math.DS

# Title: Periodic, Quasi-Periodic, Almost Periodic, Almost Automorphic, Birkhoff Recurrent and Poisson Stable Solutions for Stochastic Differential Equations

Abstract: The paper is dedicated to studying the problem of Poisson stability (in particular stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, Bohr almost automorphy, Birkhoff recurrence, almost recurrence in the sense of Bebutov, Levitan almost periodicity, pseudo-periodicity, pseudo-recurrence, Poisson stability) of solutions for semi-linear stochastic equation $$dx(t)=(Ax(t)+f(t,x(t)))dt +g(t,x(t))dW(t)\quad (*)$$ with exponentially stable linear operator $A$ and Poisson stable in time coefficients $f$ and $g$. We prove that if the functions $f$ and $g$ are appropriately "small", then equation $(*)$ admits at least one solution which has the same character of recurrence as the functions $f$ and $g$.
 Comments: 28 pages, no figures Subjects: Dynamical Systems (math.DS); Probability (math.PR) MSC classes: Primary: 34C25, 34C27, 37B20, 60H10, Secondary: 34D20 Cite as: arXiv:1702.02718 [math.DS] (or arXiv:1702.02718v1 [math.DS] for this version)

## Submission history

From: Zhenxin Liu [view email]
[v1] Thu, 9 Feb 2017 06:25:26 GMT (28kb)

Link back to: arXiv, form interface, contact.
{"ft_lang_label":"__label__en","ft_lang_prob":0.8128225,"math_prob":0.9556807,"size":1221,"snap":"2020-24-2020-29","text_gpt3_token_len":309,"char_repetition_ratio":0.12571898,"word_repetition_ratio":0.0,"special_character_ratio":0.23423423,"punctuation_ratio":0.16129032,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99162,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-06-02T07:59:17Z\",\"WARC-Record-ID\":\"<urn:uuid:dc83283b-8e78-43b7-83f6-dbd7db8d916f>\",\"Content-Length\":\"15740\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a1a8f5c8-4e69-4694-873b-00c548164c0a>\",\"WARC-Concurrent-To\":\"<urn:uuid:28e2a625-f0bc-48d0-9686-03f196aa82e3>\",\"WARC-IP-Address\":\"128.84.21.203\",\"WARC-Target-URI\":\"https://export.arxiv.org/abs/1702.02718\",\"WARC-Payload-Digest\":\"sha1:WQYUWRCOV5A4VBAT34Z6TNO52EI6DHSP\",\"WARC-Block-Digest\":\"sha1:7DDWN7YOH32WAIVJPHTWCEPLIOBVIFJ6\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347423915.42_warc_CC-MAIN-20200602064854-20200602094854-00146.warc.gz\"}"}