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A Multi-Layer Extension of the Stochastic Heat Equation

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.

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Correspondence to Neil O’Connell.

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Communicated by H. Spohn

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O’Connell, N., Warren, J. A Multi-Layer Extension of the Stochastic Heat Equation. Commun. Math. Phys. 341, 1–33 (2016). https://doi.org/10.1007/s00220-015-2541-3

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Keywords

  • Markov Property
  • Darboux Transformation
  • Brownian Bridge
  • Asymmetric Simple Exclusion Process
  • Directed Polymer