Reciprocal Time Relation of Noncolliding Brownian Motion with Drift

Abstract

We consider an N-particle system of noncolliding Brownian motion starting from x ₁≤x ₂≤…≤x _N with drift coefficients ν _j , 1≤j≤N satisfying ν ₁≤ν ₂≤…≤ν _N . When all of the initial points are degenerated to be zero, x _j =0, 1≤j≤N, the equivalence is proved between a dilatation with factor 1/t of this drifted process and the noncolliding Brownian motion starting from ν ₁≤ν ₂≤…≤ν _N without drift observed at reciprocal time 1/t, for arbitrary t>0. Using this reciprocal time relation, we study the determinantal property of the noncolliding Brownian motion with drift having finite and infinite numbers of particles.

Appendices

Appendix A: Asymptotics of Determinants

By the Schur function expansion, we can prove that [19, 20] for b∈ℂ^N

$$ \det_{1 \leq j, k \leq N} \bigl[e^{a_j b_k}\bigr] =\frac{h_N(\boldsymbol{a}) h_N(\boldsymbol{b})}{\prod_{j=1}^N \varGamma(j)} \times\bigl \{1+\mathcal{O}\bigl(|\boldsymbol{a}|\bigr) \bigr\} \quad\mbox{as } |\boldsymbol{a}|\to0. $$

(A.1)

Then

$$ \lim_{|\boldsymbol{\nu}| \to0} \frac{ { \det_{1 \leq j, k \leq N}[e^{\nu_j y_k}]}}{ { \det_{1 \leq j, k \leq N}[e^{\nu_j x_k}]}} =\frac{h_N(\boldsymbol {y})}{h_N(\boldsymbol{x})}, $$

(A.2)

which implies the fact that (1.7) is reduced from (1.13) by taking ν _j →0,1≤j≤N.

Similarly, we can see that

$$\lim_{|\boldsymbol{x}| \to0} \frac{ { \det_{1 \leq j, k \leq N} [e^{x_j y_k/t}]}}{ { \det_{1 \leq j, k \leq N} [e^{\nu_j x_k}]}} = \lim_{|\boldsymbol{x}| \to0} \frac{h_{N}(\boldsymbol{x}/\sqrt{t}) h_N(\boldsymbol{y}/\sqrt{t})}{ h_N(\boldsymbol{\nu}) h_N(\boldsymbol{x})} =\frac{h_N(\boldsymbol {y}/t)}{h_N(\boldsymbol{\nu})}. $$

Since q _N (t,y|x) given by (1.8) is equal to $(2 \pi t)^{-N/2} e^{-(|\boldsymbol{x}|^{2}+|\boldsymbol{y}|^{2})/2t} \det_{1 \leq j, k \leq N} [e^{x_{j} y_{k}/t}]$, (1.13) gives

$$\lim_{|\boldsymbol{x}| \to0} p_N^{\boldsymbol{\nu}}(t, \boldsymbol{y}| \boldsymbol{x}) =e^{-t|\boldsymbol{\nu}|^2/2} \det_{1 \leq j, k \leq N}\bigl[e^{\nu_j y_k}\bigr] (2\pi t)^{-N/2} e^{-|\boldsymbol{y}|^2/2t} \frac{h_N(\boldsymbol {y}/t)}{h_N(\boldsymbol{\nu})}, $$

which is equal to (1.21).

Appendix B: On the O’Connell Process with Drift ν

Since the formula (1.17) is not found in [30], here we give explanation for it and then prove (1.20). Note that the derivation of (1.17) with ν=0 was given in [15, 16].

For a>0,ν∈ℝ^N, we consider the following partial differential equation

$$ \biggl[ \frac{\partial}{\partial t} +\mathcal{H}_N-\frac{1}{a} \boldsymbol{\nu}\cdot\nabla\biggr] u_a^{\boldsymbol{\nu}}(t, \boldsymbol{x})=0, \quad\boldsymbol{x}\in\mathbb{R}^N, \ t \in[0, \infty), $$

(B.1)

where

$$ \mathcal{H}_N=-\frac{1}{2} \Delta+\frac{1}{a^2} \sum _{j=1}^{N-1} e^{-(x_{j+1}-x_j)/a} $$

(B.2)

is identified with the Hamiltonian of an open quantum Toda lattice [30]. Assume that $\boldsymbol{x}, \boldsymbol{\nu}\in\mathbb{W}_{N}$. Then by the Feynman-Kac formula (see, for instance, [13]) the stationary solution $u^{\boldsymbol{\nu}}_{a}(\boldsymbol{x})$ of (B.1) with (B.2), which is uniquely determined by imposing the condition $\lim_{|\boldsymbol{x}| \to\infty, \boldsymbol{x}\in\mathbb {W}_{N}} u_{a}^{\boldsymbol{\nu}}(\boldsymbol{x})=1$, $\boldsymbol{\nu}\in\mathbb{W}_{N}$, is given by

$$ u_{a}^{\boldsymbol{\nu}}(\boldsymbol{x})=\mathbf{E}^{\boldsymbol {x}} \Biggl[ \exp\Biggl( - \frac{1}{a^2} \sum_{j=1}^{N-1} \int_0^{\infty} e^{-\{\widehat{B}_{j+1}(s)-\widehat{B}_j(s)\}/a} ds \Biggr) \Biggr], $$

(B.3)

where E ^x[ ⋅ ] denotes the expectation with respect to the Brownian motion (1.5) with drift $\boldsymbol{\nu}\in\mathbb{W}_{N}$ and starting from $\boldsymbol{x}\in\mathbb{W}_{N}$. We note that for 0≤T<∞

$$ \mathcal{N}_N^{\boldsymbol{\nu}, \, a}(T, \boldsymbol{x}) = \mathbf{E}^{\boldsymbol{x}} \Biggl[ \exp\Biggl( - \frac{1}{a^2} \sum _{j=1}^{N-1} \int_0^{T} e^{-\{\widehat{B}_{j+1}(s)-\widehat{B}_j(s)\}/a} ds \Biggr) \Biggr] $$

(B.4)

expresses the probability that, in the mutually killing N-particle system with the killing term $-(1/a^{2}) \sum_{j=1}^{N-1} e^{-(x_{j+1}-x_{j})/a}$ [16], all N Brownian particles with drifts $\{\widehat{B}_{j}(t)\}_{j=1}^{N}$ starting from $\boldsymbol{x}\in \mathbb{W}_{N}$ survive up to time T and that (B.3) is its long-term limit, $\lim_{T \to\infty} \mathcal{N}_{N}^{\boldsymbol{\nu}, \, a}(T, \boldsymbol{x})$, $\boldsymbol{x}, \boldsymbol{\nu}\in\mathbb{W}_{N}$. On the other hand, the class-one Whittaker function $\psi^{(N)}_{\boldsymbol{\nu}}(\boldsymbol{x})$ given by (1.15) is an eigenfunction of the Hamiltonian (B.2) with the eigenvalue $-\sum_{j=1}^{N} \nu_{j}^{2}/2$, ν∈ℂ^N. By the method of separation of variables, we can show that u _a (x) is also expressed by using $\psi^{(N)}_{\boldsymbol{\nu}}(\boldsymbol{x}/a)$. Then the equality

$$ \lim_{T \to\infty} \mathcal{N}_N^{\boldsymbol{\nu}, \, a}(T, \boldsymbol{x}) =c_N(\boldsymbol{\nu}) e^{-\boldsymbol{\nu}\cdot\boldsymbol{x}/a} \psi^{(N)}_{\boldsymbol{\nu}}( \boldsymbol{x}/a), \quad\boldsymbol{x}, \boldsymbol{\nu}\in\mathbb{W}_N, $$

(B.5)

is established, where c _N (ν)=∏_1≤j<k≤N {sinπ(ν _k −ν _j )}/π [31].

We find that $Q_{N}^{a}(t, \boldsymbol{y}|\boldsymbol{x})$ given by (1.18) solves (B.1) with ν=0 and $Q_{N}^{a}(0,\boldsymbol{y}|\boldsymbol {x})=\delta(\boldsymbol{x}-\boldsymbol{y})$, and then we can regard it as a transition probability density of the mutually killing Brownian motions with duration t∈[0,∞) from x∈ℝ^N to y∈ℝ^N, preserving the number of particles [15, 16]. Then the transition probability density of the mutually killing Brownian motions with drift ν conditioned that all particle survive is given by

$$ P_N^{\boldsymbol{\nu},\, a}(t, \boldsymbol{y}|\boldsymbol{x}) = \lim_{T \to\infty} \frac{\mathcal{N}_N^{\boldsymbol{\nu}, \, a}(T-t, \boldsymbol{y})}{ \mathcal{N}_N^{\boldsymbol{\nu}, \, a}(T, \boldsymbol{x})} Q_N^{\boldsymbol{\nu}, \, a}(t, \boldsymbol{y}|\boldsymbol{x}) $$

(B.6)

with the drift transform of (1.18) with parameter a>0

$$ Q_N^{\boldsymbol{\nu}, \, a}(t, \boldsymbol{y}|\boldsymbol{x}) = \exp\biggl\{ - \frac{t|\boldsymbol{\nu}|^2}{2 a^2} +\frac{\boldsymbol{\nu}}{a} \cdot(\boldsymbol{y}-\boldsymbol{x}) \biggr\} Q_N^{a}(t,\boldsymbol{y}|\boldsymbol{x}). $$

(B.7)

If $\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{\nu}\in\mathbb {W}_{N}$, by (B.5), (B.6) is equal to (1.17), which should solve (1.16) with the initial condition $P_{N}^{\boldsymbol{\nu}, \, a}(0, \boldsymbol{y}|\boldsymbol {x})=\delta(\boldsymbol{x}-\boldsymbol{y})$.

The class-one Whittaker function has the alternating sum formula [1],

$$ \psi^{(N)}_{\boldsymbol{\nu}}(\boldsymbol{x})=c_N(\boldsymbol{ \nu})^{-1} \sum_{\sigma\in\mathcal{S}_N} \operatorname {sgn}(\sigma) m^{(N)}\bigl(\boldsymbol{x}, \sigma(\boldsymbol{\nu})\bigr), $$

(B.8)

where σ(ν)=(ν _σ(1),…,ν _σ(N)) for each permutation $\sigma\in\mathcal{S}_{N}$. Here m ^(N)(x,ν) is the fundamental Whittaker function, which is normalized here as lim _a→0 m ^(N)(x/a,a ν)=e ^ν⋅x for $\boldsymbol{x}\in\mathbb{W}_{N}$. Since c _N (a ν)≃a ^N(N−1)/2 h _N (ν) as a→0, we have

$$ \lim_{a \to0} a^{N(N-1)/2} \psi^{(N)}_{a \boldsymbol{\nu }}( \boldsymbol{x}/a) =\frac{ {\det_{1 \leq j, k \leq N}[e^{\nu_j x_k}]}}{h_N(\boldsymbol{\nu})} \quad\mbox{for }\boldsymbol{x},\boldsymbol{\nu}\in\mathbb{W}_N. $$

(B.9)

Moreover, the density of Sklyanin measure (1.19) has the asymptotics in a→0 as s _N (a k)≃a ^N(N−1)(h _N (k))²/{(2π)^N N!}, and thus (1.18) gives [16]

(B.10)

Then (1.20) is concluded. Note that the first equality in (B.10) can be interpreted in terms of the Slater determinants used in quantum mechanics [33].