{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:38:04Z","timestamp":1776728284422,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"238","license":[{"start":{"date-parts":[[2002,10,25]],"date-time":"2002-10-25T00:00:00Z","timestamp":1035504000000},"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":"<p>\n                    Using the main ideas of Tanaka, the measure-solution\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace upper P Subscript t Baseline right-brace Subscript t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>P<\/mml:mi>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:msub>\n                              <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{P_t\\}_t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of a\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3\">\n                        <mml:semantics>\n                          <mml:mn>3<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace upper P Subscript t Baseline right-brace Subscript t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>P<\/mml:mi>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:msub>\n                              <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{P_t\\}_t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of solutions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace upper P Subscript t Superscript l Baseline right-brace Subscript t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>P<\/mml:mi>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>l<\/mml:mi>\n                            <\/mml:msubsup>\n                            <mml:msub>\n                              <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{P^l_t\\}_t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of approximating Boltzmann equations with cutoff is proved. Then, a result of Graham-M\u00e9l\u00e9ard is used and allows us to approximate\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace upper P Subscript t Superscript l Baseline right-brace Subscript t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>P<\/mml:mi>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>l<\/mml:mi>\n                            <\/mml:msubsup>\n                            <mml:msub>\n                              <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{P^l_t\\}_t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with the empirical measure\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace mu Subscript t Superscript l comma n Baseline right-brace Subscript t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>\n                                \u03bc\n                                \n                              <\/mml:mi>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>l<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:msub>\n                              <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{\\mu ^{l,n}_t\\}_t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of an easily simulable interacting particle system. Precise rates of convergence are given. A numerical study lies at the end of the paper.\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01339-4","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"583-604","source":"Crossref","is-referenced-by-count":27,"title":["A stochastic particle numerical method for 3D Boltzmann equations without cutoff"],"prefix":"10.1090","volume":"71","author":[{"given":"Nicolas","family":"Fournier","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sylvie","family":"M\u00e9l\u00e9ard","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2001,10,25]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1137\/0726004","article-title":"A convergence proof for Nanbu\u2019s simulation method for the full Boltzmann equation","volume":"26","author":"Babovsky, Hans","year":"1989","journal-title":"SIAM J. 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