I have a markov chain with $Q(u,v)$ as transition probability matrix and $\pi(u)$ as stationary distribution defined on state space $\Omega$. The dimension of matrix $Q$ is $nxn$ and vector $\pi$ is $1xn$.

I need to construct a vorticity matrix $\Gamma (u,v)$ of dimension $nxn$ which has below properties

  1. $\Gamma$ is skew symmetric matrix i.e, $$\Gamma (u,v) = -\Gamma (v,u) \quad ,\forall \, u,v \in \Omega $$

  2. Row sum of $\Gamma$ is zero for every row i.e, $$ \sum_v \Gamma (u,v) = 0 \quad ,\forall \, u \in \Omega $$

  3. Third property is, $$\Gamma(u,v) > -\pi (v)Q(v,u) \quad ,\forall \, u,v \in \Omega $$

My question is : How to construct vorticity matrix $\Gamma (u,v)$ which satisfies above three properties? I need to construct at least one such matrix.

Is there any systematic way to build such matrices

NOTE: Transition probability matrix $P$, and stationary distribution $\pi$ has below properties

Row sum of $P$ is one for each row, $$\sum_v P(u,v)=1 \quad ,\forall \, u \in \Omega$$ $\pi$ is probability distribution hence, $$\sum_v \pi(v) = 1$$ Stationary distribution condition for $\pi$, $$\sum_u \pi(u) P(u,v) = \pi(v) \quad ,\forall \, v \in \Omega $$

  • $\begingroup$ Changing third property to $\Gamma \geq \epsilon + \dots$ for some small $\epsilon$, you can use LP for this. $\endgroup$ – Eugene Aug 4 '19 at 19:42

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