Variational representations of Varadhan functionals (Q2718980)

Let \((X,d)\) be a Polish space with metric \(d\) and denote by \(C_a(X)\) the space of all bounded and continuous functions \(F:X\rightarrow {\mathbf R}\). A functional \({\mathbf L}:C_a(X)\rightarrow {\mathbf R}\) is a Varadhan functional if: (i) if \(F\leq G\), then \({\mathbf L}(F)\leq {\mathbf L}(G)\) and (ii) \({\mathbf L}(F+\text{const})= {\mathbf L}(F)+\text{const}\) for all \(F\in C_a(X)\), \(\text{const}\in {\mathbf R}\). A Varadhan functional is maximal if \({\mathbf L}(F\vee G)= {\mathbf L}(F)\vee{\mathbf L}( G)\). Finally such a functional is \(\sigma\)-continuous if: if \(F_n\downarrow 0\), then \({\mathbf L}(F_n)\rightarrow {\mathbf L}(0)\). All the Varadhan functionals defined on a compact space \(X\) are \(\sigma\)-continuous. The main theorem of this work establishes the following variational representation: If a maximal Varadhan functional \({\mathbf L}\) is \(\sigma\)-continuous, then there is \(L_0\in {\mathbf R}\) such that NEWLINE\[NEWLINE{\mathbf L}(F)=L_0+\sup_{x\in X}\{F(x)-{\mathbf I}(x)\}\tag{1}NEWLINE\]NEWLINE and the rate function \({\mathbf I}:X\rightarrow[0,\infty]\) is given by the dual representation NEWLINE\[NEWLINE{\mathbf I}(x)={\mathbf L}(0)+\sup_{F\in C_a(X)}\{F(x)-{\mathbf L}(F)\}.\tag{2}NEWLINE\]NEWLINE Furthermore \({\mathbf I}\) is a tight rate function, i.e. \({\mathbf I}^{-1}([0,a])\) is a compact set for all \(a>0\). This theorem provides a reverse method to prove the large deviations results for measures \(\mu_n\) by using the variational representation \((1)\) and \((2)\), given by the theorem, of the functional \({\mathbf L}\) defined by NEWLINE\[NEWLINE{\mathbf L}(F)=\lim_{n\rightarrow\infty}\frac{1}{n}\log \int_{X}\exp(nF(x)) d\mu_n.NEWLINE\]NEWLINE The methods of the paper can be considered abstractly in the sense that only uses probability theory as motivation.