probability distribution for several variables - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:56:01Z http://mathoverflow.net/feeds/question/57255 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57255/probability-distribution-for-several-variables probability distribution for several variables SMH 2011-03-03T15:24:17Z 2011-03-03T16:06:42Z <p>The Fokker-Planck equation for several variables is :</p> <p>$\frac{\partial W}{\partial t} = L_{FP}W$</p> <p>where </p> <p>$L_{FP} = -\frac{\partial}{\partial x_i}D_i({x})+\frac{\partial^2}{\partial x_i \partial x_j}D_{ij}({x}).$</p> <p>The summation convention for Latin indices is used here. The drift vector $D_i$ and the diffusion tensor $D_{ij}$ generally depend on the N variables $x_1,...,x_N = {x}$. The Fokker-Planck equation is an equation for the distribution function $W({x},t)$.</p> <p>According to [Risken 1989 ch6], If drift &amp; diffusion coefficients do not depend on time &amp; $D_{ij}$ is positive definite everywhere &amp; if the drift coefficient has no singularities, a stationary solution $W_{st}$</p> <p>$L_{FP} W_{st} = 0$,</p> <p>may exist.</p> <p>If one solves the above equation, a possible stationary solution can be</p> <p>$W_{st} =\frac{a}{D_{ij}}exp(\int^{x_j}_0 \frac{D_i}{D_{ij}} dt_j)$</p> <p>Where a is a normalization constant. Now I want to expand this probability distribution for i=1,2. If I use the Einstein summation convention, it becomes</p> <p>$W_{st} ={\frac{a}{D_{11}}exp(\int^{x_1}_0 \frac{D_1}{D_{11}} dt_1)+\frac{a}{D_{12}}exp(\int^{x_2}_0 \frac{D_1}{D_{12}} dt_2)+\frac{a}{D_{21}}exp(\int^{x_1}_0 \frac{D_2}{D_{21}} dt_1)+\frac{a}{D_{22}}exp(\int^{x_2}_0 \frac{D_2}{D_{22}} dt_2)}$.</p> <p>It seems very strange to me. Is it a really correct probability distribution or I made a mistake somewhere? And if it is correct how can I normalize it? Can anyone help?</p>