Geodesic
$d$-dimensional pressureless
Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 3, pp. 610-614

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Let $x\in R^d\to u(x,0)$ be a continuous bounded function and $\rho(dx,0)$ a probability measure on $R^d$. For all random variables $X_0$ with probability distribution $\rho(dx,0)$, we show that the stochastic differential equation (SDE) $$ X_t = X_0 + \int_0^t E\big[u(X_0,0)\,|\, X_s\big]\,ds,\qquad t\ge 0, $$ has a solution which is a $\sigma(X_0)$-measurable Markov process. We derive a weak solution for the pressureless gas equation for $d \ge 1$, with initial distribution of masses $\rho(dx,0)$ and initial velocity $u(\cdot,0)$. We show for $d = 1$ the existence of a unique Markov process $(X_t)$ solution of our SDE.
Keywords: pressureless gas equations
Mots-clés : variational principles. 
@article{TVP_2004_49_3_a11,
     author = {A. Dermoune},
     title = {$d$-dimensional pressureless},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {610--614},
     publisher = {mathdoc},
     volume = {49},
     number = {3},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2004_49_3_a11/}
}
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A. Dermoune. $d$-dimensional pressureless. Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 3, pp. 610-614. http://geodesic.mathdoc.fr/item/TVP_2004_49_3_a11/