Tamtam Proceedings - lamsin

SPDE for phytoplankton aggregatios 531We have the following convergence result:Theorem 3 If N n −→ +∞ as n −→ +∞ , the sequence of rescaled processes {Y (n). } n≥1converges weakly (in distribution) in the space D([0, +∞[, M F (R)) to a measure-valuedcontinuous process {Y t , t ≥ 0}, whose distribution P satisfies the following martingaleproblem:∀ϕ ∈ C 2 b (R), M t (ϕ) 〈Y t , ϕ〉 − 〈Y 0 , ϕ〉 −is a P -martingale with the quadratic variation〈M(ϕ)〉 t = µ∫ t0∫ t0〈Y s , Dϕ ′′ + (F a ∗ η s )·ϕ ′ 〉ds (2)〈Y s , ϕ 2 〉ds. (3)For the claim, the proof of this theorem is in [4].From (2) and (3), we recognize that {Y t , t ≥ 0} is a generalization of the Dawson-Watanabe superprocess. We can prove the following result:Theorem 4 The process {Y t , t ≥ 0} is the unique solution of the non linear stochasticdifferential equation in a space of measures:dY t = D d2dx 2 Y tdt − ddx [Y t(F a ∗ Y t )]dt + dM t ,where M t is a continuous martingale measure (at the sens of [17]), with covariancemeasure µY s (dx)ds.The proof of this theorem is presented in [4].Heuristically, if we suppose that for every t > 0, Y t is absolutely continuous withrespect to the Lebesgue measure on R, then the density f(t, x) of Y t will satisfy:Theorem 5 The density f(t, x) of Y t , with respect to the Lebesgue measure, is a weaksolution to the following SPDE:∂f(t, x)∂twith Ẇ defined by:W= D ∂2 f(t, x)∂x 2− ∂∂x [f(t, x)(F a ∗ f(t, .))(x)] + √ µf(t, x)Ẇ (t, x)(dt, dx) = Ẇ (t, x)dtdxwhere W (dt, dx) is a Gaussian white noise (at the sens of [17]).TAMTAM –Tunis– 2005