Measure-valued Flows: Girsanov Transformations

Fixρ∈(0,1), for any k≥1, define the following operators:For any k≥1, letbe the unique A_k (?) diffusion process, where denotes the position of Y^k (?) at time t starting at (z₁,…,z_k). Notice {Y^k}_k≥1 (?) is a consistent exchangeable family. Write {V_t^k}_t≥0(? ) for the semigroup of Y^k (? ) . Then there are two M₁ (R¹) valued diffusion processes X = (X_t)_t≥0 and (?) whose semigroups {T_t}_t≥0and (?) are determined uniquely by the following formulae respectively: X(?) is called a measure-valued flow given {Y^k}_k≥1 (?),briefly, measure-valued flow. For any k≥1, (z₁,…, z_k)∈R^k, there is a Girsanovtransformation between (Y_t¹(z₁),…,Y_t^k(z_k))_t≥0 and (?). Let P_μ(?) be the law of X (?) with initial valueμ∈M₁ (R¹). Notice the expressions of {T_t}_t≥0 and (?) . The following question is natural: For anyμ∈M₁ (R¹), is there a Girsanov transformation between P_μand (?)?Let (F_t)_t≥0 be the natural filtration for the coordinate process X = (X_t)_t≥0 on C_M1(R¹), and Q|_Ft the restriction of Q∈M₁ (C_R¹) to F_t.Theorem. If c(·) is a nonzero constant c (a.e. with respect to the Lebesgue measure), then for anyμ∈M₁ (R¹), there are a positive continuous P_μ-martingale (α_t)_t≥0 and a positive continuous (?)-martingale (β_t)_t≥0 on C _{M1 (R¹)} with respect to the filtration (F_t)_t≥0 such that for any t∈[0,∞) and any r∈(0,∞),Further, if∫_R¹|x|μ(dx) <∞, then∫_R¹X_t (dx) is continuous in t∈[0,∞), P_μ(?) - a.s.; If c(·) does not satisfy the condition of the theorem, then we discuss why we conjecture there is no Girsanov theorem between (?) and P_μ, namely, for any t∈(0,∞), (?) is not equivalent to P_μ|_Ft.