Probabilistic Solution of the Porous Medium Equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T14:19:40Z http://mathoverflow.net/feeds/question/69380 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69380/probabilistic-solution-of-the-porous-medium-equation Probabilistic Solution of the Porous Medium Equation Alexander Moll 2011-07-03T04:17:58Z 2011-07-18T13:41:58Z <p>It is well known that the transition density for standard Brownian motion $B_t$ in $\mathbb{R}^d$ yields a solution to the global Cauchy problem for the heat equation $$u_t = \Delta u$$ with initial condition given by the Dirac distribution $\delta_0$.</p> <p>Unlike the heat equation, the porous medium equation $$u_t = \Delta(u^m)$$ with exponent $m>1$ has finite speed of propagation.</p> <ul> <li><p>When is the infinitesimal generator of a stochastic process linear?</p></li> <li><p>Is there a probabilistic solution of this non-linear diffusion equation?</p></li> </ul> http://mathoverflow.net/questions/69380/probabilistic-solution-of-the-porous-medium-equation/70431#70431 Answer by AndrĂ© Schlichting for Probabilistic Solution of the Porous Medium Equation AndrĂ© Schlichting 2011-07-15T12:30:35Z 2011-07-15T12:30:35Z <p>A probabilisitc solution is given by <a href="http://www.ams.org/mathscinet-getitem?mr=1469575" rel="nofollow">MR1469575</a> a nonlinear diffusion $$Y_t = Y_0 + \int_0^t u^{\frac{m-1}{2}}(s,Y_s)\; \mathrm{d}W_s , \qquad \mathrm{law}(Y_0) = u(0,\cdot)$$ then $$\mathrm{law}(Y_t) = u(t,\cdot) .$$ This is true for a general class of nonlinear diffusion equations. The best references I've found are <a href="http://www.ams.org/mathscinet-getitem?mr=1775228" rel="nofollow">MR1775228</a> and <a href="http://www.ams.org/mathscinet-getitem?mr=2722788" rel="nofollow">MR2722788</a>.</p> http://mathoverflow.net/questions/69380/probabilistic-solution-of-the-porous-medium-equation/70620#70620 Answer by Mark Peletier for Probabilistic Solution of the Porous Medium Equation Mark Peletier 2011-07-18T13:28:53Z 2011-07-18T13:41:58Z <p>The processes described by Andre work by having the interaction act at the level of the mobility of the particles. </p> <p>There are other ways, too. One is in the work of <a href="http://sfb611.iam.uni-bonn.de/uploads/232-komplett.pdf" rel="nofollow">Philipowski</a> (see also <a href="http://alea.impa.br/articles/v4/04-09.pdf" rel="nofollow">Figalli &amp; Philipowski</a>). Here the idea is to take interactions of potential type, i.e. for instance</p> <p>$dX^i = -\sum_{j\not=i} \nabla W_\epsilon(X^i-X^j)\, dt + \delta \, dB^i.$</p> <p>The parameter $\epsilon$ is the spatial range of $W$, and in the limit $\epsilon\to0$ the interaction becomes purely local, and leads to a nonlinear diffusion term. If one also lets $\delta\to0$, then the purely Brownian contribution also vanishes. Only the nonlinear diffusion is then left. </p>