Lipschitz continuity for solutions of Hamilton-Jacobi equation with Ornstein-Uhlenbeck operator

Transcription

1 Lipschitz continuity for solutions of Hamilton-Jacobi equation with Ornstein-Uhlenbeck operator Thi Tuyen Nguyen Ph.D student of University of Rennes 1 Joint work with: Prof. E. Chasseigne(University of Tours) and Prof. O. Ley(Insa-Rennes) March 23, 2016 hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 1 / 16

2 Outline of the presentation. Stochastic control problems and the associated PDE. Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 2 / 16

3 Outline of the presentation. Stochastic control problems and the associated PDE. Motivations to obtain Lipschitz regularity. Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 2 / 16

4 Outline of the presentation. Stochastic control problems and the associated PDE. Motivations to obtain Lipschitz regularity. Stationary problem and assumptions on the datas. Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 2 / 16

5 Outline of the presentation. Stochastic control problems and the associated PDE. Motivations to obtain Lipschitz regularity. Stationary problem and assumptions on the datas. Statement and application of Lipschitz regularity result. Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 2 / 16

6 Outline of the presentation. Stochastic control problems and the associated PDE. Motivations to obtain Lipschitz regularity. Stationary problem and assumptions on the datas. Statement and application of Lipschitz regularity result. Proof of the Lipschitz regularity theorem. Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 2 / 16

7 Stochastic control problems and the associated PDE We consider the stochastic differential equation { dx t = (g(x t, γ t) αx t)dt + 2σ(X t)dw t X 0 = x R N, (W t) is an N-dimentional standard Brownian motion defined on a filtered probability space (Ω, F, F t, P), the control γ t takes value on a compact set A R N which is defined as the collection of all F t-progressively measurable, g is a continuous vector on R N, σ is a continuous N N matrix and α-terms is called Ornstein-Uhlenbeck operator (α > 0). hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 3 / 16

8 Stochastic control problems and the associated PDE We consider the stochastic differential equation { dx t = (g(x t, γ t) αx t)dt + 2σ(X t)dw t X 0 = x R N, (W t) is an N-dimentional standard Brownian motion defined on a filtered probability space (Ω, F, F t, P), the control γ t takes value on a compact set A R N which is defined as the collection of all F t-progressively measurable, g is a continuous vector on R N, σ is a continuous N N matrix and α-terms is called Ornstein-Uhlenbeck operator (α > 0). Value functions ( infinite horizon: u λ (x) = inf E γ 0 ( t finite horizon: u(x, t) = inf E γ 0 ) e λs L(X s, γ s)ds, (1) ) L(X s, γ s)ds, (2) where L is a running cost or Lagrangian function defined L(X s, γ s) = l(x s, γ x) + f (X s), l(x,.) is a bounded function in x and f is possibly an unbounded function. hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 3 / 16

9 Stochastic control problems and the associated PDE We consider the stochastic differential equation { dx t = (g(x t, γ t) αx t)dt + 2σ(X t)dw t X 0 = x R N, (W t) is an N-dimentional standard Brownian motion defined on a filtered probability space (Ω, F, F t, P), the control γ t takes value on a compact set A R N which is defined as the collection of all F t-progressively measurable, g is a continuous vector on R N, σ is a continuous N N matrix and α-terms is called Ornstein-Uhlenbeck operator (α > 0). Value functions ( infinite horizon: u λ (x) = inf E γ 0 ( t finite horizon: u(x, t) = inf E γ 0 ) e λs L(X s, γ s)ds, (1) ) L(X s, γ s)ds, (2) where L is a running cost or Lagrangian function defined L(X s, γ s) = l(x s, γ x) + f (X s), l(x,.) is a bounded function in x and f is possibly an unbounded function. By the Dynamic Programming Principle, u λ (x), u(x, t) are viscosity solution of λu λ (x) tr(σ(x)σ T (x)d 2 u λ (x)) + αx.du λ (x) + H(x, Du λ (x)) = f (x) u t(x, t) tr(σ(x)σ T (x)d 2 u(x, t)) + αx.du(x, t) + H(x, Du(x, t)) = f (x) respectively, where in R N (CP) H(x, p) = max{ g(x, a).p l(x, a)}. (3) a A hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 3 / 16

10 Motivations Motivations to obtain Lipschitz regularity By letting λ 0 in (1) or t in (2), it is called ergodic control problem. Question: Under which conditions on (g, σ), Question 1: λu λ? as λ 0, u(., t) Question 2:? as t. t Some references: Arisawa(1997), Arisawa-Lions (1998), Lions-Papanicolaou-Varadhan, Fujita-Ishii-Loreti(2006),... Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 4 / 16

11 Stationary problem and assumptions on the datas Stationary problem For λ (0, 1), we consider the stationary problem λu λ (x) tr(σ(x)σ T (x)d 2 u λ (x)) + αx.du λ (x) + H(x, Du λ (x)) = f (x) in R N. (SP) Here σ is a diffusion matrix. Let µ > 0 we define the function φ µ C (R N ) by φ µ(x) = e µ x 2 +1 for x R N. hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 5 / 16

12 Stationary problem and assumptions on the datas Stationary problem For λ (0, 1), we consider the stationary problem λu λ (x) tr(σ(x)σ T (x)d 2 u λ (x)) + αx.du λ (x) + H(x, Du λ (x)) = f (x) in R N. (SP) Here σ is a diffusion matrix. Let µ > 0 we define the function φ µ C (R N ) by We introduce the class of solution for (SP) φ µ(x) = e µ x 2 +1 for x R N. E µ(r N ) = {v : R N R : lim x + Hereafter we use φ instead of φ µ for simplicity of notation. v(x) φ µ(x) = 0}. hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 5 / 16

13 Stationary problem and assumptions on the datas Assumptions on the datas Diffusion matrix: { σ C(R N, M N ), there exists C σ, L σ > 0 such that σ(x) C σ, σ(x) σ(y) L σ x y x R N. { (ellipticity) There exists ν > 0 such that νi σ(x)σ(x) T x R N, (4) (5) ellipticity means that the diffusion is nondegenerate in the stochastic differential equation. hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 6 / 16

14 Stationary problem and assumptions on the datas Assumptions on the datas Diffusion matrix: { σ C(R N, M N ), there exists C σ, L σ > 0 such that σ(x) C σ, σ(x) σ(y) L σ x y x R N. { (ellipticity) There exists ν > 0 such that νi σ(x)σ(x) T x R N, (4) (5) ellipticity means that the diffusion is nondegenerate in the stochastic differential equation. Assumption on H: H(x, p) C H (1 + p ), x, p R N. (6) This is very general assumption for the Hamiltonian H. From the definition H in (3), it is enough to make sure that g and l are bounded in x. Assumption on f : { For f E µ(r N ), there exists C f, µ > 0 such that f (x) f (y) C f (φ µ(x) + φ µ(y)) x y, x, y R N. (7) hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 6 / 16

15 Statement and application of Lipschitz regularity result Statement of Lipschitz regularity result Theorem 1 Let µ > 0, u λ C(R N ) E µ(r N ) be a solution of (SP). Assume that (4), (5), (6) and (7) hold. For any α > 0, there exists a constant C > 0 independent of λ such that u λ (x) u λ (y) C x y (φ(x) + φ(y)), x, y R N, λ (0, 1) (LR). Some recent results: Fujita-Ishii-Loreti (2006) Fujita-Loreti (2009) Bardi-Cesaroni-Ghilli (2015). hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 7 / 16

16 Statement and application of Lipschitz regularity result Application to ergodic peoblem We consider the ergodic control problem c tr(σ(x)σ T (x)d 2 v(x)) + αx.dv(x) + H(x, Dv(x)) = f (x) in R N, (EP) where the unknown is a pair of a constant c and a function v. Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 8 / 16

17 Statement and application of Lipschitz regularity result Application to ergodic peoblem We consider the ergodic control problem c tr(σ(x)σ T (x)d 2 v(x)) + αx.dv(x) + H(x, Dv(x)) = f (x) in R N, (EP) where the unknown is a pair of a constant c and a function v. a) Answer question 1. Theorem 2 Under the assumption of Theorem 1, there is a solution (c, v) R C(R N ) of (EP). Let (c 1, v 1), (c 2, v 2) R C(R N ) are two solutions of (EP), then c 1 = c 2 and there is a constant C R such that v 1 v 2 = C in R N. hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 8 / 16

18 Statement and application of Lipschitz regularity result Application to ergodic peoblem We consider the ergodic control problem c tr(σ(x)σ T (x)d 2 v(x)) + αx.dv(x) + H(x, Dv(x)) = f (x) in R N, (EP) where the unknown is a pair of a constant c and a function v. a) Answer question 1. Theorem 2 Under the assumption of Theorem 1, there is a solution (c, v) R C(R N ) of (EP). Let (c 1, v 1), (c 2, v 2) R C(R N ) are two solutions of (EP), then c 1 = c 2 and there is a constant C R such that v 1 v 2 = C in R N. Proof: 1. Existence of solution. We first prove that there exists a constant C > 0 independent of λ such that We consider λu λ (x) C on balls of R N. (8) max R N {uλ (x) φ(x)} = u λ (y) φ(y), for some y R N. hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 8 / 16

19 Statement and application of Lipschitz regularity result Proof (Cont.) Since u λ is a viscosity solution and hence subsolution of (SP). Then, at the maximum point y, we have λu λ (y) tr(σ(y)σ T (y)d 2 φ(y)) + αy.dφ(y) + H(y, Dφ(y)) f (y). Moreover, φ satisfies tr(σ(y)σ T (y)d 2 φ(y)) + αy.dφ(y) C H Dφ(y) φ(y) B, here C H, B > 0. Therefore, using the sublinearity of H we obtain λu λ (y) f (y) φ(y) + B + C H. hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 9 / 16

20 Statement and application of Lipschitz regularity result Proof (Cont.) Since u λ is a viscosity solution and hence subsolution of (SP). Then, at the maximum point y, we have λu λ (y) tr(σ(y)σ T (y)d 2 φ(y)) + αy.dφ(y) + H(y, Dφ(y)) f (y). Moreover, φ satisfies tr(σ(y)σ T (y)d 2 φ(y)) + αy.dφ(y) C H Dφ(y) φ(y) B, here C H, B > 0. Therefore, using the sublinearity of H we obtain λu λ (y) f (y) φ(y) + B + C H. Since y is a maximum point of u(x) φ(x) for x R N, then using the above inequality we have λu λ (x) λφ(x) + λu λ (y) λφ(y) x R N λφ(x) φ(y) + B + C H + f (y) λφ(y) λ (0, 1) φ(x) + B + C H + f (y) φ(y) C x R N. The proof for the opposite inequality is the same by considering min R N {u λ (x) + φ(x)}. hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 9 / 16

21 Statement and application of Lipschitz regularity result Proof (Cont.) Since u λ is a viscosity solution and hence subsolution of (SP). Then, at the maximum point y, we have λu λ (y) tr(σ(y)σ T (y)d 2 φ(y)) + αy.dφ(y) + H(y, Dφ(y)) f (y). Moreover, φ satisfies tr(σ(y)σ T (y)d 2 φ(y)) + αy.dφ(y) C H Dφ(y) φ(y) B, here C H, B > 0. Therefore, using the sublinearity of H we obtain λu λ (y) f (y) φ(y) + B + C H. Since y is a maximum point of u(x) φ(x) for x R N, then using the above inequality we have λu λ (x) λφ(x) + λu λ (y) λφ(y) x R N λφ(x) φ(y) + B + C H + f (y) λφ(y) λ (0, 1) φ(x) + B + C H + f (y) φ(y) C x R N. The proof for the opposite inequality is the same by considering min R N {u λ (x) + φ(x)}. Now we set v λ (x) = u λ (x) u λ (0) and using (LR) we have v λ (x) C x (φ(x) + 1) v λ (x) v λ (y) C x y (φ(x) + φ(y)). Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. Ley(Insa-Rennes) March 23, operator 2016 ) 9 / 16

22 Statement and application of Lipschitz regularity result Proof (Cont.) Therefore {v λ } λ (0,1) is a uniformly bounded and equi-continuous family on any balls of R N. Then by Ascoli s theorem, we can choose a sequence {λ j } j N (0, 1) such that And from (8) we have Note that v λ j is a solution of v λ j v in C(R N ) as λ j 0. λ j u λ j (x) c R uniformly on balls of R N. λ j v λ j tr(σ(x)σ T (x)d 2 v λ j (x)) + αx.dv λ j (x) + H(x, Dv λ j (x)) = f (x) λ j u λ j (0) λ j 0 tr(σ(x)σ T (x)d 2 v) + αx.dv(x) + H(x, Dv(x)) = f (x) c. By stability of viscosity solutions we find that (c, v) is a solution of ergodic problem (EP). 2. Uniqueness. See in Fujita-Ishii-Loreti (2006). Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 10 / 16

23 Statement and application of Lipschitz regularity result Proof (Cont.) Therefore {v λ } λ (0,1) is a uniformly bounded and equi-continuous family on any balls of R N. Then by Ascoli s theorem, we can choose a sequence {λ j } j N (0, 1) such that And from (8) we have Note that v λ j is a solution of v λ j v in C(R N ) as λ j 0. λ j u λ j (x) c R uniformly on balls of R N. λ j v λ j tr(σ(x)σ T (x)d 2 v λ j (x)) + αx.dv λ j (x) + H(x, Dv λ j (x)) = f (x) λ j u λ j (0) λ j 0 tr(σ(x)σ T (x)d 2 v) + αx.dv(x) + H(x, Dv(x)) = f (x) c. By stability of viscosity solutions we find that (c, v) is a solution of ergodic problem (EP). 2. Uniqueness. See in Fujita-Ishii-Loreti (2006). b) Answer question 2. Theorem 3 Let u C(R N [0, T )) be a solution of (CP) with initial data u(x, 0) = u 0(x). Let (c, v) R C(R N ) be a solution of (EP). Let L > 0 such that u 0(x) v(x) L in R N. Then u(x, t) c, as t. t Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 10 / 16

24 Proof. Statement and application of Lipschitz regularity result We easily check that v(x) + ct L and v(x) + ct + L are solutions of (CP) with the initial condition replaced v(x) ± L. Then applying comparison principle for (CP) we get This implies v(x) + ct L u(x, t) v(x) + ct + L. v(x) L t u(x, t) t c v(x) + L. t Sending t to we obtain the result. Remark: More precise result-long time behavior. Let u is a solution of (CP), (c, v) is a solution of (EP). There exists a constant a R such that u(x, t) (ct + v(x)) a as t locally uniformly in R N. Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 11 / 16

25 Proof of Lipschitz regumarity theorem Proof of Theorem 1 Let δ, A, C 1 > 0 and a C 2 concave and increasing function ψ : R + R + with ψ(0) = 0 which is constructed as following hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 12 / 16

26 Proof of Lipschitz regumarity theorem Proof of Theorem 1 Let δ, A, C 1 > 0 and a C 2 concave and increasing function ψ : R + R + with ψ(0) = 0 which is constructed as following We consider M δ,a,c1 = sup x,y R N { u λ (x) u λ (y) } δ C 1(ψ( x y ) + δ)(φ(x) + φ(y) + A). (9) hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 12 / 16

27 Proof of Lipschitz regumarity theorem Proof of Theorem 1(Cont.) If M δ,a,c1 0 for some good choice of A, C 1, ψ independent of δ > 0, then we get (LR) by letting δ 0. hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 13 / 16

28 Proof of Lipschitz regumarity theorem Proof of Theorem 1(Cont.) If M δ,a,c1 0 for some good choice of A, C 1, ψ independent of δ > 0, then we get (LR) by letting δ 0. We argue by contradiction, assuming that for δ small enough M δ,a,c1 > 0. Set Φ(x, y) = C 1(φ(x) + φ(y) + A), ϕ(x, y) = δ + (ψ( x y ) + δ)φ(x, y). Suppose that the supremum is achieved at some point (x, y) with x y. Since u λ is a viscosity solution of (SP). In view of viscosity theory in Crandall-Ishii-Lions(1992), there exist X, Y S N such that the following viscosity inequality holds hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 13 / 16

29 Proof of Lipschitz regumarity theorem Proof of Theorem 1(Cont.) If M δ,a,c1 0 for some good choice of A, C 1, ψ independent of δ > 0, then we get (LR) by letting δ 0. We argue by contradiction, assuming that for δ small enough M δ,a,c1 > 0. Set Φ(x, y) = C 1(φ(x) + φ(y) + A), ϕ(x, y) = δ + (ψ( x y ) + δ)φ(x, y). Suppose that the supremum is achieved at some point (x, y) with x y. Since u λ is a viscosity solution of (SP). In view of viscosity theory in Crandall-Ishii-Lions(1992), there exist X, Y S N such that the following viscosity inequality holds λ(u λ (x) u λ (y)) (tr(σ(x)σ T (x)x ) tr(σ(y)σ T (y)y )) + αx.d xϕ αy.( D y ϕ) +H(x, D xϕ) H(y, D y ϕ) f (x) f (y). (10) hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 13 / 16

30 Proof of Lipschitz regumarity theorem Proof of Theorem 1(Cont.) If M δ,a,c1 0 for some good choice of A, C 1, ψ independent of δ > 0, then we get (LR) by letting δ 0. We argue by contradiction, assuming that for δ small enough M δ,a,c1 > 0. Set Φ(x, y) = C 1(φ(x) + φ(y) + A), ϕ(x, y) = δ + (ψ( x y ) + δ)φ(x, y). Suppose that the supremum is achieved at some point (x, y) with x y. Since u λ is a viscosity solution of (SP). In view of viscosity theory in Crandall-Ishii-Lions(1992), there exist X, Y S N such that the following viscosity inequality holds λ(u λ (x) u λ (y)) (tr(σ(x)σ T (x)x ) tr(σ(y)σ T (y)y )) + αx.d xϕ αy.( D y ϕ) +H(x, D xϕ) H(y, D y ϕ) f (x) f (y). (10) Since M δ,a,c1 > 0 and using the assumptions (4), (5), (6) and (7) to estimate for all the different terms, inequality (10) becomes hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 13 / 16

31 Proof of Lipschitz regumarity theorem Proof of Theorem 1(Cont.) λ δ 4νψ ( x y ) + αψ ( x y ) x y (2C H + (1 + µ)c σ)ψ ( x y ) (11) +2(ψ( x y ) + δ) + 2C H + C f x y. We droped Φ-terms on both sides of the above inequality to simplify of explaination. Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 14 / 16

32 Proof of Lipschitz regumarity theorem Proof of Theorem 1(Cont.) λ δ 4νψ ( x y ) + αψ ( x y ) x y (2C H + (1 + µ)c σ)ψ ( x y ) (11) +2(ψ( x y ) + δ) + 2C H + C f x y. We droped Φ-terms on both sides of the above inequality to simplify of explaination. Goal: reach a contradiction in (11). Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 14 / 16

33 Proof of Lipschitz regumarity theorem Proof of Theorem 1(Cont.) λ δ 4νψ ( x y ) + αψ ( x y ) x y (2C H + (1 + µ)c σ)ψ ( x y ) (11) +2(ψ( x y ) + δ) + 2C H + C f x y. We droped Φ-terms on both sides of the above inequality to simplify of explaination. Goal: reach a contradiction in (11). Case 1: r := x y r 0. The function ψ above was chosen to be strictly concave for small r r 0, we take profit of the ellipticity of the equation 4νψ (r) in (11) to control all the others terms. Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 14 / 16

34 Proof of Lipschitz regumarity theorem Proof of Theorem 1(Cont.) λ δ 4νψ ( x y ) + αψ ( x y ) x y (2C H + (1 + µ)c σ)ψ ( x y ) (11) +2(ψ( x y ) + δ) + 2C H + C f x y. We droped Φ-terms on both sides of the above inequality to simplify of explaination. Goal: reach a contradiction in (11). Case 1: r := x y r 0. The function ψ above was chosen to be strictly concave for small r r 0, we take profit of the ellipticity of the equation 4νψ (r) in (11) to control all the others terms. Case 2: r := x y r 0. In this case, the second derivative ψ (r) of the increasing concave function ψ is small and the ellipticity is not powerful enough to control the bad terms. Instead, we use the positive term αψ (r)r coming from the Ornstein-Uhlenbeck operator to control everything. Note that by using the good terms as explained in the two above cases, we can choose all of parameters independently of λ to reach a contradiction in (11). hi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 14 / 16

35 Proof of Lipschitz regumarity theorem M. Arisawa Ergodic problem for the Hamilton-Jacobi-Bellman equation. I. Ann. Inst. H. Poincaré Anal. Non Linéaire, no. 4, , M. Arisawa and P.-L. Lions. On ergodic stochastic control. Comm. Partial Differential Equations, 11-12, , M. Bardi, A. Cesaroni, and D. Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete Contin. Dyn. Syst., no. 9, , M. G. Crandall, H. Ishii, and P.-L. Lions. User s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.), 27(1):1 67, Y. Fujita, H. Ishii, and P. Loreti. Asymptotic solutions of viscous Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Comm. Partial Differential Equations, 31(4-6): , Y. Fujita and P. Loreti. Long-time behavior of solutions to Hamilton-Jacobi equations with quadratic gradient term. NoDEA Nonlinear Differential Equations Appl, no. 6, , Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 15 / 16

36 Proof of Lipschitz regumarity theorem Thank you for your attention! Thi Tuyen Nguyen (Ph.D student of University of Rennes Lipschitz 1 Joint continuity work with: for solutions Prof. E. Chasseigne(University of Hamilton-Jacobi equation of Tours) with and Ornstein-Uhlenbeck Prof. O. March Ley(Insa-Rennes) 23, operator 2016 ) 16 / 16