{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T18:48:37Z","timestamp":1778525317487,"version":"3.51.4"},"reference-count":35,"publisher":"American Mathematical Society (AMS)","issue":"266","license":[{"start":{"date-parts":[[2009,10,7]],"date-time":"2009-10-07T00:00:00Z","timestamp":1254873600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n In this paper we study the approximation of the distribution of\n \n \n \n \n X<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n X_t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n Hilbert\u2013valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as\n \n \\[\n \n \n \n \n d<\/mml:mi>\n <\/mml:mrow>\n \n X<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n +<\/mml:mo>\n A<\/mml:mi>\n \n X<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msub>\n \n \n d<\/mml:mi>\n <\/mml:mrow>\n t<\/mml:mi>\n =<\/mml:mo>\n \n Q<\/mml:mi>\n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n d<\/mml:mi>\n <\/mml:mrow>\n W<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n ,<\/mml:mo>\n \n \n X<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n x<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n H<\/mml:mi>\n ,<\/mml:mo>\n \n t<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n T<\/mml:mi>\n ]<\/mml:mo>\n ,<\/mml:mo>\n <\/mml:mrow>\n \\mathrm {d} X_t+AX_t \\, \\mathrm {d} t = Q^{1\/2} \\mathrm {d} W(t), \\quad X_0=x \\in H, \\quad t\\in [0,T],<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n driven by a Gaussian space time noise whose covariance operator\n \n \n \n Q<\/mml:mi>\n Q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is given. We assume that\n \n \n \n \n A<\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n \n \u03b1\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n A^{-\\alpha }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a finite trace operator for some\n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\alpha >0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and that\n \n \n \n Q<\/mml:mi>\n Q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is bounded from\n \n \n \n H<\/mml:mi>\n H<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n into\n \n \n \n \n D<\/mml:mi>\n (<\/mml:mo>\n \n A<\/mml:mi>\n \n \u03b2\n \n <\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n D(A^\\beta )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for some\n \n \n \n \n \n \u03b2\n \n <\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\beta \\geq 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . It is not required to be nuclear or to commute with\n \n \n \n A<\/mml:mi>\n A<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>\n

\n The discretization is achieved thanks to finite element methods in space (parameter\n \n \n \n \n h<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n h>0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ) and a\n \n \n \n \n \u03b8\n \n <\/mml:mi>\n \\theta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -method in time (parameter\n \n \n \n \n \n \u0394\n \n <\/mml:mi>\n t<\/mml:mi>\n =<\/mml:mo>\n T<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n N<\/mml:mi>\n <\/mml:mrow>\n \\Delta t=T\/N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ). We define a discrete solution\n \n \n \n \n X<\/mml:mi>\n h<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msubsup>\n X^n_h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and for suitable functions\n \n \n \n \n \u03c6\n \n <\/mml:mi>\n \\varphi<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n defined on\n \n \n \n H<\/mml:mi>\n H<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we show that\n \n \\[\n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n \n E<\/mml:mi>\n <\/mml:mrow>\n \n \n \u03c6\n \n <\/mml:mi>\n (<\/mml:mo>\n \n X<\/mml:mi>\n h<\/mml:mi>\n N<\/mml:mi>\n <\/mml:msubsup>\n )<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n E<\/mml:mi>\n <\/mml:mrow>\n \n \n \u03c6\n \n <\/mml:mi>\n (<\/mml:mo>\n \n X<\/mml:mi>\n T<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n \n |<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n O<\/mml:mi>\n (<\/mml:mo>\n \n h<\/mml:mi>\n \n 2<\/mml:mn>\n \n \u03b3\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n \n t<\/mml:mi>\n \n \u03b3\n \n <\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n |\\mathbb {E} \\, \\varphi (X^N_h) - \\mathbb {E} \\, \\varphi (X_T) | = O(h^{2\\gamma } + \\Delta t^\\gamma )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n where\n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n ><\/mml:mo>\n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n \n \u03b1\n \n <\/mml:mi>\n +<\/mml:mo>\n \n \u03b2\n \n <\/mml:mi>\n <\/mml:mrow>\n \\gamma >1- \\alpha + \\beta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations.\n <\/p>","DOI":"10.1090\/s0025-5718-08-02184-4","type":"journal-article","created":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T13:09:23Z","timestamp":1259672963000},"page":"845-863","source":"Crossref","is-referenced-by-count":74,"title":["Weak order for the discretization of the stochastic heat equation"],"prefix":"10.1090","volume":"78","author":[{"given":"Arnaud","family":"Debussche","sequence":"first","affiliation":[]},{"given":"Jacques","family":"Printems","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,10,7]]},"reference":[{"issue":"1-2","key":"1","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1080\/17442509808834159","article-title":"Finite element and difference approximation of some linear stochastic partial differential equations","volume":"64","author":"Allen, E. 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