{
  "abstract": "This paper is concerned with the connection between parabolic equations with memory (MPEs, for short) and backward stochastic differential equations (BSDEs, for short). To provide a probabilistic representation of MPEs, we introduce a new class of BSDEs, called BSDEs with memory (MBSDEs, for short). The main feature of MBSDEs is that, at the current time $s$, the generator depends not only on the current value $Y^{t,x}(s)\\equiv Y^{s,X^{t,x}(s)}(s)$ but also on the future path $r\\mapsto Y^{r,X^{t,x}(s)}(r)$, $r\\in[s,T]$. This structure reveals a fundamental difference from classical BSDEs: in the classical setting, the time and space variables are coupled and inseparable, whereas in MBSDEs they are completely independent. We prove the well-posedness of these MBSDEs and show that their adapted solutions provide a probabilistic representation for the solutions of MPEs. Furthermore, under a monotonicity condition, we establish a comparison principle for one-dimensional MBSDEs, which enables us to develop a viscosity solution theory for MPEs. This generalizes the celebrated nonlinear Feynman--Kac formula of Peng [25] and Pardoux--Peng [22] to the case with memory, in both the classical and viscosity solution senses.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01591",
  "author_total": 3,
  "authors": [
    {
      "name": "Hanxiao Wang"
    },
    {
      "name": "Jiongmin Yong"
    },
    {
      "name": "Qi Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0nt62wfs352vj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01591",
  "fulltext_url": null,
  "id": "0nt62wfs352vj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01591",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01591",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0nt62wfs352vj.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01591",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01591"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Feynman--Kac Representation for Parabolic Equations with Memory: A New Class of BSDEs",
  "title_text": "Feynman--Kac Representation for Parabolic Equations with Memory: A New Class of BSDEs",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/0nt62wfs352vj",
  "venue": "arXiv",
  "versions": []
}