Communications in Mathematical Physics

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

A Multi-Layer Extension of the Stochastic Heat Equation

  • Neil O’ConnellEmail author
  • Jon Warren
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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Authors and Affiliations

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

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