Staff View: :: Library Catalog

Saved in:

Bibliographic Details
Main Author:	Chen, Hongyi
Format:	Preprint
Published:	2026
Subjects:	Probability Dynamical Systems 60H15, 60K37, 31C35
Online Access:	https://arxiv.org/abs/2602.16126
Tags:	Add Tag No Tags, Be the first to tag this record!

_version_	1866915971420127232
author	Chen, Hongyi
author_facet	Chen, Hongyi
contents	We study invariant random fields of nonlinear multiplicative stochastic heat equations in the weak disorder regime. Under a natural second-moment condition, we show that positive invariant fields are in one-to-one correspondence with bounded positive harmonic functions of the underlying space. This implies that the space of invariant fields inherits the structure of the Martin boundary. We also show that whenever the deterministic heat flow converges to a bounded harmonic function, the stochastic evolution converges to the corresponding invariant field. The results apply to many settings with nontrivial Martin boundary, such as negatively curved manifolds and trees.
format	Preprint
id	arxiv_https___arxiv_org_abs_2602_16126
institution	arXiv
publishDate	2026
record_format	arxiv
spellingShingle	Martin Boundary and the Nonlinear Multiplicative Stochastic Heat Equation in Weak Disorder Chen, Hongyi Probability Dynamical Systems 60H15, 60K37, 31C35 We study invariant random fields of nonlinear multiplicative stochastic heat equations in the weak disorder regime. Under a natural second-moment condition, we show that positive invariant fields are in one-to-one correspondence with bounded positive harmonic functions of the underlying space. This implies that the space of invariant fields inherits the structure of the Martin boundary. We also show that whenever the deterministic heat flow converges to a bounded harmonic function, the stochastic evolution converges to the corresponding invariant field. The results apply to many settings with nontrivial Martin boundary, such as negatively curved manifolds and trees.
title	Martin Boundary and the Nonlinear Multiplicative Stochastic Heat Equation in Weak Disorder
topic	Probability Dynamical Systems 60H15, 60K37, 31C35
url	https://arxiv.org/abs/2602.16126

if (filemtime($filename) + $maxlifetime < time()) { unlink($filename); $count++; } } return $count; } /** * A function that is called internally when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ protected function saveSession($sessId, $data): bool { $sessFile = $this->path . '/sess_' . $sessId; if ($handle = fopen($sessFile, 'w')) { $return = false; // Lock the file for exclusive access to avoid issues with multiple // processes writing session simultaneously: if (flock($handle, LOCK_EX)) { $return = fwrite($handle, $data); // Make sure that there's no trailing data by truncating the file to // the correct length: ftruncate($handle, strlen($data)); fflush($handle); flock($handle, LOCK_UN); } fclose($handle); if ($return !== false) { return true; } } // If we got this far, something went wrong with the file output... // It is tempting to throw an exception here, but this code is called // outside of the context of exception handling, so all we can do is // echo a message.

// Anecdotal testing Today and Yesterday seems to indicate destroy() // is called by the garbage collector and everything is good. // Something to keep in mind though. return 0; } /** * Write function that is called when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ public function write($sessId, $data): bool { if ($this->writesDisabled) { return true; } return $this->saveSession($sessId, $data); } /** * A function that is called internally when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ abstract protected function saveSession($sessId, $data): bool; }

* * @return void */ public function writeClose() { // The assumption is that we're using PHP's ext/session. // session_write_close() will actually overwrite $_SESSION with an // empty array on completion -- which leads to a mismatch between what // is in the storage object and $_SESSION. To get around this, we // temporarily reset $_SESSION to an array, and then re-link it to // the storage object. // // Additionally, while you _can_ write to $_SESSION following a // session_write_close() operation, no changes made to it will be // flushed to the session handler. As such, we now mark the storage // object isImmutable. $storage = $this->getStorage(); if (! $storage->isImmutable()) { $_SESSION = $storage->toArray(true); session_write_close(); $storage->fromArray($_SESSION); $storage->markImmutable(); } } /** * Attempt to set the session name * * If the session has already been started, or if the name provided fails * validation, an exception will be raised. * * @param string $name * @return SessionManager * @throws Exception\InvalidArgumentException */ public function setName($name) { if ($this->sessionExists()) { throw new Exception\InvalidArgumentException( 'Cannot set session name after a session has already started'

Environment & details:

GET Data empty

POST Data empty

Files empty

Key	Value
ui	"standard"
VUFIND_SESSION	"8bc3d4399f8a9476e17c6addc5822bd4"
language	"en"

Session

Key	Value
__Laminas	array:2 [`"_REQUEST_ACCESS_TIME" => 1784548826.6531 "_VALID" => array:1 [ "Laminas\Session\Validator\Id" => "8bc3d4399f8a9476e17c6addc5822bd4" ]`]
SessionState	Laminas\Stdlib\ArrayObject {#938}
FlashMessenger	Laminas\Stdlib\ArrayObject {#1583}
SessionHelper	Laminas\Stdlib\ArrayObject {#1775}

Server/Request Data

Key	Value
REDIRECT_VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
REDIRECT_HTTPS	"on"
REDIRECT_SSL_TLS_SNI	"info.umar.mx"
REDIRECT_STATUS	"200"
VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
HTTPS	"on"
SSL_TLS_SNI	"info.umar.mx"
HTTP_ACCEPT	"/"
HTTP_USER_AGENT	"Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)"
HTTP_COOKIE	"ui=standard; VUFIND_SESSION=8bc3d4399f8a9476e17c6addc5822bd4; language=en"
HTTP_ACCEPT_ENCODING	"gzip, br, zstd, deflate"
HTTP_HOST	"info.umar.mx"
PATH	"/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/snap/bin"
SERVER_SIGNATURE	"<address>Apache/2.4.66 (Ubuntu) Server at info.umar.mx Port 443</address>\n"
SERVER_SOFTWARE	"Apache/2.4.66 (Ubuntu)"
SERVER_NAME	"info.umar.mx"
SERVER_ADDR	"200.23.222.19"
SERVER_PORT	"443"
REMOTE_ADDR	"216.73.216.85"
DOCUMENT_ROOT	"/usr/local/vufind/public"
REQUEST_SCHEME	"https"
CONTEXT_PREFIX	""
CONTEXT_DOCUMENT_ROOT	"/usr/local/vufind/public"
SERVER_ADMIN	"admin@localhost"
SCRIPT_FILENAME	"/usr/local/vufind/public/index.php"
REMOTE_PORT	"51664"
REDIRECT_URL	""" /Record/\n arxiv_https___arxiv_org_abs_2602_16126/Details """
GATEWAY_INTERFACE	"CGI/1.1"
SERVER_PROTOCOL	"HTTP/1.1"
REQUEST_METHOD	"GET"
QUERY_STRING	""
REQUEST_URI	"/Record/%0Aarxiv_https___arxiv_org_abs_2602_16126/Details"
SCRIPT_NAME	"/index.php"
PHP_SELF	"/index.php"
REQUEST_TIME_FLOAT	1784548826.6483
REQUEST_TIME	1784548826

Environment Variables empty

Registered Handlers

0. Whoops\Handler\PrettyPageHandler

Environment & details:

GET Data empty

POST Data empty

Files empty

Key	Value
ui	"standard"
VUFIND_SESSION	"8bc3d4399f8a9476e17c6addc5822bd4"
language	"en"

Session

Key	Value
__Laminas	array:2 [`"_REQUEST_ACCESS_TIME" => 1784548826.6531 "_VALID" => array:1 [ "Laminas\Session\Validator\Id" => "8bc3d4399f8a9476e17c6addc5822bd4" ]`]
SessionState	Laminas\Stdlib\ArrayObject {#938}
FlashMessenger	Laminas\Stdlib\ArrayObject {#1583}
SessionHelper	Laminas\Stdlib\ArrayObject {#1775}

Server/Request Data

Key	Value
REDIRECT_VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
REDIRECT_HTTPS	"on"
REDIRECT_SSL_TLS_SNI	"info.umar.mx"
REDIRECT_STATUS	"200"
VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
HTTPS	"on"
SSL_TLS_SNI	"info.umar.mx"
HTTP_ACCEPT	"/"
HTTP_USER_AGENT	"Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)"
HTTP_COOKIE	"ui=standard; VUFIND_SESSION=8bc3d4399f8a9476e17c6addc5822bd4; language=en"
HTTP_ACCEPT_ENCODING	"gzip, br, zstd, deflate"
HTTP_HOST	"info.umar.mx"
PATH	"/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/snap/bin"
SERVER_SIGNATURE	"<address>Apache/2.4.66 (Ubuntu) Server at info.umar.mx Port 443</address>\n"
SERVER_SOFTWARE	"Apache/2.4.66 (Ubuntu)"
SERVER_NAME	"info.umar.mx"
SERVER_ADDR	"200.23.222.19"
SERVER_PORT	"443"
REMOTE_ADDR	"216.73.216.85"
DOCUMENT_ROOT	"/usr/local/vufind/public"
REQUEST_SCHEME	"https"
CONTEXT_PREFIX	""
CONTEXT_DOCUMENT_ROOT	"/usr/local/vufind/public"
SERVER_ADMIN	"admin@localhost"
SCRIPT_FILENAME	"/usr/local/vufind/public/index.php"
REMOTE_PORT	"51664"
REDIRECT_URL	""" /Record/\n arxiv_https___arxiv_org_abs_2602_16126/Details """
GATEWAY_INTERFACE	"CGI/1.1"
SERVER_PROTOCOL	"HTTP/1.1"
REQUEST_METHOD	"GET"
QUERY_STRING	""
REQUEST_URI	"/Record/%0Aarxiv_https___arxiv_org_abs_2602_16126/Details"
SCRIPT_NAME	"/index.php"
PHP_SELF	"/index.php"
REQUEST_TIME_FLOAT	1784548826.6483
REQUEST_TIME	1784548826

Environment Variables empty

Registered Handlers

0. Whoops\Handler\PrettyPageHandler

Similar Items

Environment & details:

Environment & details: