Stochastic Nonlinear Schrödinger Equations with Linear Multiplicative Noise: Rescaling Approach

Abstract

We prove well-posedness results for stochastic nonlinear Schrödinger equations with linear multiplicative Wiener noise, including the nonconservative case. Our approach is different from the standard literature on stochastic nonlinear Schrödinger equations. By a rescaling transformation we reduce the stochastic equation to a random nonlinear Schrödinger equation with lower-order terms and treat the resulting equation by a fixed point argument based on generalizations of Strichartz estimates proved by Marzuola et al. (J Funct Anal 255(6):1479–1553, 2008). This approach makes it possible to improve earlier well-posedness results obtained in the conservative case by a direct approach to the stochastic Schrödinger equation. In contrast to the latter, we obtain well-posedness in the full range $[1, 1 + 4/d)$ of admissible exponents in the nonlinear part (where $d$ is the dimension of the underlying Euclidean space), i.e., in exactly the same range as in the deterministic case.

Appendices

Appendix

Proof of Lemma 4.1

Under assumption (H1), the coefficients $c,b$ defined in (3.4) satisfy (1.4)–(1.6) in Marzuola et al. (2008) on $[0,T]\times {\mathbb {R}}^d$. Recall that in Marzuola et al. (2008) [e.g., (1.1)], the common notation $D_t=-i{\partial }_t,\,D_j=-i{\partial }_{x_j}$ is used. Then, by Theorem 1.13 in Marzuola et al. (2008) and, more precisely, by estimate (1.24) (Remark 1.17 in Marzuola et al. 2008), we have

$$\begin{aligned} \Vert u\Vert _{L^{q_1}(0,T;L^{p_1})}\le C\left( |u_0|_{2}+\Vert f\Vert _{L^{q'_2}(0,T;L^{p'_2})} +\Vert u\Vert _{L^2(0,T;L^2(|\xi |\le 2R))}\right) \end{aligned}$$

(6.3)

for $R$ sufficiently large.

We will prove first that (4.3) holds for $T$ sufficiently small. To this end, we note that

$$\begin{aligned} \Vert u\Vert ^2_{L^2(0,T;L^2(|\xi |\le 2R))}&\le (m(B_{2R}))^{\frac{p_1-2}{p_1}} \int \limits _0^T|u(t)|^2_{L^{p_1}}\hbox {d}t \\&\le (m(B_{2R}))^{\frac{p_1-2}{p_1}} T^{\frac{q_1-2}{q_1}} \Vert u\Vert ^2_{L^{q_1}(0,T;L^{p_1})}, \end{aligned}$$

where $m(B_{2R})$ is the volume of the ball $B_{2R}$ of radius $2R$. For simplicity, we assume that $q_1>2$, which is, in fact, the case in the application of Lemma 4.1 to problem (3.5). Then for

$$\begin{aligned} 0<T=\left( (2C)^{-2}(m(B_{2R})) ^{-\frac{p_1-2}{p_1}}\right) ^{\frac{q_1}{q_1-2}} \end{aligned}$$

(6.4)

we obtain by (6.3) that

$$\begin{aligned} \Vert u\Vert _{L^{q_1}(0,T;L^{p_1})} \le 2C\left( |u_0|_2+\Vert f\Vert _{L^{q'_2}(0,T;L^{p'_2})}\right) . \end{aligned}$$

(6.5)

For $q_1={\infty },\ p_1=2$ we obtain in a similar way

$$\begin{aligned} \Vert u\Vert _{L^{\infty }(0,T;L^{2})}\le 2C\left( |u_0|_2 +\Vert f\Vert _{L^{q'_2}(0,T;L^{p'_2})}\right) \end{aligned}$$

for $0<T<(2C)^{-2}$. Reiterating (6.5) on the interval $(T,2T)$, we therefore obtain

$$\begin{aligned} \Vert u\Vert _{L^{q_1}(T,2T;L^{p_1})}&\le 2C\left( |u(T)|_2+\Vert f\Vert _{L^{q'_2}(T,2T; L^{p'_2})}\right) \\&\le 2C\left[ 2C(|u_0|_2+ \Vert f\Vert _{L^{q'_2}(0,T;L^{p'_2})}) +\Vert f\Vert _{L^{q'_2}(T,2T;L^{p'_2})}\right] \\&\le 2C\left[ 2C|u_0|_2 +(2C+1)\Vert f\Vert _{L^{q'_2}(0,2T;L^{p'_2})}\right] ,\\&\le 4C(C+1)\left( |u_0|_2+\Vert f\Vert _{L^{q'_2}(0,2T;L^{p'_2})}\right) . \end{aligned}$$

Hence,

$$\begin{aligned} \Vert u\Vert _{L^{q_1}(0,2T;L^{p_1})} \le 8C(C+1)\left( |u_0|_2+\Vert f\Vert _{L^{q'_2} (0,2T;L^{p'_2})}\right) . \end{aligned}$$

Then, after a finite number of steps, we obtain estimate (4.3) on an arbitrary bounded interval, as claimed.

Furthermore, for each $t\in [0,T]$ we may take

$$\begin{aligned} C_t&= \sup \{\Vert U(t,0)u_0\Vert _{L^{q_1}(0,t;L^{p_1})} ;|u_0|_2\le 1\}\\&\quad + \sup \left\{ \left\| \int \limits _0^tU(t,s)f(s)\hbox {d}s\right\| _{L^{q_1}(0,t;L^{p_1})}; \Vert f\Vert _{L^{q'_2}(0,t;L^{p'_2})}=1\right\} . \end{aligned}$$

Obviously, the function $t\rightarrow C_t$ is monotonically increasing, $C_0=0$, and it follows by (4.3) and standard arguments that it is continuous. Since by separability the sup in the definition of $C_t$ is a sup over countably many $u_0\in L^2$ and $f\in L^{q'_2}(0,t;L^{p'_2})\subset L^1(0,t;H^{-1})$ (by Sobolev embedding) and since, as seen earlier in Lemma 3.3, $t\rightarrow U(t,0)u_0,\,t\rightarrow \int _0^tU(t,s)f(s)\hbox {d}s$ is adapted, we conclude that $t\rightarrow C_t$ is adapted to the filtration $(\mathcal {F}_t)_{t\ge 0}$. But then, as a continuous process, $C_t$ is $(\mathcal {F}_t)$-progressively measurable, thereby completing the proof. $\square$

Lemma 6.1

• (i) Let $y=y(t),\ t\in [0,T]$, be an $L^2$-valued $(\mathcal {F}_t)$-adapted process with continuous sample paths satisfying (3.3), (3.6), and (3.7). Then $X:=e^Wy$ is a solution to (2.1).

• (ii) Suppose $X=X(t),\,t\in [0,T]$ is an $L^2$-valued $(\mathcal {F}_t)$-adapted process with continuous sample paths satisfying (2.1), (2.2), and (2.3). Then $y:=e^{-W}X$ satisfies (3.3) [equivalently, (3.5)].

Before going to the proof of Lemma 6.1, a few remarks are in order concerning the formal calculation given at the beginning of Sect. 3 to link (2.1) and (3.2). In fact, it is purely heuristic since we applied Itô’s product rule to $y$, though it is not of bounded variation in $L^2$. Furthermore, taking into account that the exponential is an operator of the Nemitsky type in $L^2$ that is not differentiable, the infinite-dimensional Itô formula in $L^2$ is not justified. Also, when we try to apply Itô’s product rule to real-valued stochastic processes after evaluating the $L^2$-valued processes $X,\,W,\,y$ at $\xi \in {\mathbb {R}}^d$, which by itself is delicate since $L^2$ consists of equivalence classes of functions, we run into problems since, for example, again $X(t,\xi ),\,y(t,\xi ),\,t\in [0,T]$, might not be semimartingales.

The following proof is based on the stochastic Fubini theorem and uses the stochastic calculus for complex-valued processes and their products in $\mathbb {C}$. (We refer the reader to Kendal and Price, Sect. 2, as background literature in this regard.)

Proof of Lemma 7.1

We only prove (i) since (ii) can be proved analogously. Let ${\varphi }\in H^2({\mathbb {R}}^d)$. Then, for every $t\in [0,T]$, we have

$$\begin{aligned} \left<{\varphi },e^{W(t)}y(t)\right>_2=\sum ^{\infty }_{j=1} \left<\overline{e^{W(t)}}{\varphi },f_j\right>_2\left<f_j,y(t)\right>_2, \end{aligned}$$

where $\{f_j\}^{\infty }_{j=1}$ is an orthonormal basis in $L^2;\ f_j\in H^2({\mathbb {R}}^d)$.

By Itô’s formula, we have for all $\xi \in {\mathbb {R}}^d,\,t\in [0,T]$,

$$\begin{aligned} e^{W(t,\xi )}=1+\int \limits _0^t e^{W(s,\xi )}\hbox {d}W(s,\xi ) +\widetilde{\mu }(\xi )\int \limits _0^t e^{W(s,\xi )}\hbox {d}s. \end{aligned}$$

Fix $j\in {\mathbb {N}}$. Then we have $\mathbb {P}\text{-a.s. }$, for all $t\in [0,T],$

$$\begin{aligned}&\left<\overline{e^{W(t)}}{\varphi },f_j\right>_2 \\&\quad =\!\left<{\varphi },f_j\right>_2\!+\!\sum ^N_{k=1}\overline{\mu }_k \int \limits _{{\mathbb {R}}^d}{\varphi }(\xi )e_k(\xi ) \bar{f}_j(\xi )\hbox {d}\xi \int \limits _0^t\overline{e^{W(s,\xi )}}\hbox {d}{\beta }_k(s) \!+\!\int \limits _0^t\left<\overline{\widetilde{\mu }}\,\overline{e^{W(s)}} {\varphi },f_j\right>_2\hbox {d}s\\&\quad =\left<{\varphi },f_j\right>_2+\sum ^N_{k=1}\overline{\mu }_k \int \limits _0^t\left< e_k\,\overline{e^{W(s)}}{\varphi },f_j\right>_2\mathrm{d}{\beta }_k(s) +\int \limits _0^t\left<\overline{\widetilde{\mu }}\,\overline{e^{W(s)}}{\varphi },f_j\right>_2\hbox {d}s. \end{aligned}$$

(Here we have used the stochastic Fubini theorem in the second equality.)

Now we set $A_0=i{\Delta }$, $D(A_0)=H^2({\mathbb {R}}^d)$, and $J_{\varepsilon }=(I+{\varepsilon }A_0)^{-1}$.

Let $y_{\varepsilon }=J_{\varepsilon }(y)$. Then $y_{\varepsilon }\in C([0,T],H^2({\mathbb {R}}^d))$ and

$$\begin{aligned} \frac{{\partial }y_{\varepsilon }}{{\partial }t}&= -i J_{\varepsilon }(e^{-W}{\Delta }(e^Wy))-J_{\varepsilon }((\mu +\widetilde{\mu }) y)\nonumber \\&\quad -{\lambda }iJ_{\varepsilon }(|e^{({\alpha }-1)W}|\,|y|^{{\alpha }-1}y),\quad t\in (0,T),\\ y_{\varepsilon }(0)&= J_{\varepsilon }(x)=x_{\varepsilon }.\nonumber \end{aligned}$$

(7.4)

Since $f_j\in H^2({\mathbb {R}}^d)$, for each $j,\,\left<f_j,y_{\varepsilon }(t)\right>_2,\ t\in [0,T]$, is of bounded variation. Hence, we can apply Itô’s product rule (for scalar-valued processes) to obtain

$$\begin{aligned}&\left<\overline{e^{W(t)}}{\varphi },f_j\right>_2\left<f_j,y_{\varepsilon }(t)\right>_2 =\left<{\varphi },f_j\right>_2\left<f_j,x_{\varepsilon }\right>_2\\&\quad +i\int \limits _0^t\left<\overline{e^{W(s)}}{\varphi },f_j\right>_2 \left<f_j,J_{\varepsilon }(e^{-W(s)}{\Delta }(e^{W(s)}y(s)))\right>_2\hbox {d}s\\&\quad -\int \limits _0^t\left<\overline{e^{W(s)}}{\varphi },f_j\right>_2 \left<f_j,J_{\varepsilon }((\mu +\widetilde{\mu }) y(s))\right>_2\hbox {d}s\\&\quad +{\lambda }i\int \limits _0^t\left<\overline{e^{W(s)}}{\varphi },f_j\right>_2 \left<f_j,J_{\varepsilon }(|e^{({\alpha }-1) W(s)}|\,|y(s)|^{{\alpha }-1}y(s))\right>_2\hbox {d}s\\&\quad +\sum ^N_{k=1}\overline{\mu }_k\int \limits _0^t \left<f_j,y_{\varepsilon }(s)\right>_2 \left<e_k\,\overline{e^{W(s)}}{\varphi },f_j\right>_2\mathrm{d}{\beta }_k(s)\\&\quad +\int \limits _0^t\left<f_j,y_{\varepsilon }(s)\right>_2\left<\overline{\widetilde{\mu }}\, \overline{e^{W(s)}}{\varphi },f_j\right>_2\hbox {d}s. \end{aligned}$$

(We note that, since $J_{\varepsilon }(e^{-W}{\Delta }(e^Wy))\in C([0,T];L^2)$, the second integral in the preceding equality makes sense.)

Now, summing over $j\in {\mathbb {N}}$ and interchanging the infinite sum with the integrals, we obtain $\mathbb {P}\text{-a.s. }$, for all $t\in [0,T]$,

$$\begin{aligned}&\left<{\varphi },e^{W(t)}y_{\varepsilon }(t)\right>_2=\left<{\varphi },x_{\varepsilon }\right>_2 +i \int \limits _0^t\left<{\varphi },e^{W(s)} J_{\varepsilon }(e^{-W(s)}{\Delta }(e^{W(s)}y(s)))\right>_2\hbox {d}s\\&\quad - \int \limits _0^t\!\!\left<{\varphi },e^{W(s)}J_{\varepsilon }((\mu +\widetilde{\mu }) y)\right>_2\hbox {d}s +{\lambda }i \int \limits _0^t\!\!\left<{\varphi },e^{W(s)} J_{\varepsilon }(|e^{({\alpha }-1)W}|\,|y(s)|^{{\alpha }-1}y(s))\right>_2\hbox {d}s\\&\quad + \sum ^N_{k=1}\int \limits _0^t \left<{\varphi },\mu _k e_ke^{W(s)}y_{\varepsilon }(s)\right>_2 \hbox {d}{\beta }_k(s) + \int \limits _0^t\left<{\varphi },\widetilde{\mu }e^{W(s)}y_{\varepsilon }(s)\right>_2\hbox {d}s. \end{aligned}$$

On the other hand, we have, for ${\varepsilon }\rightarrow 0$,

$$\begin{aligned} J_{\varepsilon }(f)\rightarrow f \text{ strongly } \text{ in } H^k\text{. } \end{aligned}$$

Furthermore,

$$\begin{aligned} \Vert J_{\varepsilon }(f)\Vert _{H^k}\le \Vert f\Vert _{H^k}, \end{aligned}$$

where $f\in H^k$ and $k=0,1,2$. Then we may pass to the limit ${\varepsilon }\rightarrow 0$ in the previous equality to obtain

$$\begin{aligned} \left<{\varphi },e^{W(t)}y(t)\right>_2&= \left<{\varphi },x\right>_2 +i\int \limits _0^t\left<{\varphi },{\Delta }(e^{W(s)}y(s))\right>\hbox {d}s -\int \limits _0^t\left<{\varphi },\mu e^{W(s)}y(s)\right>_2\hbox {d}s\\&\quad +{\lambda }i\int \limits _0^t\left<{\varphi },e^{W(s)}|e^{({\alpha }-1)W(s)}|\, |y(s)|^{{\alpha }-1}y(s)\right>\hbox {d}s\\&\quad +\sum ^N_{k=1}\int \limits _0^t\left<{\varphi },\mu _k e_k e^{W(s)}y(s)\right>_2\mathrm{d}{\beta }_k(s),\quad \forall t\in [0,T], \end{aligned}$$

which implies the fact that $X(t)=e^{W(t)}y(t)$ is the solution to (2.1), as claimed. In the preceding equality, $\left<\cdot ,\cdot \right>$ is the pairing between $L^2$, $H^2$, and $H^{-2}$ or, equivalently,

$$\begin{aligned} \left<{\varphi },{\Delta }(e^Wy)\right>=\int \limits _{{\mathbb {R}}^d} {\Delta }{\varphi }\overline{e^W}\bar{y}\,\hbox {d}\xi ,\quad {\varphi }\in H^2. \end{aligned}$$

This completes the proof. $\square$

Lemma 6.2

Let $\tau _{n+1}$ be defined as in Step 2 in the proof of Lemma 4.2. Then $y_{n+1}$ is adapted to $(\mathcal {F}_t)$.

Proof

We first note that, by $z_{n+1}=F_n(z_{n+1})$ and Banach’s fixed point theorem, there exists a sequence $\{v_{n+1,m}\}_{m\ge 1}$, adapted to $(\mathcal {F}_{\tau _n+t})$, satisfying $v_{n+1,m+1}=F_n(v_{n+1,m})$ for $m\ge 1,\,v_{n+1,1}(t)=U(\tau _n+t,\tau _n)y_n(\tau _n)$, and $z_{n+1}=\lim _{m\rightarrow {\infty }}v_{n+1,m}$ in $C([0,t];L^2)\cap L^q(0,t;L^{{\alpha }+1})),\,t\in [0,\sigma _n]$. Now we define

$$\begin{aligned} u_{n+1,m}(t)=\left\{ \begin{array}{l@{\quad }l} y_n(t),&{}t\in [0,\tau _n],\\ v_{n+1,m}(t-\tau _n),&{}t\in (\tau _n,{\infty }). \end{array}\right. \end{aligned}$$

Thus, $y_{n+1}=\lim _{m\rightarrow {\infty }}u^{\tau _{n+1}}_{n+1,m}$ in $C([0,T];L^2)$. In what follows, we show that $u_{n+1,m}$ is adapted to $(\mathcal {F}_t)$. In fact, let $f_j,\,j\in {\mathbb {N}}$, be an orthonormal basis of $L^2$. We have, for each $a>0,\,\{|\left<u_{n+1,m}(t),f_j\right>_2|<a\}=J_{1,a}\cup J_{2,a}$, where $J_{1,a}=\{|\left<y_n(t),f_j\right>_2|<a,\, t\le \tau _n\}$ and $J_{2,a}=\{|\left<v_{n+1,m}(t-\tau _n),f_j\right>_2|<a,\, \tau _n< t\}$. Since $y_n$ is adapted to $(\mathcal {F}_t)$ and $\tau _n$ is an $(\mathcal {F}_t)$ stopping time, it follows that $J_{1,a}\in \mathcal {F}_t$.

By the continuity of $t\mapsto |\left<v_{n+1,m}(t-\tau _n),f_j\right>_2|$, we see that

$$\begin{aligned} J_{2,a}=\bigcup _{^{q\in Q}_{q<a}}\bigcup _{h\in {\mathbb {N}}}\bigcap _{s\in Q}J_{q,h,s}, \end{aligned}$$

where $J_{q,h,s}=\left\{ |\left<v_{n+1,m}(s),f_j\right>_2|<q,\ t-\tau _n- \frac{1}{h}<s<t-\tau _n,\tau _n<t\right\}$.

Taking into account that $\{|\left<v_{n+1,m}(s),f_j\right>_2|<q\}\in \mathcal {F}_{\tau _n+s}$ and $\tau _n+s<t$, we have $J_{q,h,s}\in \mathcal {F}_t$, which implies that $J_{2,a}\in \mathcal {F}_t$.

Collecting the preceding results, we obtain that, for any $j\in {\mathbb {N}}$ and $a>0$, $\{|\left<u_{n+1,m}(t),f_j\right>_2|<a\}\in \mathcal {F}_t$. This is enough to imply that $u_{n+1,m}$ is adapted to $(\mathcal {F}_t)$. Therefore, as the limit of $u^{\tau _{n+1}}_{n+1,m},\,y_{n+1}$ is also adapted to $(\mathcal {F}_t)$. This completes the proof. $\square$

Note Added in Proof

After this paper was in print we learned by the work of A. de Bouard and R. Fukuitumi: Representation formula for Stochastic Schrödinger equation and applications, Nonlinearity, 25, 2993–3022 (2012) where a similar rescaling method is used without however a large overlap with present paper.