A Multi-Layer Extension of the Stochastic Heat Equation

Neil O'Connell; Jon Warren

doi:10.1007/s00220-015-2541-3

A Multi-Layer Extension of the Stochastic Heat Equation

Neil O'Connell, Jon Warren

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Abstract

Motivated by recent developments on solvable directed polymer models, we define a ‘multi-layer’ extension of the stochastic heat equation involving non-intersecting Brownian motions. By developing a connection with Darboux transformations and the two-dimensional Toda equations, we conjecture a Markovian evolution in time for this multi-layer process. As a first step in this direction, we establish an analogue of the Karlin-McGregor formula for the stochastic heat equation and use it to prove a special case of this conjecture.

Original language	English
Pages (from-to)	1-33
Number of pages	33
Journal	Communications in Mathematical Physics
Volume	341
Issue number	1
Early online date	24 Dec 2015
DOIs	https://doi.org/10.1007/s00220-015-2541-3
Publication status	Published - 1 Jan 2016

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title = "A Multi-Layer Extension of the Stochastic Heat Equation",

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AB - Motivated by recent developments on solvable directed polymer models, we define a ‘multi-layer’ extension of the stochastic heat equation involving non-intersecting Brownian motions. By developing a connection with Darboux transformations and the two-dimensional Toda equations, we conjecture a Markovian evolution in time for this multi-layer process. As a first step in this direction, we establish an analogue of the Karlin-McGregor formula for the stochastic heat equation and use it to prove a special case of this conjecture.

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A Multi-Layer Extension of the Stochastic Heat Equation

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