#### Date of Original Version

8-2011

#### Type

Article

#### Rights Management

© Institute of Mathematical Statistics, 2011

#### Abstract or Description

We study the dissipation mechanism of a stochastic particle system for the Burgers equation. The velocity field of the viscous Burgers and Navier–Stokes equations can be expressed as an expected value of a stochastic process based on noisy particle trajectories [Constantin and Iyer *Comm. Pure Appl. Math.* **3**(2008) 330–345]. In this paper we study a particle system for the viscous Burgers equations using a Monte–Carlo version of the above; we consider *N* copies of the above stochastic flow, each driven by independent Wiener processes, and replace the expected value with 1/*N* times the sum over these copies. A similar construction for the Navier–Stokes equations was studied by Mattingly and the first author of this paper [Iyer and Mattingly *Nonlinearity* **21** (2008) 2537–2553].

Surprisingly, for any finite *N*, the particle system for the Burgers equations shocks almost surely in finite time. In contrast to the full expected value, the empirical mean 1/*N* ∑_{1}^{N} does not regularize the system enough to ensure a time global solution. To avoid these shocks, we consider a resetting procedure, which at first sight should have no regularizing effect at all. However, we prove that this procedure prevents the formation of shocks for any *N* ≥ 2, and consequently as *N*→ ∞ we get convergence to the solution of the viscous Burgers equation on long time intervals.

#### DOI

10.1214/10-AOP586

#### Published In

Annals of Applied Probability, 39, 4, 1468-1501.