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Let $M_d$ be the set of $d\times d$ complex matrices and $S_d\subset M_d$ be its subset of density matrices, i.e. $A\in S_d$ iff $A\ge 0$, $A^*=A$ and $tr(A)=1$, where $A^*$ denotes the conjugate transpose of $A$ and $tr(A)$ denotes its trace. Consider

$$d\rho_t= \left(-i[H,\rho_t]+L\rho_tL^*-\frac{1}{2}\{L^*L,\rho_t\}\right)dt+ \left(L\rho_t+\rho_tL^*-tr\big((L+L^*)\rho_t\big)\rho_t\right)dW_t,\quad \forall t>0,$$

where $i:=\sqrt {-1}$, $H\in M_d$ is a Hamiltonian matrix, $L\in M_d$, $(W_t)_{t\ge 0}$ is a Brownian motion and

$$[H,\rho_t]:=H\rho_t-\rho_t H \quad \mbox{and} \quad \{L^*L,\rho_t\}:=L^*L\rho_t+\rho_tL^*L.$$

For any given $\rho\in S_d$, it is known that there exists a unique global solution $(\rho_t)_{t\ge 0}$ taking values in $S_d$ to the above SDE such that $\rho_0=\rho$. What is the corresponding Fokker-Planck equation for the "density function" of $\rho_t$? More precisely, does there exist a reference measure, denoted by $dx$, on $S_d$ such that $Law(\rho_t):=p(t,x)dx$ and $p$ satisfies some PDE?

PS : In view of the structure of this SDE, if we consider $\rho_t$ as an vector of $\mathbb C^{d^2}$, denoted by $\Phi_t$, then there must exist $B, C\in M_{d^2}$ and $K\in \mathbb C^{d^2}$ such that

$$d\Phi_t= B\Phi_tdt+ \left(C\Phi_t-\langle K,\Phi_t\rangle\Phi_t \right)dW_t,\quad \forall t>0.$$

Roughly speaking, we fall into the classical SDE (by separating the real and imaginary parts of each term), and we obtain the density of $\Phi_t$ $P(t,x)$ with respect to the Lebesgue measure on $\mathbb C^{d^2}$. We may conclue that $P$ is supported on some subspace of $\mathbb C^{d^2}$ of dimension at most $d(d+1)/2$ (as the dimension of $S_d$ is at most $d(d+1)/2$). In particular, $P$ is almost surely equal to zero under Lebesgue measure. This suggests me to consider some "reference measure" on $S_d$ (instead of Lebesgue measure on $\mathbb C^{d^2}$).

Let $M_d$ be the set of $d\times d$ complex matrices and $S_d\subset M_d$ be its subset of density matrices, i.e. $A\in S_d$ iff $A\ge 0$, $A^*=A$ and $tr(A)=1$, where $A^*$ denotes the conjugate transpose of $A$ and $tr(A)$ denotes its trace. Consider

$$d\rho_t= \left(-i[H,\rho_t]+L\rho_tL^*-\frac{1}{2}\{L^*L,\rho_t\}\right)dt+ \left(L\rho_t+\rho_tL^*-tr\big((L+L^*)\rho_t\big)\rho_t\right)dW_t,\quad \forall t>0,$$

where $i:=\sqrt {-1}$, $H\in M_d$ is a Hamiltonian matrix, $L\in M_d$, $(W_t)_{t\ge 0}$ is a Brownian motion and

$$[H,\rho_t]:=H\rho_t-\rho_t H \quad \mbox{and} \quad \{L^*L,\rho_t\}:=L^*L\rho_t+\rho_tL^*L.$$

For any given $\rho\in S_d$, it is known that there exists a unique global solution $(\rho_t)_{t\ge 0}$ taking values in $S_d$ to the above SDE such that $\rho_0=\rho$. What is the corresponding Fokker-Planck equation for the "density function" of $\rho_t$? More precisely, does there exist a reference measure, denoted by $dx$, on $S_d$ such that $Law(\rho_t):=p(t,x)dx$ and $p$ satisfies some PDE?

In view of the structure of this SDE, if we consider $\rho_t$ as an vector of $\mathbb C^{d^2}$, denoted by $\Phi_t$, then there must exist $B, C\in M_{d^2}$ and $K\in \mathbb C^{d^2}$ such that

$$d\Phi_t= B\Phi_tdt+ \left(C\Phi_t-\langle K,\Phi_t\rangle\Phi_t \right)dW_t,\quad \forall t>0.$$

Roughly speaking, we fall into the classical SDE (by separating the real and imaginary parts of each term), and we obtain the density of $\Phi_t$ $P(t,x)$ with respect to the Lebesgue measure on $\mathbb C^{d^2}$. We may conclue that $P$ is supported on some subspace of $\mathbb C^{d^2}$ of dimension at most $d(d+1)/2$ (as the dimension of $S_d$ is at most $d(d+1)/2$). In particular, $P$ is almost surely equal to zero under Lebesgue measure. This suggests me to consider some "reference measure" on $S_d$ (instead of Lebesgue measure on $\mathbb C^{d^2}$).

Let $M_d$ be the set of $d\times d$ complex matrices and $S_d\subset M_d$ be its subset of density matrices, i.e. $A\in S_d$ iff $A\ge 0$, $A^*=A$ and $tr(A)=1$, where $A^*$ denotes the conjugate transpose of $A$ and $tr(A)$ denotes its trace. Consider

$$d\rho_t= \left(-i[H,\rho_t]+L\rho_tL^*-\frac{1}{2}\{L^*L,\rho_t\}\right)dt+ \left(L\rho_t+\rho_tL^*-tr\big((L+L^*)\rho_t\big)\rho_t\right)dW_t,\quad \forall t>0,$$

where $i:=\sqrt {-1}$, $H\in M_d$ is a Hamiltonian matrix, $L\in M_d$, $(W_t)_{t\ge 0}$ is a Brownian motion and

$$[H,\rho_t]:=H\rho_t-\rho_t H \quad \mbox{and} \quad \{L^*L,\rho_t\}:=L^*L\rho_t+\rho_tL^*L.$$

For any given $\rho\in S_d$, it is known that there exists a unique solution $(\rho_t)_{t\ge 0}$ taking values in $S_d$ to the above SDE such that $\rho_0=\rho$. What is the corresponding Fokker-Planck equation for the "density function" of $\rho_t$? More precisely, does there exist a reference measure, denoted by $dx$, on $S_d$ such that $Law(\rho_t):=p(t,x)dx$ and $p$ satisfies some PDE?

PS : In view of the structure of this SDE, if we consider $\rho_t$ as an vector of $\mathbb C^{d^2}$, denoted by $\Phi_t$, then there must exist $B, C\in M_{d^2}$ and $K\in \mathbb C^{d^2}$ such that

$$d\Phi_t= B\Phi_tdt+ \left(C\Phi_t-\langle K,\Phi_t\rangle\Phi_t \right)dW_t,\quad \forall t>0.$$

Roughly speaking, we fall into the classical SDE (by separating the real and imaginary parts of each term), and we obtain the density of $\Phi_t$ $P(t,x)$ with respect to the Lebesgue measure on $\mathbb C^{d^2}$. We may conclue that $P$ is supported on some subspace of $\mathbb C^{d^2}$ of dimension at most $d(d+1)/2$ (as the dimension of $S_d$ is at most $d(d+1)/2$). In particular, $P$ is almost surely equal to zero under Lebesgue measure. This suggests me to consider some "reference measure" on $S_d$ (instead of Lebesgue measure on $\mathbb C^{d^2}$).

Let $M_d$ be the set of $d\times d$ complex matrices and $S_d\subset M_d$ be its subset of density matrices, i.e. $A\in S_d$ iff $A\ge 0$, $A^*=A$ and $tr(A)=1$, where $A^*$ denotes the conjugate transpose of $A$ and $tr(A)$ denotes its trace. Consider

$$d\rho_t= \left(-i[H,\rho_t]+L\rho_tL^*-\frac{1}{2}\{L^*L,\rho_t\}\right)dt+ \left(L\rho_t+\rho_tL^*-tr\big((L+L^*)\rho_t\big)\rho_t\right)dW_t,\quad \forall t>0,$$

where $i:=\sqrt {-1}$, $H\in M_d$ is a Hamiltonian matrix, $L\in M_d$, $(W_t)_{t\ge 0}$ is a Brownian motion and

$$[H,\rho_t]:=H\rho_t-\rho_t H \quad \mbox{and} \quad \{L^*L,\rho_t\}:=L^*L\rho_t+\rho_tL^*L.$$

For any given $\rho\in S_d$, it is known that there exists a unique global solution $(\rho_t)_{t\ge 0}$ taking values in $S_d$ to the above SDE such that $\rho_0=\rho$. What is the corresponding Fokker-Planck equation for the "density function" of $\rho_t$? More precisely, does there exist a reference measure, denoted by $dx$, on $S_d$ such that $Law(\rho_t):=p(t,x)dx$ and $p$ satisfies some PDE?

PS : In view of the structure of this SDE, if we consider $\rho_t$ as an vector of $\mathbb C^{d^2}$, denoted by $\Phi_t$, then there must exist $B, C\in M_{d^2}$ and $K\in \mathbb C^{d^2}$ such that

$$d\Phi_t= B\Phi_tdt+ \left(C\Phi_t-\langle K,\Phi_t\rangle\Phi_t \right)dW_t,\quad \forall t>0.$$

Roughly speaking, we fall into the classical SDE (by separating the real and imaginary parts of each term), and we obtain the density of $\Phi_t$ $P(t,x)$ with respect to the Lebesgue measure on $\mathbb C^{d^2}$. We may conclue that $P$ is supported on some subspace of $\mathbb C^{d^2}$ of dimension at most $d(d+1)/2$ (as the dimension of $S_d$ is at most $d(d+1)/2$). In particular, $P$ is almost surely equal to zero under Lebesgue measure. This suggests me to consider some "reference measure" on $S_d$ (instead of Lebesgue measure on $\mathbb C^{d^2}$).

Let $M_d$ be the set of $d\times d$ complex matrices and $S_d\subset M_d$ be its subset of density matrices, i.e. $A\in S_d$ iff $A\ge 0$, $A^*=A$ and $tr(A)=1$, where $A^*$ denotes the conjugate transpose of $A$ and $tr(A)$ denotes its trace. Consider

$$d\rho_t= \left(-i[H,\rho_t]+L\rho_tL^*-\frac{1}{2}\{L^*L,\rho_t\}\right)dt+ \left(L\rho_t+\rho_tL^*-tr\big((L+L^*)\rho_t\big)\rho_t\right)dW_t,\quad \forall t>0,$$

where $i:=\sqrt {-1}$, $H\in M_d$ is a Hamiltonian matrix, $L\in M_d$, $(W_t)_{t\ge 0}$ is a Brownian motion and

$$[H,\rho_t]:=H\rho_t-\rho_t H \quad \mbox{and} \quad \{L^*L,\rho_t\}:=L^*L\rho_t+\rho_tL^*L.$$

For any given $\rho\in S_d$, it is known that there exists a unique solution $(\rho_t)_{t\ge 0}$ taking values in $S_d$ to the above SDE such that $\rho_0=\rho$. What is the corresponding Fokker-Planck equation for the "density function" of $\rho_t$? More precisely, does there exist a reference measure, denoted by $dx$, on $S_d$ such that $Law(\rho_t):=p(t,x)dx$ and $p$ satisfies some PDE?

PS : In view of the structure of this SDE, if we consider $\rho_t$ as an vector of $\mathbb C^{d^2}$, denoted by $\Phi_t$, then there must exist $B, C\in M_{d^2}$ and $K\in \mathbb C^{d^2}$ such that

$$d\Phi_t= B\Phi_tdt+ \left(C\Phi_t-\langle K,\Phi_t\rangle\Phi_t \right)dW_t,\quad \forall t>0.$$

Roughly speaking, we fall into the classical SDE (by separating the real and imaginary parts of each term), and we obtain the density of $\Phi_t$ $P(t,x)$ with respect to the Lebesgue measure on $\mathbb C^{d^2}$. We may conclue that $P$ is supported on some subspace of $\mathbb C^{d^2}$ of dimension at most $d(d+1)/2$ (as the dimension of $S_d$ is at most $d(d+1)/2$). In particular, $P$ is almost surely equal to zero under Lebesgue measure. How this may happen?

Let $M_d$ be the set of $d\times d$ complex matrices and $S_d\subset M_d$ be its subset of density matrices, i.e. $A\in S_d$ iff $A\ge 0$, $A^*=A$ and $tr(A)=1$, where $A^*$ denotes the conjugate transpose of $A$ and $tr(A)$ denotes its trace. Consider

$$d\rho_t= \left(-i[H,\rho_t]+L\rho_tL^*-\frac{1}{2}\{L^*L,\rho_t\}\right)dt+ \left(L\rho_t+\rho_tL^*-tr\big((L+L^*)\rho_t\big)\rho_t\right)dW_t,\quad \forall t>0,$$

where $i:=\sqrt {-1}$, $H\in M_d$ is a Hamiltonian matrix, $L\in M_d$, $(W_t)_{t\ge 0}$ is a Brownian motion and

$$[H,\rho_t]:=H\rho_t-\rho_t H \quad \mbox{and} \quad \{L^*L,\rho_t\}:=L^*L\rho_t+\rho_tL^*L.$$

For any given $\rho\in S_d$, it is known that there exists a unique solution $(\rho_t)_{t\ge 0}$ taking values in $S_d$ to the above SDE such that $\rho_0=\rho$. What is the corresponding Fokker-Planck equation for the "density function" of $\rho_t$? More precisely, does there exist a reference measure, denoted by $dx$, on $S_d$ such that $Law(\rho_t):=p(t,x)dx$ and $p$ satisfies some PDE?

PS : In view of the structure of this SDE, if we consider $\rho_t$ as an vector of $\mathbb C^{d^2}$, denoted by $\Phi_t$, then there must exist $B, C\in M_{d^2}$ and $K\in \mathbb C^{d^2}$ such that

$$d\Phi_t= B\Phi_tdt+ \left(C\Phi_t-\langle K,\Phi_t\rangle\Phi_t \right)dW_t,\quad \forall t>0.$$

Roughly speaking, we fall into the classical SDE (by separating the real and imaginary parts of each term), and we obtain the density of $\Phi_t$ $P(t,x)$ with respect to the Lebesgue measure on $\mathbb C^{d^2}$. We may conclue that $P$ is supported on some subspace of $\mathbb C^{d^2}$ of dimension at most $d(d+1)/2$ (as the dimension of $S_d$ is at most $d(d+1)/2$). In particular, $P$ is almost surely equal to zero under Lebesgue measure. This suggests me to consider some "reference measure" on $S_d$ (instead of Lebesgue measure on $\mathbb C^{d^2}$).