Hlder continuity for the nonlinear stochastic heat equation with - PowerPoint PPT Presentation

Hölder continuity for the nonlinear stochastic heat equation with rough initial conditions Le CHEN Department of Mathematics University of Utah Joint work with Prof. Robert C. DALANG To appear in Stochastic Partial Differential Equations: Analysis and Computations, 2014 18–20, May 2014 Frontier Probability Days Tucson, Arizona 1 / 12

Stochastic Heat Equation (SHE) � ∂  � ∂ 2 ∂ t − ν u ( t , x ) = ρ ( u ( t , x )) ˙ W ( t , x ) , x ∈ R , t ∈ R ∗  + ,  ∂ x 2 2 (SHE)   u ( 0 , · ) = µ ( · ) , ˙ W is the space-time white noise; ρ is Lipschitz continuous; µ is the initial measure (to be specified). �� u ( t , x ) = J 0 ( t , x ) + ρ ( u ( s , y )) G ν ( t − s , x − y ) W ( d s , d y ) . [ 0 , t ] × R � � − x 2 1 G ν ( t , x ) = √ exp 2 t 2 πν t J 0 ( t , x ) := ( µ ∗ G ν ( t , · ))( x ) 2 / 12

Definition of random field solution �� u ( t , x ) = J 0 ( t , x ) + ρ ( u ( s , y )) G ν ( t − s , x − y ) W ( d s , d y ) . (SHE) [ 0 , t ] × R � �� � := I ( t , x ) Definition (Random field solution) u = ( u ( t , x ) : ( t , x ) ∈ R ∗ + × R ) is called a random field solution to (SHE) if (1) u is adapted, i.e., for all ( t , x ) ∈ R ∗ + × R , u ( t , x ) is F t -measurable; (2) u is jointly measurable with respect to B ( R ∗ + × R ) × F ; � � G 2 ν ⋆ || ρ ( u ) || 2 ( t , x ) < + ∞ for all ( t , x ) ∈ R ∗ (3) + × R , and 2 ( t , x ) �→ I ( t , x ) : R ∗ + × R �→ L 2 (Ω) is continuous; (4) u satisfies (SHE) almost surely, for all ( t , x ) ∈ R ∗ + × R . � t � � � G 2 ν ⋆ || ρ ( u ) || 2 ( t , x ) := d s d y G 2 ν ( t − s , x − y ) || ρ ( u ( s , y )) || 2 2 . 2 0 R 3 / 12

Rough initial data Initial data has a bounded density function (Walsh theory [2]) , (Bounded initial data) i.e., µ ( d x ) = f ( x ) d x with f ∈ L ∞ ( R ) . Measure-valued initial data (Ch. & Dalang [1]) . � � � e − ax 2 | µ | ( d x ) < + ∞ , ∀ a > 0 M H ( R ) := signed Borel meas. µ , s.t. R � 1 e − ( x − y ) 2 ( | µ | ∗ G ν ( t , · )) ( x ) := | µ | ( d y ) < + ∞ ∀ t > 0 , ∀ x ∈ R . √ 2 ν t 2 πν t R Initial data cannot go beyond measures. No random field solution for δ ′ 0 . [1] L. Chen, R. Dalang, Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions, Ann. Probab. , (accepted, pending revision), 2014. [2] J. B. Walsh. An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV—1984 , pp. 265–439. Springer, Berlin, 1986. 4 / 12

Rough initial data Initial data has a bounded density function (Walsh theory [2]) , (Bounded initial data) i.e., µ ( d x ) = f ( x ) d x with f ∈ L ∞ ( R ) . Measure-valued initial data (Ch. & Dalang [1]) . � � � e − ax 2 | µ | ( d x ) < + ∞ , ∀ a > 0 M H ( R ) := signed Borel meas. µ , s.t. R � 1 e − ( x − y ) 2 ( | µ | ∗ G ν ( t , · )) ( x ) := | µ | ( d y ) < + ∞ ∀ t > 0 , ∀ x ∈ R . √ 2 ν t 2 πν t R Initial data cannot go beyond measures. No random field solution for δ ′ 0 . [1] L. Chen, R. Dalang, Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions, Ann. Probab. , (accepted, pending revision), 2014. [2] J. B. Walsh. An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV—1984 , pp. 265–439. Springer, Berlin, 1986. 4 / 12

Rough initial data Initial data has a bounded density function (Walsh theory [2]) , (Bounded initial data) i.e., µ ( d x ) = f ( x ) d x with f ∈ L ∞ ( R ) . Measure-valued initial data (Ch. & Dalang [1]) . � � � e − ax 2 | µ | ( d x ) < + ∞ , ∀ a > 0 M H ( R ) := signed Borel meas. µ , s.t. R � 1 e − ( x − y ) 2 ( | µ | ∗ G ν ( t , · )) ( x ) := | µ | ( d y ) < + ∞ ∀ t > 0 , ∀ x ∈ R . √ 2 ν t 2 πν t R Initial data cannot go beyond measures. No random field solution for δ ′ 0 . [1] L. Chen, R. Dalang, Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions, Ann. Probab. , (accepted, pending revision), 2014. [2] J. B. Walsh. An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV—1984 , pp. 265–439. Springer, Berlin, 1986. 4 / 12

Rough initial data Initial data has a bounded density function (Walsh theory [2]) , (Bounded initial data) i.e., µ ( d x ) = f ( x ) d x with f ∈ L ∞ ( R ) . Measure-valued initial data (Ch. & Dalang [1]) . � � � e − ax 2 | µ | ( d x ) < + ∞ , ∀ a > 0 M H ( R ) := signed Borel meas. µ , s.t. R � 1 e − ( x − y ) 2 ( | µ | ∗ G ν ( t , · )) ( x ) := | µ | ( d y ) < + ∞ ∀ t > 0 , ∀ x ∈ R . √ 2 ν t 2 πν t R Initial data cannot go beyond measures. No random field solution for δ ′ 0 . J 0 ( t , x ) ∈ C ∞ ( R ∗ + × R ) I ( t , x ) ∈ C ? , ? ( R ∗ + × R ) 4 / 12

Some notation for locally Hölder continuous functions Given a subset D ⊆ R + × R and positive constants β 1 , β 2 , denote by C β 1 ,β 2 ( D ) the set of functions v : R + × R → R with the following property: For each compact subset ˜ D ⊂ D , ∃ C s.t. for all ( t , x ) and ( s , y ) ∈ ˜ D , � | t − s | β 1 + | x − y | β 2 � | v ( t , x ) − v ( s , y ) | ≤ C . � � C β 1 − ,β 2 − ( D ) := C α 1 ,α 2 ( D ) . 0 <α 1 <β 1 0 <α 2 <β 2 5 / 12

u ( t , x ) = J 0 ( t , x ) + I ( t , x ) � � � e − ax 2 | µ | ( d x ) < + ∞ , ∀ a > 0 M H ( R ) := signed Borel meas. µ , s.t. R Theorem (1) If µ ∈ M H ( R ) , then I ∈ C 1 2 − ( R ∗ + × R ) a.s. Therefore, 4 − , 1 2 − ( R ∗ u ∈ C 1 + × R ) , a.s. 4 − , 1 6 / 12

u ( t , x ) = J 0 ( t , x ) + I ( t , x ) � � � e − ax 2 | µ | ( d x ) < + ∞ , ∀ a > 0 M H ( R ) := signed Borel meas. µ , s.t. R � � | f ( x ) | e −| x | a < + ∞ M ∗ µ ( d x ) = f ( x ) d x , s.t. ∃ a ∈ ] 1 , 2 [ , sup H ( R ) := . x ∈ R Theorem (1) If µ ∈ M H ( R ) , then I ∈ C 1 2 − ( R ∗ + × R ) a.s. Therefore, 4 − , 1 2 − ( R ∗ u ∈ C 1 + × R ) , a.s. 4 − , 1 (2) If µ ∈ M ∗ H ( R ) with µ ( d x ) = f ( x ) d x, then I ∈ C 1 2 − ( R + × R ) , a.s. 4 − , 1 Moreover, (i) If f is continuous, then u ∈ C ( R + × R ) ∩ C 1 2 − ( R ∗ a.s. + × R ) , 4 − , 1 (ii) If f is α -Hölder continuous, then u ∈ C ( α 2 ) − ( R + × R ) ∩ C 1 2 − ( R ∗ a.s. + × R ) , 2 ∧ 1 4 ) − , ( α ∧ 1 4 − , 1 6 / 12

Difficulties with rough initial data Conventional method: For p > 1 and q = p / ( p − 1 ) , t < t ′ ( ρ ( u ) = u ), Set G ν ( t − s , x − y ; t ′ − s , x ′ − y ) = G ν ( t − s , x − y ) − G ν ( t ′ − s , x ′ − y ) . � � � � 2 p �� � � � � �� � �� � � 2 p G ν ( t − s , x − y ; t ′ − s , x ′ − y ) u ( s , y ) W ( d s d y ) � I ( t , x ) − I ( t ′ , x ′ ) � � � � 2 p = � � � � [ 0 , t ′ ] × R � � � � 2 p �� t ′ � p / q � t ′ � � � � 1 + || u ( s , y ) || 2 p ≤ C G ν ( · · · ) 2 d s d y G 2 d s d y ν · 2 p 0 0 R R � p � �� t ′ � � 1 + || u ( s , y ) || 2 p G ν ( · · · ) 2 d s d y ≤ C sup sup 2 p s ∈ [ 0 , t ′ ] y ∈ R 0 R � � � | t ′ − t | p / 2 + | x ′ − x | p � 1 + || u ( s , y ) || 2 p ≤ C sup sup 2 p s ∈ [ 0 , t ′ ] y ∈ R [1] Robert C. Dalang. The stochastic wave equation. In A minicourse on stochastic partial differential equations , volume 1962 of Lecture Notes in Math. Springer, Berlin, 2009. [2] Marta Sanz-Solé and Mònica Sarrà. Hölder continuity for the stochastic heat equation with spatially correlated noise. In Seminar on Stochastic Analysis, Random Fields and Applications, III , volume 52 of Progr. Probab. . Birkhäuser, Basel, 2002. [3] Tokuzo Shiga. Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canad. J. Math., 46(2):415–437, 1994. 7 / 12

Difficulties with rough initial data Conventional method: For p > 1 and q = p / ( p − 1 ) , t < t ′ ( ρ ( u ) = u ), Set G ν ( t − s , x − y ; t ′ − s , x ′ − y ) = G ν ( t − s , x − y ) − G ν ( t ′ − s , x ′ − y ) . � � � � 2 p �� � � � � �� � � �� � 2 p G ν ( t − s , x − y ; t ′ − s , x ′ − y ) u ( s , y ) W ( d s d y ) � I ( t , x ) − I ( t ′ , x ′ ) � � � � 2 p = � � � � [ 0 , t ′ ] × R � � � � 2 p �� t ′ � p / q � t ′ � � � � 1 + || u ( s , y ) || 2 p ≤ C G ν ( · · · ) 2 d s d y G 2 d s d y ν · 2 p 0 0 R R � p � �� t ′ � � 1 + || u ( s , y ) || 2 p G ν ( · · · ) 2 d s d y ≤ C sup sup 2 p s ∈ [ 0 , t ′ ] y ∈ R 0 R � � � | t ′ − t | p / 2 + | x ′ − x | p � 1 + || u ( s , y ) || 2 p ≤ C sup sup 2 p s ∈ [ 0 , t ′ ] y ∈ R Tails ⇒ integrability of x at ±∞ . Measure ⇒ integrability of t at 0: e.g., µ = δ 0 , = C s e − y 2 1 || u ( s , y ) || 2 2 p ≥ || u ( s , y ) || 2 2 ≥ G ν 2 ( s , y ) ν s ⇒ p < 3 / 2 . √ 4 πν s 7 / 12