Communications in Mathematical Physics

, Volume 341, Issue 1, pp 1–33 | Cite as

A Multi-Layer Extension of the Stochastic Heat Equation

Open Access


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.


Markov Property Darboux Transformation Brownian Bridge Asymmetric Simple Exclusion Process Directed Polymer 
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Department of StatisticsUniversity of WarwickCoventryUK

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