Analyticity of semigroups generated by Fleming-Viot type operators

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1 Analyticity of semigroups generated by Fleming-Viot type operators Elisabetta Mangino, in collaboration with A. Albanese Università del Salento, Lecce, Italy

2 s Au(x) = x i (δ ij x j )D ij u + b i (x)d i u i,j=1 i=1 d=1: S d = {(x 1,..., x d ) R d i {1,..., d} x i 0; Au(x) = x(1 x)u (x) + b(x)u (x) x i 1} d=2: Au(x, y) = x(1 x)u xx + y(1 y)u yy 2xyu xy + b 1(x, y)u x + b 2(x, y)u y. The problem u t = Au arises as a continuous approximation of a Markov process i=1 (p 1(k),..., p d (k), 1 p 1(k) p d (k)) k N describing the frequency of (d+1) types of genetic character at the k-th generation of a population (Wright-Fischer model). Ewens, Mathematical Population Genetics, 1979 (geneticist points of view and motivations) Ethier, Kurtz, Markov processes, 1985 ( mathematical points of view).

3 s Au(x) = x i (δ ij x j )D ij u + b i (x)d i u i,j=1 i=1 d=1: S d = {(x 1,..., x d ) R d i {1,..., d} x i 0; Au(x) = x(1 x)u (x) + b(x)u (x) x i 1} d=2: Au(x, y) = x(1 x)u xx + y(1 y)u yy 2xyu xy + b 1(x, y)u x + b 2(x, y)u y. The problem u t = Au arises as a continuous approximation of a Markov process i=1 (p 1(k),..., p d (k), 1 p 1(k) p d (k)) k N describing the frequency of (d+1) types of genetic character at the k-th generation of a population (Wright-Fischer model). Ewens, Mathematical Population Genetics, 1979 (geneticist points of view and motivations) Ethier, Kurtz, Markov processes, 1985 ( mathematical points of view).

4 C 0 -semigroups If X is a Banach space, A : D(A) X X a linear operator and x X we can consider the abstract Cauchy problem { u (t) = Au(t) t 0 (ACP) (1) u(0) = x. (A, D(A)) closed, D(A) = X λ > 0 (λi A) 1 = R(λ, A) λ > 0 (λi A) 1 1 λ ( ) Setting, for every t 0, T tx = u(t, x), it holds that: x D(A) u(, x) C 1 ([0, [, X ) s.t. t 0 u(t, x) D(A) and (ACP) T t L(X ), T t+s = T t T s, lim Ttx = x. t 0 + (T t) t 0 is the strongly continuous semigroup generated by (A, D(A)). Remark. If A is a differential operator, (*) is a a-priori estimate of the solution of the equation λu Au = f : λ > 0, u D(A) u X λu Au X. λ

5 C 0 -semigroups If X is a Banach space, A : D(A) X X a linear operator and x X we can consider the abstract Cauchy problem { u (t) = Au(t) t 0 (ACP) (1) u(0) = x. (A, D(A)) closed, D(A) = X λ > 0 (λi A) 1 = R(λ, A) λ > 0 (λi A) 1 1 λ ( ) Setting, for every t 0, T tx = u(t, x), it holds that: x D(A) u(, x) C 1 ([0, [, X ) s.t. t 0 u(t, x) D(A) and (ACP) T t L(X ), T t+s = T t T s, lim Ttx = x. t 0 + (T t) t 0 is the strongly continuous semigroup generated by (A, D(A)). Remark. If A is a differential operator, (*) is a a-priori estimate of the solution of the equation λu Au = f : λ > 0, u D(A) u X λu Au X. λ

6 Sectorial operators and analytic semigroups (A, D(A)) sectorial there exists ω R, M > 0 such that { λ C, Reλ > ω (λi A) 1 λ C, Reλ > ω (λi A) 1 M λ ( ) A closed sectorial operator with dense domain generates an analytic semigroup, i.e. the semigroup generated by A extends to a semigroup (T z) z Σδ where δ ]0, π/2[ and Σ δ = {z C Arg(z) < δ}, such that z T z is analytic in Σ 0,δ, lim z 0,z Σω,δ T zx = x per ogni δ < δ.

7 Agmon-Douglis-Niremberg estimates A(x, D) = N N a ij (x)d ij + b i (x)d i i,j=1 Ω R N bounded open set with C 2 -boundary, a ij, b i continuous on Ω, a ij = a ji and there exists ν > 0 such that i=1 x Ω ξ R N N i,j=1 a ij (x)ξ i ξ j ν ξ 2. If X = L p (Ω), 1 < p <, D(A) = W 2,p (Ω) W 1,p 0 (Ω): ω p, M p > 0 λ C, Reλ > ω p : u D(A) λ u p + λ 1 2 Du p + D 2 u p M p λu Au p. (A, D(A)) is a sectorial operator. Remark. Analogous estimates, due to Stewart, hold in continuous function spaces.

8 A d u(x) = x i (δ ij x j )D ij u + b i (x) D i u Setting: C(S d ) (Feller s theory). i,j=1 i=1 Q(x) = (x i (δ ij x j )) i,j=1,...,d det Q(x) = x 1 x d (1 x 1... x d ), If b i = 0: d=1: A 1u(x) = x(1 x)u (x) d=2: A 2u(x, y) = x(1 x)u xx + y(1 y)u yy 2xyu xy = ( ) 2 = x(1 x y) 2 u + y(1 x y) 2 u + xy x 2 y 2 x y u.

9 A d u(x) = x i (δ ij x j )D ij u + b i (x) D i u Setting: C(S d ) (Feller s theory). i,j=1 i=1 Q(x) = (x i (δ ij x j )) i,j=1,...,d det Q(x) = x 1 x d (1 x 1... x d ), If b i = 0: d=1: A 1u(x) = x(1 x)u (x) d=2: A 2u(x, y) = x(1 x)u xx + y(1 y)u yy 2xyu xy = ( ) 2 = x(1 x y) 2 u + y(1 x y) 2 u + xy x 2 y 2 x y u.

10 Generation results 1 (Ethier 1977, Ethier-Kurtz 1986) If b = (b 1,..., b d ) is a Lipschitz function and (b, detq) 0 on S, then (A d, C 2 (S)) generates a strongly continuous semigroup (T t) t 0 on C(S). Moreover T t(c k (S d )) C k (S d ), if b C k+2 (S d ) (k N). d = 1: b(0) 0, b(1) 0 d = 2: b 1(x, 0) 0, b 2(0, y) 0, b 1(x, 1 x) + b 2(x, 1 x) 0 2 (Feller 1954, Clement-Timmermans 1986) If d = 1, then D(A 1) = {u C 2 (]0, 1[) C([0, 1]) lim x(1 x)u (x) = 0}. x 0 +,1 3 (Shimakura, 1977, 1981, 1992) If b = 0, the eigenvalues of A d are λ m = m(m 1) 2, m N.

11 Generation results 1 (Ethier 1977, Ethier-Kurtz 1986) If b = (b 1,..., b d ) is a Lipschitz function and (b, detq) 0 on S, then (A d, C 2 (S)) generates a strongly continuous semigroup (T t) t 0 on C(S). Moreover T t(c k (S d )) C k (S d ), if b C k+2 (S d ) (k N). d = 1: b(0) 0, b(1) 0 d = 2: b 1(x, 0) 0, b 2(0, y) 0, b 1(x, 1 x) + b 2(x, 1 x) 0 2 (Feller 1954, Clement-Timmermans 1986) If d = 1, then D(A 1) = {u C 2 (]0, 1[) C([0, 1]) lim x(1 x)u (x) = 0}. x 0 +,1 3 (Shimakura, 1977, 1981, 1992) If b = 0, the eigenvalues of A d are λ m = m(m 1) 2, m N.

12 Generation results 1 (Ethier 1977, Ethier-Kurtz 1986) If b = (b 1,..., b d ) is a Lipschitz function and (b, detq) 0 on S, then (A d, C 2 (S)) generates a strongly continuous semigroup (T t) t 0 on C(S). Moreover T t(c k (S d )) C k (S d ), if b C k+2 (S d ) (k N). d = 1: b(0) 0, b(1) 0 d = 2: b 1(x, 0) 0, b 2(0, y) 0, b 1(x, 1 x) + b 2(x, 1 x) 0 2 (Feller 1954, Clement-Timmermans 1986) If d = 1, then D(A 1) = {u C 2 (]0, 1[) C([0, 1]) lim x(1 x)u (x) = 0}. x 0 +,1 3 (Shimakura, 1977, 1981, 1992) If b = 0, the eigenvalues of A d are λ m = m(m 1) 2, m N.

13 Regularity results 1 Metafune, Metafune-Campiti, 1998: (A 1, D(A 1)) generates an analytic semigroup. 2 Shimakura 1977, Cerrai-Clement 2001, Albanese-Campiti-Mangino 2007, Albanese-Mangino 2009: ( ) Au(x) = x i (δ ij x j )D ij u + (ω i ω j x i )D i u(x) ( ) i,j=1 i=1 where ω i ]0, + [ per ogni i = 0, 1,..., d. Then the semigroup (T (t)) t 0 generated by ( ) in C(S d ) is differentiable, i.e. for all f C(S d ) and t 0 > 0 the map t T (t)f is differentiable in t 0. - Sesquilinear Forms, Log-Sobolev inequalities. j=0

14 Regularity results 1 Metafune, Metafune-Campiti, 1998: (A 1, D(A 1)) generates an analytic semigroup. 2 Shimakura 1977, Cerrai-Clement 2001, Albanese-Campiti-Mangino 2007, Albanese-Mangino 2009: ( ) Au(x) = x i (δ ij x j )D ij u + (ω i ω j x i )D i u(x) ( ) i,j=1 i=1 where ω i ]0, + [ per ogni i = 0, 1,..., d. Then the semigroup (T (t)) t 0 generated by ( ) in C(S d ) is differentiable, i.e. for all f C(S d ) and t 0 > 0 the map t T (t)f is differentiable in t 0. - Sesquilinear Forms, Log-Sobolev inequalities. j=0

15 Albanese, Mangino: Analyticity of a class of degenerate evolution equations on the canonical simplex of R d arising from Fleming-Viot processes, JMAA 2011 Theorem A d u(x) = x i (δ ij x j ) x 2 i x j u(x), i,j=1 The closure of (A d, C 2 (S d )) generates an analytic semigroup in C(S d ).

16 Induction over d Statement: (A d, C 2 (S d )) satisfies the following properties. 1 it generates an analytic C 0 semigroup (T (t)) t 0 on C(S d ). 2 There exist C > 0 and R > 1 such that, for every λ C, Reλ > R, i = 1,..., d and u C(S d ), we have x i (1 x i ) xi [R(λ, A d )u] Sd C λ u Sd.

17 Induction over d Statement: (A d, C 2 (S d )) satisfies the following properties. 1 it generates an analytic C 0 semigroup (T (t)) t 0 on C(S d ). 2 There exist C > 0 and R > 1 such that, for every λ C, Reλ > R, i = 1,..., d and u C(S d ), we have x i (1 x i ) xi [R(λ, A d )u] Sd C λ u Sd.

18 Proposition Consider the differential operator B d u(x) = x i (δ ij x j ) x 2 i x j u(x), u C 2 ([0, 1] d ) i,j=1 1 The closure of (B d, C 2 ([0, 1] d )) generates an analytic semigroup (T (t)) t 0 in C([0, 1] d ). 2 There exists K > 0 such that for every t > 0, we have x i (1 x i ) xi (T (t)u) K max{ 1 t, 1} u, u C([0, 1] d ), 3 There exist C > 0 and R > 1 such that, for every λ C, Reλ > R, x i (1 x i ) xi (R(λ, B d )u) C λ u, u C([0, 1] d ).

19 1-dimensional case Proposition. The differential operator (A 1, D(A 1)) satisfies the following properties. (1) There exist ε > 0, C > 0 and D > 0 such that, for every 0 < ε < ε and u C([0, 1]) C 2 (]0, 1[) with A 1u C([0, 1]), we have x(1 x)u C u + Dε A1u. ε (2) There exists K > 0 such that for every t > 0, we have x(1 x)(t (t)u) K max{ 1 t, 1} u, u C([0, 1]), (3) There exist C > 0 and R > 1 such that, for every λ C, Reλ > R, x(1 x)(r(λ, A 1)u) C λ u, u C([0, 1]).

20 2-dimensional case B 2u(x, y) = x(1 x) 2 x u(x, y) + y(1 y) 2 y u(x, y), u C 2 ([0, 1] 2 ), B 1u(x) = x(1 x)u (x), x [0, 1], B 2v(y) = y(1 y)v (y), y [0, 1], D(B 1) = D(B 2) = {u C([0, 1]) C 2 (]0, 1[) lim x 0 +,1 x(1 x)u (x) = 0} (B i, D(B i )) generate a contractive analytic C 0 semigroup (S i (t)) t 0 on C([0, 1]), i = 1, 2.

21 Injective tensor product of semigroups The injective tensor product (S 1(t) ε S 2(t)) t 0 is also a contractive analytic C 0 semigroup on C([0, 1] 2 ) = C([0, 1]) ε C([0, 1]), generated by the closure of the operator ((B 1 I y ) + (I y B 2), D(B 1) D(B 2)), where I x and I y denote the identity map on C([0, 1]) with respect to the variables x and y respectively. Observe that B 2u = (B 1 I y )u + (I y B 2)u, u D(B 1) D(B 2), and that C 2 ([0, 1]) C 2 ([0, 1] is dense in D(B 1) D(B 2) with respect to the graph norm of B 2. Thus (B 2, C 2 ([0, 1] 2 )) = (B 1 I y + I y B 2, D(B 1) D(B 2)). and therefore the semigroup (T (t)) t 0 generated by (B 2, C 2 ([0, 1] 2 )) is exactly (S 1(t) ε S 2(t)) t 0.

22 Injective tensor product of semigroups The injective tensor product (S 1(t) ε S 2(t)) t 0 is also a contractive analytic C 0 semigroup on C([0, 1] 2 ) = C([0, 1]) ε C([0, 1]), generated by the closure of the operator ((B 1 I y ) + (I y B 2), D(B 1) D(B 2)), where I x and I y denote the identity map on C([0, 1]) with respect to the variables x and y respectively. Observe that B 2u = (B 1 I y )u + (I y B 2)u, u D(B 1) D(B 2), and that C 2 ([0, 1]) C 2 ([0, 1] is dense in D(B 1) D(B 2) with respect to the graph norm of B 2. Thus (B 2, C 2 ([0, 1] 2 )) = (B 1 I y + I y B 2, D(B 1) D(B 2)). and therefore the semigroup (T (t)) t 0 generated by (B 2, C 2 ([0, 1] 2 )) is exactly (S 1(t) ε S 2(t)) t 0.

23 Gradient estimates for the semigroup x(1 x) xs 1(t) L(C([0, 1])), x(1 x) xs 1(t) K max{t 1 2, 1}, (t > 0) ( x(1 x) xs 1(t)) ε S 2(t) L(C([0, 1] 2 ), ( x(1 x) xs 1(t)) ε S 2(t) K max{t 1 2, 1} The assertion follows by observing that for every u C([0, 1] 2 ) x(1 x) x(t (t)u) = (( x(1 x) xs 1(t)) ε S 2(t))(u),

24 Gradient estimates for the resolvent λ R, λ > ω: x(1 x)d ( + and hence, 0 ) e λt T (t)udt = x(1 x) x(r(λ, B)u) [0,1] 2 K u [0,1] 2 K λ u [0,1] 2. 0 e λt x(1 x) x(t (t)u)dt ( + 0 ) max{t 1/2, 1}e λt dt

25 Under preparation Proposition. A 2u(x, y) = x(1 x)u xx + y(1 y)u yy 2xyu xy + b 1(x, y)u x + b 2(x, y)u y. If b 1(0, y), b 2(x, 0), b 1(x, 1 x) + b 2(x, 1 x) are costant functions on [0, 1], then the closure of (A, C 2 (S 2)) generates an analytic semigroup in C(S 2).

Elliptic Operators with Unbounded Coefficients

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