where G ( t , x ) = ( 4 π t ) − d / 2 e −| x | 2 / ( 4 t ) , t > 0 , x ∈ R d Remark: G ( t − · , x − · ) ∈ P ( d ) R d ( 1 + | ξ | 2 ) − 1 µ ( d ξ ) < ∞ . � if and only if 0 R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 10 / 23
The Framework The equation and its solution Stochastic heat equation Stochastic heat equation driven by ˙ M � u t = ∆ u + ˙ [ 0 , T ] × R d M in . u ( 0 , x ) = 0 Mild solution � t � R d G ( t − s , x − y ) M ( ds , dy ) , u ( t , x ) = 0 where G ( t , x ) = ( 4 π t ) − d / 2 e −| x | 2 / ( 4 t ) , t > 0 , x ∈ R d Remark: G ( t − · , x − · ) ∈ P ( d ) R d ( 1 + | ξ | 2 ) − 1 µ ( d ξ ) < ∞ . � if and only if 0 R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 10 / 23
The Framework The equation and its solution Stochastic heat equation Stochastic heat equation driven by ˙ M � u t = ∆ u + ˙ [ 0 , T ] × R d M in . u ( 0 , x ) = 0 Mild solution � t � R d G ( t − s , x − y ) M ( ds , dy ) , u ( t , x ) = 0 where G ( t , x ) = ( 4 π t ) − d / 2 e −| x | 2 / ( 4 t ) , t > 0 , x ∈ R d Remark: G ( t − · , x − · ) ∈ P ( d ) R d ( 1 + | ξ | 2 ) − 1 µ ( d ξ ) < ∞ . � if and only if 0 R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 10 / 23
RKHS General characterization Outline History of the problem 1 The Framework 2 The noise The stochastic integral The equation and its solution RKHS 3 General characterization Bessel kernel Riesz kernel Germ Markov property 4 The necessary and sufficient condition Main result R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 11 / 23
RKHS General characterization The Reproducing kernel Hilbert space H u Row of isometries → H M = H u → H u P ( d ) 0 ϕ �→ M ( ϕ ) = Y �→ h Y ( t , x ) = E ( Yu ( t , x )) H u = span of { u ( t , x ) : ( t , x ) ∈ [ 0 , T ] × R d } in L 2 (Ω) H M = { M ( ϕ ); ϕ ∈ P ( d ) } 0 Definition of H u : H u = { h ( t , x ) = E ( M ( ϕ ) u ( t , x )) : ϕ ∈ P ( d ) } 0 and � h , g � H u = E ( M ( ϕ ) M ( ψ )) = � ϕ, ψ � 0 , R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 12 / 23
RKHS General characterization The Reproducing kernel Hilbert space H u Row of isometries → H M = H u → H u P ( d ) 0 ϕ �→ M ( ϕ ) = Y �→ h Y ( t , x ) = E ( Yu ( t , x )) H u = span of { u ( t , x ) : ( t , x ) ∈ [ 0 , T ] × R d } in L 2 (Ω) H M = { M ( ϕ ); ϕ ∈ P ( d ) } 0 Definition of H u : H u = { h ( t , x ) = E ( M ( ϕ ) u ( t , x )) : ϕ ∈ P ( d ) } 0 and � h , g � H u = E ( M ( ϕ ) M ( ψ )) = � ϕ, ψ � 0 , R. Balan and D. Kim (University of Ottawa) SPDE and Markov property Large Deviations, Ann Arbor 12 / 23
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