{"id":"https://openalex.org/W4389559720","doi":"https://doi.org/10.3390/sym15122181","title":"Insight into Spatially Colored Stochastic Heat Equation: Temporal Fractal Nature of the Solution","display_name":"Insight into Spatially Colored Stochastic Heat Equation: Temporal Fractal Nature of the Solution","publication_year":2023,"publication_date":"2023-12-10","ids":{"openalex":"https://openalex.org/W4389559720","doi":"https://doi.org/10.3390/sym15122181"},"language":"en","primary_location":{"id":"doi:10.3390/sym15122181","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15122181","pdf_url":"https://www.mdpi.com/2073-8994/15/12/2181/pdf?version=1702199138","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/15/12/2181/pdf?version=1702199138","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100720710","display_name":"Wensheng Wang","orcid":"https://orcid.org/0000-0002-1768-1995"},"institutions":[{"id":"https://openalex.org/I50760025","display_name":"Hangzhou Dianzi University","ror":"https://ror.org/0576gt767","country_code":"CN","type":"education","lineage":["https://openalex.org/I50760025"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Wensheng Wang","raw_affiliation_strings":["School of Economics, Hangzhou Dianzi University, Hangzhou 310018, China"],"affiliations":[{"raw_affiliation_string":"School of Economics, Hangzhou Dianzi University, Hangzhou 310018, China","institution_ids":["https://openalex.org/I50760025"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5100720710"],"corresponding_institution_ids":["https://openalex.org/I50760025"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.22770718,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"15","issue":"12","first_page":"2181","last_page":"2181"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11270","display_name":"Complex Systems and Time Series Analysis","score":0.9921000003814697,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T11270","display_name":"Complex Systems and Time Series Analysis","score":0.9921000003814697,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9900000095367432,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9890000224113464,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7061166167259216},{"id":"https://openalex.org/keywords/hausdorff-dimension","display_name":"Hausdorff dimension","score":0.6606711149215698},{"id":"https://openalex.org/keywords/iterated-function-system","display_name":"Iterated function system","score":0.6026540994644165},{"id":"https://openalex.org/keywords/heat-equation","display_name":"Heat equation","score":0.5655872225761414},{"id":"https://openalex.org/keywords/fractal","display_name":"Fractal","score":0.5384840965270996},{"id":"https://openalex.org/keywords/fractal-dimension","display_name":"Fractal dimension","score":0.5197669863700867},{"id":"https://openalex.org/keywords/law-of-the-iterated-logarithm","display_name":"Law of the iterated logarithm","score":0.5187947154045105},{"id":"https://openalex.org/keywords/colored","display_name":"Colored","score":0.5173084735870361},{"id":"https://openalex.org/keywords/point-process","display_name":"Point process","score":0.5035881400108337},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4651625454425812},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.4620586633682251},{"id":"https://openalex.org/keywords/stochastic-process","display_name":"Stochastic process","score":0.4547523558139801},{"id":"https://openalex.org/keywords/iterated-function","display_name":"Iterated function","score":0.43608516454696655},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.43564146757125854},{"id":"https://openalex.org/keywords/hausdorff-space","display_name":"Hausdorff space","score":0.42908990383148193},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.41626983880996704},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.3889835774898529},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.22437942028045654},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.13013407588005066},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.07771649956703186}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7061166167259216},{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.6606711149215698},{"id":"https://openalex.org/C168460219","wikidata":"https://www.wikidata.org/wiki/Q1502167","display_name":"Iterated function system","level":3,"score":0.6026540994644165},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.5655872225761414},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.5384840965270996},{"id":"https://openalex.org/C26546657","wikidata":"https://www.wikidata.org/wiki/Q1412452","display_name":"Fractal dimension","level":3,"score":0.5197669863700867},{"id":"https://openalex.org/C139907963","wikidata":"https://www.wikidata.org/wiki/Q198740","display_name":"Law of the iterated logarithm","level":3,"score":0.5187947154045105},{"id":"https://openalex.org/C2778307483","wikidata":"https://www.wikidata.org/wiki/Q5149038","display_name":"Colored","level":2,"score":0.5173084735870361},{"id":"https://openalex.org/C88871306","wikidata":"https://www.wikidata.org/wiki/Q7208287","display_name":"Point process","level":2,"score":0.5035881400108337},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4651625454425812},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4620586633682251},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.4547523558139801},{"id":"https://openalex.org/C140479938","wikidata":"https://www.wikidata.org/wiki/Q5254619","display_name":"Iterated function","level":2,"score":0.43608516454696655},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.43564146757125854},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.42908990383148193},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.41626983880996704},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3889835774898529},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.22437942028045654},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.13013407588005066},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.07771649956703186},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym15122181","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15122181","pdf_url":"https://www.mdpi.com/2073-8994/15/12/2181/pdf?version=1702199138","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:ac0fcd2f7f6346d6ae48e8c846e430ef","is_oa":true,"landing_page_url":"https://doaj.org/article/ac0fcd2f7f6346d6ae48e8c846e430ef","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 12, p 2181 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym15122181","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15122181","pdf_url":"https://www.mdpi.com/2073-8994/15/12/2181/pdf?version=1702199138","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1231421488","display_name":null,"funder_award_id":"under","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3744313044","display_name":null,"funder_award_id":"Social","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5666573507","display_name":null,"funder_award_id":"11671115","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5994120800","display_name":null,"funder_award_id":"Natural","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4389559720.pdf","grobid_xml":"https://content.openalex.org/works/W4389559720.grobid-xml"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W593630733","https://openalex.org/W1277962881","https://openalex.org/W1482940592","https://openalex.org/W1977159695","https://openalex.org/W1980299817","https://openalex.org/W1991365442","https://openalex.org/W2009781729","https://openalex.org/W2027689690","https://openalex.org/W2030193783","https://openalex.org/W2056268316","https://openalex.org/W2092173086","https://openalex.org/W2097710222","https://openalex.org/W2105464579","https://openalex.org/W2127368952","https://openalex.org/W2138087464","https://openalex.org/W2146276429","https://openalex.org/W2150769797","https://openalex.org/W2160463528","https://openalex.org/W2169770193","https://openalex.org/W2207773095","https://openalex.org/W2477464792","https://openalex.org/W2498994366","https://openalex.org/W2920689604","https://openalex.org/W2950571254","https://openalex.org/W3201514062","https://openalex.org/W3205317034","https://openalex.org/W4240810995","https://openalex.org/W4300246981","https://openalex.org/W4301425558","https://openalex.org/W6802616291"],"related_works":["https://openalex.org/W2898687288","https://openalex.org/W1981559745","https://openalex.org/W1788369290","https://openalex.org/W2002505525","https://openalex.org/W2734644987","https://openalex.org/W2184734332","https://openalex.org/W2390681877","https://openalex.org/W2606913028","https://openalex.org/W2120496796","https://openalex.org/W1995639825"],"abstract_inverted_index":{"In":[0],"this":[1,64,87],"paper,":[2],"the":[3,31,46,54,74,93,96,107,112],"solution":[4,16],"to":[5],"a":[6,18,26,71],"spatially":[7],"colored":[8],"stochastic":[9],"heat":[10],"equation":[11],"(SHE)":[12],"is":[13,17,51,70,78],"studied.":[14],"This":[15],"random":[19,33,72],"function":[20,34],"of":[21,35,45,56,76,86,95,111],"time":[22,36],"and":[23,43,66,100],"space.":[24],"For":[25],"fixed":[27],"point":[28],"in":[29,63],"space,":[30],"resulting":[32],"has":[37],"exact,":[38],"dimension-dependent,":[39],"global":[40],"continuity":[41],"moduli,":[42],"laws":[44],"iterated":[47],"logarithm":[48],"(LILs).":[49],"It":[50],"obtained":[52],"that":[53],"set":[55,114],"fast":[57],"points":[58,85],"at":[59],"which":[60,77],"LILs":[61],"fail":[62],"process,":[65],"occur":[67],"infinitely":[68],"often,":[69],"fractal,":[73],"size":[75],"evaluated":[79],"by":[80,106],"its":[81],"Hausdorff":[82],"dimension.":[83],"These":[84],"process":[88],"are":[89,104],"everywhere":[90],"dense":[91],"with":[92],"power":[94],"continuum":[97],"almost":[98],"surely,":[99],"their":[101],"hitting":[102],"probabilities":[103],"determined":[105],"packing":[108],"dimension":[109],"dimP(E)":[110],"target":[113],"E.":[115]},"counts_by_year":[],"updated_date":"2026-04-18T07:56:08.524223","created_date":"2025-10-10T00:00:00"}