For an adapted cadlag process \((X_ t)\) on \((\Omega,F,P,F_ t)\), set \(\Delta (n,i,x)=X_{(i+1)/2^ n}-X_{i/2^ n},\) \(S(n,X)=E[X^ 2_ 0]+\sum_{i}E[\Delta (n,i,x)^ 2],\) \(\| X\|^ 2_ e=\sup_{n}S(n,X)\) and \(Q(X)= \limsup_{n}S(n,X).\) The authors prove that, if \(\| X\|_ e<\infty\) then X can be decomposed into the sum of a martingale M and a previsible process A such that \(\lim_{j}E[A_ 0Y_ 0+\sum \Delta (n_ j,i,A)\Delta (n_ j,i,Y)]=0\) for all square integrable martingales Y. If A can be chosen so that \(Q(A)=0\) then the decomposition is unique. Let \({\mathcal H}\) be the space of processes X such that \(\| X\|_ e<\infty\) and \(Q(A)=0\), then the semimartingales are dense in \({\mathcal H}\) relative to \(\| \cdot \|_ e\)-norm and, for \(f\in C^ 1_ b\), \(f(X)\in {\mathcal H}\) and its martingale part is given by \(Y_ 0+\int^{t}_{0}f'(X_ s)dM_ s.\)
There are different definitions of energy related with Hunt processes, one of which is the energy of class (D) potentials introduced by Z. R. Pop-Stōjanovic and M. Rao [Z. Wahrscheinlichkeitstheor. Verw. Geb. 69, 593-608 (1985; Zbl 0548.60073)] and the other is the energy of additive functionals of symmetric Hunt processes introduced by M. Fukushima [Dirichlet forms and Markov processes, (1980; Zbl 0422.31007)]. The authors discuss the relations among those energies and characterize BLD functions f with compact support as \(\| f(B_ t)\|_ e<\infty\).