1 Introduction

Fractional calculus is a wide spread theory having applications in varied areas like mechanics, engineering, biochemistry and even in medicine. Many models that involve the characterization of memory effects into them cannot be modeled efficiently using integer-order equations. Fractional differential equations can help in such circumstances by incorporating the effects due to the memory into the system.

In the case of diffusion process, the mean square displacement is a quantity entitled to measure the dispersion of random particles and rate at which they diffuse. In classical models, they exhibit a linear relation with time, that is, the larger the time, the particles diffuse more faster. But in various processes, like diffusion on fractals, it can be observed that the mean square displacement develops logarithmically for large times. Since the classical integer order diffusion equation works well only in homogeneous medium, its time-fractional counterpart is more advantageous in modeling of phenomena with subdiffusion.

Fractional operators are highly efficient to model anomalous diffusion which is exhibited by systems in which displacement and time have a nonlinear relationship between them. In [1], the efficiency of fractional approach in modeling subdiffusion is validated due to natural exhibition of unusual dynamics by the system in the case of behaviors like slow dispersion, slow approach to the stationary state and memory effects. There are ample applications of such systems like in modeling growth of tumor cells which has memory effect. In [2], analysis of a system of coupled partial differential equations, which models tumor growth under the influence of subdiffusion is done. The study of diffusion of fluids in porous media with memory using time-fractional models involving subdiffusion is done in [3]. The existence, uniqueness and regularity of a mild solution of time-fractional Fokker-Planck equation is proved in [4] under the assumption of sufficient regularity for initial data. For more related works on existence, one can refer to [5,6,7] and so on. Apart from the analysis on the existence of solutions for fractional partial differential equations, a quite good research on numerical analysis is being carried out recently. For mathematical model describing Belousov–Zhabotinsky reaction, [8] discusses consequences of generalizing the model within the fractional order and also studies the boundedness, stability, existence, and other dynamical conditions.

The time-fractional diffusion equation is given by

$$\begin{aligned} \begin{aligned} \partial _t \phi&= \text{ div }(m\partial _t^{1-\alpha }\nabla \phi ) + f(t,\phi ), \end{aligned} \end{aligned}$$
(1.1)

with initial condition \(\phi (0)=\phi _0\) and homogeneous Neumann condition at the boundary. In (1.1), \(\phi \) denotes the concentration, \(m>0\) is the diffusion coefficient and f is a nonlinear source term. The derivative on right hand side is Riemann–Liouville fractional derivative and on taking convolution, the equation can be equivalently written using Caputo fractional derivative on the left hand side. The form with Riemann–Liouville derivative can simplify the estimates as done in [6]. Though in recent years there are many fractional operators being introduced, we prefer using these classical operators since either the new operators lack mathematical reasoning or they are just an extension of these classical operators [9].

The inclusion of the concept of uncertainty into models is proved to provide better approximations to their real life physical phenomenon. These types of situations arise in modeling the population of species with relative memory of distribution of resources. The basic idea is to model the random disturbances created in the environment using a stochastic term denoted by random noise. Based on the property of disturbances affecting the system, the noise term is chosen accordingly. On considering the Gaussian noise, we bring the effects of continuous disturbances into account. Some works involving the analysis of stochastic partial differential equations perturbed by Gaussian noise include [10, 11]. The stochastic counterpart of fractional differential equations help in efficiently elucidating complex dynamics exhibited due to hereditary effects of systems in areas like visco-elasticity and signal processing. The study of stochastic fractional equation remains not much explored with few works done on the existence of solutions as in [12,13,14]. In [15], the comparison between two stochastic models of European option pricing, one with time derivative replaced with fractional derivative and the other with noise term given by fractional Brownian motion, is made. The model with fractional derivative was proved to be efficient than the one with fractional Brownian motion.

In this work, we consider the time-fractional diffusion equation perturbed by Brownian-type noise which results in a stochastic version of (1.1). The novelty of this equation is that it models subdiffusion and therefore it is evident to realize the necessity to provide an analytic proof for existence of its solution. Using Galerkin approximations, the problem is projected on to a finite-dimensional space and in turn gaining an approximate solution for the projected equation. An a-priori estimate of this solution paves the way to compute the solution of our original problem.

The aim of this work is to establish the existence and uniqueness of solution of the stochastic time-fractional diffusion equation. The flow of this paper is as follows. In the next section, we introduce the basic mathematical concepts, inequalities and assumptions required for the proof. The last section is dedicated for the proof of our main result by initially establishing the energy estimates and establishing the convergence of approximations to the original solution by use of fractional Gronwall–Bellman-type inequality.

2 Mathematical model

Let \({{\mathcal {O}}}\subset {{\mathbb {R}}}^2\). The stochastic time-fractional diffusion equation perturbed by a parameter \(\varepsilon >0\) is given by

$$\begin{aligned} \partial _t \phi ^\varepsilon&= \text{ div }(m\partial _t^{1-\alpha }\nabla \phi ^\varepsilon ) + f(t,\phi ^\varepsilon )\nonumber \\ {}&\quad +\sqrt{\varepsilon }\sigma (t,\phi ^\varepsilon )\partial _tW(t),\nonumber \\ \text{ with } \phi ^\varepsilon (0)&=\phi _0\quad \text{ and } \frac{\partial {\phi ^\varepsilon }}{\partial {\nu }}=0. \end{aligned}$$
(2.1)

Here

  • \(\nu :\) outward normal to boundary \(\partial {{\mathcal {O}}}\),

  • W(t): independent Wiener process defined on a complete filtered probability space \((\Omega ,{{\mathcal {F}}},{{\mathcal {F}}}_t,{{\mathbb {P}}})\),

  • \(\sigma (t,\phi ^\varepsilon )\): noise coefficient satisfying conditions stated later.

The corresponding deterministic equation is given in (1.1). Here, the fractional order \(\alpha \) satisfies \(0<\alpha <1\). For a function F, we denote

$$\begin{aligned} \partial _t^{1-\alpha }F=\partial _t(g_{\alpha }*F), \end{aligned}$$

where \(g_\alpha \) is defined as \(g_\alpha (t)=\frac{t^{\alpha -1}}{\Gamma (\alpha )}\) (also refer [16]). The operator \(*\) denotes the convolution with respect tot the time variable is denoted by \(I^\alpha \),

$$\begin{aligned} (g_\alpha *\varphi )(t)=\int _0^t{g_\alpha (t-s)\varphi (s)\text {d}s}, \text{ for } \text{ some } \varphi \in {{\mathbb {L}}}^1(0,T). \end{aligned}$$

We now introduce the function spaces. We define the spaces V and H in the Gelfand triple \(V\hookrightarrow H \hookrightarrow V'\) as

  • The Lebesgue space \(H={{\mathbb {L}}}^2({{\mathcal {O}}})\) with the norm \(\Vert \cdot \Vert _H\) defined by

    $$\begin{aligned} (\phi ,\psi )=\int _{{{\mathcal {O}}}}{\phi (x)\psi (x)\text {d}x},\qquad \Vert \phi \Vert _H=\sqrt{(\phi ,\phi )}. \end{aligned}$$

  • The Sobolev space \(V={{\mathbb {H}}}^1({{\mathcal {O}}})\) with norm \(\Vert \cdot \Vert _V\) defined by

    $$\begin{aligned} \Vert \phi \Vert _V^2=\int _{{{\mathcal {O}}}}{|\phi (x)|^2\text {d}x}+\int _{{{\mathcal {O}}}}{|\nabla \phi (x)|^2\text {d}x}. \end{aligned}$$

Here, H and V are Hilbert spaces which naturally occur and guarantee the existence of orthonormal normal basis required for Galerkin approximations. For a Banach space X, the Bochner space [17] is defined

$$\begin{aligned}&{{\mathbb {L}}}^p(0,T;X)=\bigg \{\phi :(0,T)\rightarrow X: \phi \text{ is } \text{ strongly } \text{ measurable, } \\&\quad \bigg . \left. \Vert \phi \Vert ^p_{{{\mathbb {L}}}^p(0,T;X)}=\int _0^T{\Vert \phi (t)\Vert _X^p\text {d}s}<\infty \right\} . \end{aligned}$$

For \(p=\infty \), we define \({{\mathbb {L}}}^\infty (0,T;X)\) with \(\Vert \phi \Vert _{{{\mathbb {L}}}^\infty (0,T;X)}={\text {*}}{ess~sup}_{t\in (0,T)}\Vert \phi (t)\Vert _X\). The fractional Sobolev–Bochner space for a Banach space X is defined as

$$\begin{aligned} {{\mathbb {W}}}^{\alpha ,p}(0,T;X)=\{\phi \in {{\mathbb {L}}}^p(0,T;X):\partial _t^{1-\alpha } \phi \in {{\mathbb {L}}}^p(0,T;X)\}. \end{aligned}$$

For the case \(p=2\), \({{\mathbb {W}}}^{\alpha ,2}(0,T;X)={{\mathbb {H}}}^\alpha (0,T;X)\). Let Q be the covariance operator of H-valued Wiener process W(t) such that it is strictly positive, symmetric and trace class operator on H. Define \(H_0=Q^{1/2}H\). Then, \(H_0\) is a Hilbert space with scalar product

$$\begin{aligned} (\phi ,\psi )_0=(Q^{-1/2}\phi ,Q^{-1/2}\psi ), \quad \text{ for } \phi ,\psi \in H_0. \end{aligned}$$

Let \({{\mathcal {L}}}_Q\) be the space of linear operators S such that \(SQ^{1/2}\) is a Hilbert–Schmidt operator from H to H with the norm \(\Vert S\Vert _{{{\mathcal {L}}}_Q}=\text{ trace }(SQS^*)\). The following assumptions made, especially on the noise coefficient \(\sigma (t,\phi )\) and the nonlinear source term \(f(t,\phi )\) will be helpful for the proof of existence and uniqueness and in computing energy estimates.

Assumption 2.1

We assume the following conditions on the nonlinear term f and initial concentration \(\phi _0\).

  1. (i)

    \({{\mathcal {O}}}\subset {{\mathbb {R}}}^2\) is a bounded-Lipschitz domain and \(T>0\) is finite.

  2. (ii)

    \(f\in {{\mathbb {L}}}^\infty (0,T;H)\) is Lipschitz and the initial concentration \(\phi _0\in V.\)

Assumption 2.2

The function \(\sigma \in C([0,T]\times V;{{\mathcal {L}}}_Q(H_0;H))\) satisfy \(\forall \) \(t\in [0,T]\), \(\exists \) \(K_1>0\) and \(K_2>0\) such that

$$\begin{aligned}&\text{(A1) }\hspace{2cm}\Vert \sigma (t,\phi )\Vert ^2_{{{\mathcal {L}}}_Q}\le K_1(1+\Vert \nabla \phi \Vert _H^2),\\&\quad \text{ for } \text{ all } \phi \in V.\\&\quad \text{(A2) }\quad \Vert \sigma (t,\phi )-\sigma (t,\psi )\Vert ^2_{{{\mathcal {L}}}_Q} \le K_2\Vert \nabla (\phi -\psi )\Vert _H^2, \\&\quad \text{ for } \phi ,\psi \in V . \end{aligned}$$

3 Existence results

The results on existence and uniqueness of the solution for (2.1) are discussed in this section. The process \(\phi ^\varepsilon (t,\omega )\) is said to be a weak solution of (2.1) if it satisfies the initial condition \(\phi _0\) and for test function \(\psi \) of required regularity,

$$\begin{aligned}&(\phi ^\varepsilon (t),\psi )-(\phi _0,\psi )=\int _0^t\\ {}&\quad {\bigg [\big (\text{ div }(m\partial _t^ {1-\alpha }\nabla \phi ^\varepsilon ) + f(t,\phi ^\varepsilon ),\psi \big )\bigg ]\text {d}s}\\&\quad +\sqrt{\varepsilon }\int _0^t {(\sigma (s,\phi ^\varepsilon (s))\text {d}W,\psi )}. \end{aligned}$$

Some results required for the proof of existence are stated initially. It is well known that not all rules that apply for integer order are applicable to fractional calculus too. The chain rule of the integer order calculus cannot be considered for the fractional derivatives. The ensuing proposition gives a counterpart for the chain rule for semiconvex functions in fractional setting. A function \(f:{{\mathbb {R}}} \rightarrow {{\mathbb {R}}}\) is said to be semiconvex if for some \(\lambda \in {{\mathbb {R}}}\), the function \(x\rightarrow f(x)-\frac{\lambda }{2}|x|^2\) is convex.

Proposition 3.1

Let \({{\mathcal {V}}}\) be a Banach space such that \({{\mathcal {V}}}\hookrightarrow {{\mathbb {L}}}^2({{\mathcal {O}}})\hookrightarrow {{\mathcal {V}}}'\) forms a Gelfand triple. Let \(u\in {{\mathbb {H}}}^\alpha (0,T;{{\mathcal {V}}}')\cap {{\mathbb {L}}}^\infty (0,T;{{\mathcal {V}}})\) with \(u_0\in {{\mathbb {L}}}^2({{\mathcal {O}}})\) and \(E\in C^1({{\mathbb {R}}})\), a \(\lambda \)-convex function with \(\lambda \in {{\mathbb {R}}}\). If \(E'(u)\in {{\mathbb {L}}}^2(0,T;{{\mathcal {V}}})\), then we have for all \(t\in (0,T)\)

$$\begin{aligned}&\int _0^t{\bigg (\langle \partial _t^\alpha u,E'(u)\rangle _{{\mathcal {V}}} -\lambda \langle \partial _t^\alpha u,u\rangle _{{\mathcal {V}}}\bigg )\text {d}s}\ge g_{1-\alpha }\nonumber \\&\qquad *\int _{{\mathcal {O}}}{\big [E(u)-E(u_0)\big ]\text {d}x}\nonumber \\&\qquad +\frac{\lambda }{2}g_{1-\alpha }*\big (\Vert u\Vert _H^2-\Vert u_0\Vert _H^2\big ), \end{aligned}$$
(3.1)
$$\begin{aligned}&g_\alpha *\langle \partial _t^\alpha u,E'(u)\rangle _{{\mathcal {V}}}-\lambda g*\langle \partial _t^\alpha u,u\rangle _{{\mathcal {V}}}&\nonumber \\&\quad \ge \int _{{\mathcal {O}}}{\big [E(u)-E(u_0)\big ]\text {d}x}\nonumber \\&\qquad +\frac{\lambda }{2}\big (\Vert u\Vert _H^2-\Vert u_0\Vert _H^2\big ). \end{aligned}$$
(3.2)

The proof of the above proposition is given in [18]. For a special case of \(E(\cdot )=\frac{1}{2}|\cdot |^2\) in Hilbert space, one can refer [19]. The following lemma is a corollary of fractional Gronwall–Bellman-type inequality which is required in the proof of existence.

Lemma 3.1

Let \(u,v\in {{\mathbb {L}}}^1(0,T;{{\mathbb {R}}}_{\ge 0})\) and \(a,b>0.\) If u and v satisfy

$$\begin{aligned} u(t)+g_\alpha *v(t)\le a+b(g_\alpha *u)(t) \text{ a.e } t\in (0,T), \end{aligned}$$

then we have

$$\begin{aligned} u(t)+v(t)\le a\cdot C(\alpha ,b,T) \text{ a.e } t\in (0,T). \end{aligned}$$
(3.3)

For proof, refer [18]. We now state the Itô formula.

Theorem 3.1

(Itô formula) [20] Assume that \(\Phi \) is an \({{\mathcal {L}}}_Q\)-valued process stochastically integrable in [0, T], \(\phi \) is an H-valued predictable process Bochner integrable on [0, T], \({{\mathbb {P}}}\)-a. s., and X(0) is an \({{\mathcal {F}}}_0\)-measurable H-valued random variable. Then, the following process

$$\begin{aligned} X(t) = X(0) + \int ^t_0{\phi (s)\text {d}s} +\int _0^t{\Phi (s)\text {d}W(s)}, t \in [0, T], \end{aligned}$$

is well defined. Assume that a function \(F: [0, T]\times H \rightarrow {{\mathbb {R}}}\). Then, for all \(t \in [0, T],\)

$$\begin{aligned}&F(t, X(t))= F(0, X(0)) \nonumber \\&\quad +\int _0^t{[F_s(s, X(s)) + (F_x(s, X(s)), \phi (s))] ds}\nonumber \\&\quad +\int _0^t{(F_x(s, X(s)), \Phi (s)dW(s))}\nonumber \\&\quad +\frac{1}{2}\int _0^t{Tr[F_{xx}(s, X(s))(\Phi (s)Q^{1/2})(\Phi (s)Q^{1/2})^*]\text {d}s.} \end{aligned}$$
(3.4)

The Burkholder–Davis–Gundy inequality is used to break down the stochastic integral and produces a simplified integral term. Let \(M=(M(t),t\ge 0)\) be a Brownian integral with the drift of the form

$$\begin{aligned} M(t) = \int _0^t{F^j(s)\text {d}W^j(s)}, \end{aligned}$$

where each \(F^j\in {{\mathbb {L}}}^2[0,T] \) for all \(t \ge 0, 1\le j\le d.\) Let the quadratic variation process, denoted as \(([M,M](t), t\ge 0),\) be defined by

$$\begin{aligned}{}[M,M](t) = \sum \limits _{j=1}^d\int _0^t{F^j(s)^2 \text {d}s}. \end{aligned}$$

Lemma 3.2

(Burkholder–Davis–Gundy Inequality) For every \(p\ge 1\), there is a constant \(C_p\in (0,\infty )\) such that for any real-valued square integrable cádlág martingale M with \(M(0) = 0\), and for any \(T\ge 0,\)

$$\begin{aligned} C_p^{-1} {{\mathbb {E}}}[M,M]_T^{p/2} \le {{\mathbb {E}}}\sup \limits _{0\le t\le T} |M|^p \le C_p{{\mathbb {E}}}[M,M]_T^{p/2}. \end{aligned}$$

The following theorem is the main result of this work which states the existence and uniqueness of the solution for (2.1).

Theorem 3.2

Let Assumptions 2.1 and 2.2 hold. Then, there exists \(\varepsilon _0> 0\) such that for \(\varepsilon \in [0,\varepsilon _0]\) there exists a pathwise unique weak solution \(\phi ^\varepsilon \) for the stochastic fractional diffusion equation (2.1) such that

$$\begin{aligned} \phi ^\varepsilon \in {{\mathbb {H}}}^\alpha (0,T;V')\cap {{\mathbb {L}}}^\infty (0,T;V), \end{aligned}$$

satisfying the energy inequality

$$\begin{aligned} {{\mathbb {E}}}\biggl (\Vert \phi ^\varepsilon \Vert _{{{\mathbb {L}}}^\infty (0,T;V)}^2\biggr )\le \varepsilon C({{\mathbb {E}}}\Vert \phi _0\Vert _V^2+\Vert f\Vert ^2_{{{\mathbb {L}}}^\infty ([0,T],H)}),\nonumber \\ \end{aligned}$$
(3.5)

where C is an appropriate constant.

From here, \(\Vert \cdot \Vert \) denotes the norm in H unless specifically mentioned. We use Faedo–Galerkin method of approximation to establish the existence and uniqueness. This is an efficient tool employed to prove existence results since it reduces the problem to finite dimension and use orthonormal basis for approximating their solutions. Let \(\{\varphi _n\}_{n\ge 1}\): complete ONB of H corresponding to the Laplacian operator with Neumann boundary condition and \(H_n=span(\varphi _1,\cdots ,\varphi _n)\) and \(P_n:H\rightarrow H_n\) be an orthogonal projection onto \(H_n\). Let \(W_n=P_n W\) and \(\sigma _n=P_n \sigma \). Then for \(\psi \in H_n\), consider the equation in \(H_n\),

$$\begin{aligned}&\big (\partial _t \phi ^\varepsilon _n,\psi \big ) = \big (\text{ div }(m\partial _t^{1-\alpha }\nabla \phi ^\varepsilon _n) + f(t,\phi ^\varepsilon _n),\psi \big )\nonumber \\&\quad + \sqrt{\varepsilon }\big (\sigma _n(t,\phi ^\varepsilon _n)\,\partial _t W_n,\psi \big ), \end{aligned}$$
(3.6)

with \(\phi _n^\varepsilon (0)=P_n\phi (0)\). Here we observe that as \(n\rightarrow \infty \), \(\phi _n^\varepsilon (0)=P_n\phi (0)\rightarrow \phi _0\) in V. If \(\psi \in V\), then \(f\in {{\mathbb {L}}}^\infty (0,T;H)\) implies that by Lipschitz condition satisfied by the coefficients, from theory of fractional ODES, as in [6], we have a solution to Eq. (3.6) on \([0,T_n]\) such that

$$\begin{aligned} \phi _n^\varepsilon \in {{\mathbb {H}}}^\alpha (0,T_n;H_n)\cap {{\mathbb {L}}}^\infty (0,T_n;H_n). \end{aligned}$$

It implies that there is a stopping time \(T_n\le T\) such that for \(t<T_n\), (3.6) holds and for \(t\uparrow T_n<T\), \(|\phi ^\varepsilon _n(t)|\rightarrow \infty \). We now prove \(T_n=T\) and estimate \(\phi _n^\varepsilon \) for all n and \(\varepsilon \in [0,\varepsilon _0]\) for some \(\varepsilon _0>0\). For \(N>0\), take

$$\begin{aligned} \tau _N=\inf \{t:|\phi ^\varepsilon _n(t)|\ge N\}\wedge T. \end{aligned}$$

Proposition 3.2

Under Assumptions 2.1 and 2.2, there exists \(\varepsilon \ge 0\), such that \(T_n=T\) and there exists a unique solution \(\phi _n^\varepsilon \in {{\mathbb {H}}}^\alpha (0,T;H_n)\cap {{\mathbb {L}}}^\infty (0,T;H_n)\) satisfying (3.5) for an appropriate constant C.

Proof

We first prove the estimate (3.5) for \(\phi _n^\varepsilon \). Applying Itô’s formula for \(\Vert \phi ^\varepsilon _n\Vert ^2\), we get

$$\begin{aligned}&\Vert \phi ^\varepsilon _n(t)\Vert ^2 =\Vert \phi ^\varepsilon _n(0)\Vert ^2\\&\qquad +2\int _0^t{ \big ( \text{ div }(m\partial _s^{1-\alpha }\nabla \phi ^\varepsilon _n(s)) + f(s,\phi ^\varepsilon _n(s)),\phi ^\varepsilon _n(s)\big )\text {d}s}\\&\qquad +\varepsilon \int _0^t{\Vert \sigma _n(s,\phi ^\varepsilon _n(s))\Vert _{{{\mathcal {L}}}_Q}^2\text {d}s} \\&\qquad +2\sqrt{\varepsilon }\int _0^t{\big (\phi ^\varepsilon _n(s),\sigma _n(s,\phi ^\varepsilon _n(s))\,\text {d}W_n(s)\big )}.\\&\Vert \phi ^\varepsilon _n(t)\Vert ^2 +2\int _0^t{ \big ( m\partial _s^{1-\alpha }\nabla \phi ^\varepsilon _n(s),\nabla \phi ^\varepsilon _n\big )\text {d}s} \\&\quad =\Vert \phi ^\varepsilon _n(0)\Vert ^2+2\int _0^t{\big ( f(s,\phi ^\varepsilon _n(s)),\phi ^\varepsilon _n(s)\big )\text {d}s}\\&\qquad + \varepsilon \int _0^t{\Vert \sigma _n(s,\phi ^\varepsilon _n(s))\Vert _{{{\mathcal {L}}}_Q} ^2\text {d}s}\\&\qquad +2\sqrt{\varepsilon }\int _0^t{\big (\phi ^\varepsilon _n(s),\sigma _n (s,\phi ^\varepsilon _n(s))\,\text {d}W_n(s)\big )}. \end{aligned}$$

Using (3.1) in Proposition 3.1, for \(E(u)=\frac{|u|^2}{2}\), \(\alpha =1-\alpha \) and \(\lambda =0\), we get

$$\begin{aligned}&2 \int _0^t{ \big ( m\partial _s^{1-\alpha }\nabla \phi ^\varepsilon _n(s), \nabla \phi ^\varepsilon _n(s)\big ) \text {d}s}\ge m g_{\alpha }\\&\quad *\left( \Vert \nabla \phi ^\varepsilon _n(t)\Vert ^2- \Vert \nabla \phi _n^\varepsilon (0)\Vert ^2\right) . \end{aligned}$$

Substituting the above estimate, we have

$$\begin{aligned}&\Vert \phi ^\varepsilon _n(t)\Vert ^2 +m g_{\alpha } * \Vert \nabla \phi ^\varepsilon _n(t)\Vert ^2 \\&\quad \le \Vert \phi ^\varepsilon _n(0)\Vert ^2 + m g_\alpha * \Vert (\nabla \phi ^\varepsilon _n)(0)\Vert ^2\\&\qquad +2\int _0^t{\big ( f(s,\phi ^\varepsilon _n(s)),\phi ^\varepsilon _n(s)\big )\text {d}s}\\&\qquad + \varepsilon \int _0^t{\Vert \sigma _n(s,\phi ^\varepsilon _n(s))\Vert _{{{\mathcal {L}}}_Q}^2\text {d}s} \\&\qquad +2\sqrt{\varepsilon }\int _0^t{\big (\phi ^\varepsilon _n(s),\sigma _n(s,\phi ^\varepsilon _n(s))\,\text {d}W_n(s)\big )}. \end{aligned}$$

Using Assumptions and Young’s inequality,

$$\begin{aligned}&\Vert \phi ^\varepsilon _n(t)\Vert ^2 +m g_{\alpha } * \Vert \nabla \phi ^\varepsilon _n(t)\Vert ^2 \\&\quad \le \Vert \phi ^\varepsilon _n(0)\Vert ^2 + m g_\alpha * \Vert (\nabla \phi ^\varepsilon _n)(0)\Vert ^2\\&\qquad +\int _0^t{[\Vert f(s,\phi ^\varepsilon _n(s))\Vert ^2+\Vert \phi ^\varepsilon _n(s)\Vert ^2]\text {d}s}\\&\qquad + \varepsilon K_1\int _0^t{(1+\Vert \nabla \phi ^\varepsilon _n(s)\Vert ^2)\text {d}s}\\&\qquad +2\sqrt{\varepsilon } \int _0^t{\big (\phi ^\varepsilon _n(s),\sigma _n(s,\phi ^\varepsilon _n(s))\,\text {d}W_n(s)\big )}. \end{aligned}$$

Taking supremum over time and then taking expectation,

$$\begin{aligned}&{{\mathbb {E}}}\sup \limits _{0\le t\le T\wedge \tau _N}\bigg \{\Vert \phi ^\varepsilon _n(t)\Vert ^2 +m g_{\alpha } * \Vert \nabla \phi ^\varepsilon _n(t)\Vert ^2 \bigg \}\\&\quad \le {{\mathbb {E}}}\bigg \{\Vert \phi ^\varepsilon _n(0)\Vert ^2 + m g_\alpha * \Vert (\nabla \phi ^\varepsilon _n)(0)\Vert ^2\bigg \}\\&\quad +{{\mathbb {E}}}\int _0^{T\wedge \tau _N}{[\Vert f(s,\phi ^\varepsilon _n(s))\Vert ^2+\Vert \phi ^\varepsilon _n(s)\Vert ^2]\text {d}s}\\&\quad + \varepsilon K_1{{\mathbb {E}}}\int _0^{T\wedge \tau _N}{\!\!(1+\Vert \nabla \phi ^\varepsilon _n(s)\Vert ^2)\text {d}s}\\&\quad +2\sqrt{\varepsilon }{{\mathbb {E}}}\sup \limits _{0\le t\le T\wedge \tau _N}\int _0^t{\big (\phi ^\varepsilon _n(s),\sigma _n(s,\phi ^\varepsilon _n(s))\,\text {d}W_n(s)\big )}. \end{aligned}$$

For the stochastic integral term, using Burkholder–Davis–Gundy inequality, Young’s inequality and (A1) gives,

$$\begin{aligned}&2\sqrt{\varepsilon }{{\mathbb {E}}}\sup \limits _{0\le t\le T\wedge \tau _N}\int _0^{t}{ \!\bigl ( \sigma _n(\phi _n^\varepsilon (s))\text {d}W_n,\phi _n^\varepsilon (s)\bigr )}&\\&\qquad \le 2\sqrt{\varepsilon }C_1{{\mathbb {E}}}\bigg \{\int _0^{T\wedge \tau _N}{ \Vert \sigma _n(s,\phi _n^\varepsilon (s))\Vert ^2_{{{\mathcal {L}}}_Q}\Vert \phi _n^\varepsilon (s)\Vert ^{2}\text {d}s}\bigg \}^{1/2}&\\&\qquad \le 2\sqrt{\varepsilon }C_1{{\mathbb {E}}}\bigg \{\sup \limits _{0\le t\le T\wedge \tau _N}\Vert \phi _n^\varepsilon (t)\Vert ^{2}\int _0^{T\wedge \tau _N}{ \Vert \sigma _n(s,\phi _n^\varepsilon (s))\Vert ^2_{{{\mathcal {L}}}_Q}\text {d}s}\bigg \}^{1/2}&\\&\qquad \le \frac{1}{2}{{\mathbb {E}}}\sup \limits _{0\le t\le T\wedge \tau _N}\Vert \phi _n^\varepsilon (t)\Vert ^{2}+\varepsilon C^2K_1{{\mathbb {E}}}\int _0^{T\wedge \tau _N}{ (1+\Vert \nabla \phi _n^\varepsilon (s)\Vert ^2)\text {d}s}.&\end{aligned}$$

Combining, we get

$$\begin{aligned}&{{\mathbb {E}}}\sup \limits _{0\le t\le T\wedge \tau _N}\bigg \{\Vert \phi ^\varepsilon _n(t)\Vert ^2 +2m g_{\alpha } * \Vert \nabla \phi ^\varepsilon _n(t)\Vert ^2 \bigg \}\\&\quad \le 2{{\mathbb {E}}}\bigg \{\Vert \phi ^\varepsilon _n(0)\Vert ^2 + m g_\alpha * \Vert (\nabla \phi ^\varepsilon _n)(0)\Vert ^2\bigg \}\\&\qquad +2{{\mathbb {E}}}\int _0^{T\wedge \tau _N}{[\Vert f(s,\phi ^\varepsilon _n(s))\Vert ^2+\Vert \phi ^\varepsilon _n(s)\Vert ^2]\text {d}s}\\&\qquad + 2\varepsilon K_1(1+C^2){{\mathbb {E}}}\int _0^{T\wedge \tau _N}{\!\!(1+\Vert \nabla \phi ^\varepsilon _n(s)\Vert ^2)\text {d}s}. \end{aligned}$$

Using (3.3) from Lemma 3.1, we get

$$\begin{aligned}&{{\mathbb {E}}}\sup \limits _{0\le t\le T\wedge \tau _N}\bigg \{\Vert \phi ^\varepsilon _n(t)\Vert ^2 +2m \Vert \nabla \phi ^\varepsilon _n(t)\Vert ^2 \bigg \}\\&\quad \le C(T){{\mathbb {E}}}\bigg \{\Vert \phi ^\varepsilon _n(0)\Vert ^2 + m \Vert (\nabla \phi ^\varepsilon _n)(0)\Vert ^2\\&\qquad +\int _0^{T\wedge \tau _N}{\Vert f(s,\phi ^\varepsilon _n(s))\Vert ^2\text {d}s}+\varepsilon K_1(1+C^2)T\bigg \}. \end{aligned}$$

Then, for \(\varepsilon \ge 0\),

$$\begin{aligned}{} & {} {{\mathbb {E}}}\bigg (\Vert \phi _n^\varepsilon (t)\Vert ^{2}_{{{\mathbb {L}}}^\infty (0,T;V)}\bigg )\\ {}{} & {} \quad \le \varepsilon C\cdot {{\mathbb {E}}}\bigg (\Vert \phi ^\varepsilon _n(0)\Vert ^2_V+\Vert f\Vert ^2_{{{\mathbb {L}}}^\infty ([0,T];H)}\bigg ). \end{aligned}$$

Here \(\tau _N\rightarrow T_n\) as \(N\rightarrow \infty \) and for \(\{T_n<T\}\), \(\sup \nolimits _{0\le s\le \tau _N}|\phi _n^\varepsilon (s)|\rightarrow \infty \). Hence \({{\mathbb {P}}}(T_n<T)=0\) and so for large N, \(\tau _N=T\) and \(\phi _n^\varepsilon \in {{\mathbb {H}}}^\alpha (0,T;H_n)\cap {{\mathbb {L}}}^\infty (0,T;H_n)\). Hence the proof. \(\square \)

Proof of Theorem 3.2

Let \({{\mathcal {O}}}_T = [0, T] \times {{\mathcal {O}}}\). The theorem is proved by splitting it into several steps.

Step 1 From energy estimate obtained in Proposition 3.2, for \(\phi _n^\varepsilon \), there exist a subsequence also denoted by \(\{\phi _n^\varepsilon \}_{n\ge 0}\) and processes \(\phi ^\varepsilon \in {{\mathbb {H}}}^\alpha (0,T;V')\cap {{\mathbb {L}}}^p(0,T;H)\cap {{\mathbb {L}}}^\infty ([0,T],V)\), \(F^\varepsilon \in {{\mathbb {L}}}^2(0,T;V')\) and \(S^\varepsilon \in {{\mathbb {L}}}^2(0,T;{{\mathcal {L}}}_Q)\) such that

  1. (i)

    \(\phi _n^\varepsilon \rightarrow \phi ^\varepsilon \) strongly in \({{\mathbb {L}}}^p(0,T;H)\),

  2. (ii)

    \(\partial ^{1-\alpha }_t\nabla \phi _n^\varepsilon \rightarrow \partial ^{1-\alpha }_t\nabla \phi ^\varepsilon \) weakly in \({{\mathbb {L}}}^2(0,T;V')\),

  3. (iii)

    \(\phi _n^\varepsilon \) is \(\text{ weak}^*\)-converging to \(\phi ^\varepsilon \) in \({{\mathbb {L}}}^\infty (0,T;V)\),

  4. (iv)

    \(f(t,\phi _n^\varepsilon )\rightarrow F^\varepsilon \) in \({{\mathbb {L}}}^2({{\mathcal {O}}}_T,V')\),

  5. (v)

    \(\sigma _n(t,\phi _n^\varepsilon )\rightarrow S^\varepsilon \) in \({{\mathbb {L}}}^2({{\mathcal {O}}}_T,{{\mathcal {L}}}_Q)\).

As a consequence of Proposition 3.2, we get (i)-(iii). To prove that the limit \(\phi ^\varepsilon \) satisfies the weak formulation of (2.1), we integrate (3.6) and then decompose their terms using known inequalities. Using Young’s inequality,

$$\begin{aligned}&\int _0^T{\langle f(s,\phi _n^\varepsilon (s)),\psi \rangle \text {d}s}\\ {}&\quad \le \int _0^T{[\Vert f(s,\phi ^\varepsilon _n)\Vert ^2+\Vert \psi \Vert ^2]\text {d}s}<\infty .&\end{aligned}$$

The above estimate with (i) proves (iv). From Assumption 2.2,

$$\begin{aligned} {{\mathbb {E}}}\int _0^T{\Vert \sigma _n(s,\phi _n^\varepsilon (s))\Vert ^2_{{{\mathcal {L}}}_Q}\text {d}s}&\le K_1 {{\mathbb {E}}}\\ \int _0^T{(1+\Vert \nabla \phi _n^\varepsilon (s)\Vert ^2)\text {d}s}&<\infty . \end{aligned}$$

This implies (v). Since as \(n\rightarrow \infty \), \(P_n\phi _0=\phi ^\varepsilon _n(0)\rightarrow \phi _0\) in H, we have \(\phi ^\varepsilon \) satisfies

$$\begin{aligned} \phi ^\varepsilon (T)&=\phi _0+\int _0^T{\text{ div }(m\partial ^{1-\alpha }_s\nabla \phi ^\varepsilon (s)) \text {d}s}+\int _0^T{F^\varepsilon (s)\text {d}s}\nonumber \\&\quad +\sqrt{\varepsilon }\int _0^T{S^\varepsilon (s)\text {d}W(s)}. \end{aligned}$$
(3.7)

Step 2

It now remains to prove that \(F^\varepsilon (s)=f(s,\phi ^\varepsilon (s))\) and \(S^\varepsilon (s)=\sigma (s,\phi ^\varepsilon (s))\). By Fatou’s lemma,

$$\begin{aligned} {{\mathbb {E}}}\{\Vert \phi ^\varepsilon (T)\Vert ^2\}\le \liminf \limits _{n}{{\mathbb {E}}}\{\Vert \phi _n^\varepsilon (T)\Vert ^2\}. \end{aligned}$$
(3.8)

Applying Itô’s formula to (3.6),

$$\begin{aligned}&\Vert \phi ^\varepsilon _n(T)\Vert ^2= \Vert \phi ^\varepsilon _n(0)\Vert ^2\\ {}&\quad + 2\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s\nabla \phi ^\varepsilon _n(s)), \phi ^\varepsilon _n(s)\big )\text {d}s}\\&\quad +2\int _0^T{\big (f(s,\phi ^\varepsilon _n(s)),\phi ^\varepsilon _n(s) \big )\text {d}s}\\&\quad +\varepsilon \int _0^T{\Vert \sigma _n(s,\phi _n^\varepsilon (s))\Vert ^2_{{{\mathcal {L}}}_Q}\text {d}s}+I(t), \end{aligned}$$

where

$$\begin{aligned} I(t)=2\sqrt{\varepsilon }\int _0^T{\big (\sigma _n (s,\phi _n^\varepsilon (s))\text {d}W_n(s),\phi ^\varepsilon _n(s)\big )}. \end{aligned}$$

Here since I(t) is a local martingale with zero average,

$$\begin{aligned}&{{\mathbb {E}}}\{I(t)\}=2\sqrt{\varepsilon }{{\mathbb {E}}}\int _0^T {\big (\sigma _n(s,\phi _n^\varepsilon (s))\text {d}W_n(s),\phi ^\varepsilon _n(s)\big )}=0.&\\ \end{aligned}$$

Therefore, we get,

$$\begin{aligned}&{{\mathbb {E}}}\big (\Vert \phi ^\varepsilon _n(T)\Vert ^2\big )= {{\mathbb {E}}}\Vert \phi ^\varepsilon _n(0)\Vert ^2\\&\quad + 2{{\mathbb {E}}}\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s\nabla \phi ^\varepsilon _n(s)), \phi ^\varepsilon _n(s)\big )\text {d}s}\\&\quad +2{{\mathbb {E}}}\int _0^T{\big (f(s,\phi ^\varepsilon _n(s)), \phi ^\varepsilon _n(s)\big )\text {d}s}+\varepsilon {{\mathbb {E}}}\\&\int _0^T{\Vert \sigma _n(s, \phi _n^\varepsilon (s))\Vert ^2_{{{\mathcal {L}}}_Q}\text {d}s}.&\end{aligned}$$

Similarly from (3.7),

$$\begin{aligned}&{{\mathbb {E}}}\Vert \phi ^\varepsilon (T)\Vert ^2\le {{\mathbb {E}}}\Vert \phi ^\varepsilon (0)\Vert ^2\\&\quad + 2{{\mathbb {E}}}\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s\nabla \phi ^\varepsilon (s)), \phi ^\varepsilon (s)\big )\text {d}s}\\&\quad +2{{\mathbb {E}}}\int _0^T{\big (F^\varepsilon (s),\phi ^\varepsilon (s) \big )\text {d}s}\\&\quad +\varepsilon {{\mathbb {E}}}\int _0^T{\Vert S^\varepsilon (s)\Vert ^2_{{{\mathcal {L}}}_Q}\text {d}s}. \end{aligned}$$

Using the above two estimates in (3.8),

$$\begin{aligned}&2{{\mathbb {E}}}\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s\nabla \phi ^\varepsilon (s)), \phi ^\varepsilon (s)\big )\text {d}s}\nonumber \\&\qquad +2{{\mathbb {E}}}\int _0^T{\big (F^\varepsilon (s), \phi ^\varepsilon (s)\big )\text {d}s}+\varepsilon {{\mathbb {E}}}\int _0^ T{\Vert S^\varepsilon (s)\Vert ^2_{{{\mathcal {L}}}_Q}\text {d}s}&\nonumber \\&\quad \le \lim \inf \limits _n{{\mathbb {E}}}\bigg \{2\int _0^T{\big (\text{ div } (m\partial ^{1-\alpha }_s\nabla \phi ^\varepsilon _n(s)),\phi ^\varepsilon _n(s)\big ) \text {d}s}\nonumber \\&\qquad +2\int _0^T{\big (f(s,\phi ^\varepsilon _n(s)), \phi ^\varepsilon _n(s)\big )\text {d}s}\bigg .\nonumber \\&\qquad \bigg .+\varepsilon \int _0^T{\Vert \sigma _n(s,\phi _n^\varepsilon (s))\Vert ^2_{{{\mathcal {L}}}_Q}\text {d}s}\bigg \}. \end{aligned}$$
(3.9)

For a corresponding test function \(\psi \), we have

$$\begin{aligned}&\bigg \{2{{\mathbb {E}}}\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s\nabla (\phi ^\varepsilon _n(s) -\psi (s))),\phi ^\varepsilon _n(s)-\psi (s)\big )\text {d}s}\nonumber \\&\quad +\varepsilon {{\mathbb {E}}} \int _0^T{\Vert \sigma _n(s,\phi _n^\varepsilon (s))-\sigma _n(s,\psi (s))\Vert ^2_{{{\mathcal {L}}}_Q} \text {d}s}\bigg .\nonumber \\&\quad \bigg .+2{{\mathbb {E}}}\int _0^T{\big (f(s,\phi ^\varepsilon _n(s))-f(s,\psi (s)),\phi ^\varepsilon _n(s) -\psi (s)\big )\text {d}s}\bigg \}\le 0.&\end{aligned}$$
(3.10)

By subtracting (3.10) from right hand side of (3.9), we get

$$\begin{aligned}&2{{\mathbb {E}}}\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s\nabla \phi ^\varepsilon (s)), \phi ^\varepsilon (s)\big )\text {d}s}\\&\qquad +2{{\mathbb {E}}}\int _0^T{\big (F^\varepsilon (s),\phi ^ \varepsilon (s)\big )\text {d}s}+\varepsilon {{\mathbb {E}}}\int _0^T{\Vert S^\varepsilon (s) \Vert ^2_{{{\mathcal {L}}}_Q}\text {d}s}\\&\quad \le \lim \inf \limits _n{{\mathbb {E}}}\bigg \{2\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s\nabla \phi ^\varepsilon _n(s)),\psi (s)\big )\text {d}s}\\&\qquad +2\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_ s\nabla \psi (s)),\phi ^\varepsilon _n(s)\big )\text {d}s}\bigg .\\&\qquad -2\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s\nabla \psi (s)),\psi (s)\big ) \text {d}s}\\&\qquad +2\int _0^T{\big (f(s,\psi (s)),\phi ^\varepsilon _n(s)-\psi (s)\big )\text {d}s}\bigg .\\&\qquad \bigg .+2\int _0^T{\big (f(s,\phi ^\varepsilon _n(s)),\psi (s)\big )\text {d}s}\\&\qquad +\varepsilon \int _0^T{\big (2\sigma _n(s,\phi _n^\varepsilon (s))-\sigma _n(s,\psi (s)), \sigma _n(s,\psi (s))\big )\text {d}s}\bigg \}. \end{aligned}$$

Applying limit as \(n\rightarrow \infty \),

$$\begin{aligned}&2{{\mathbb {E}}}\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s\nabla \phi ^\varepsilon (s)), \phi ^\varepsilon (s)\big )\text {d}s}\\&\qquad +2{{\mathbb {E}}}\int _0^T{\big (F^\varepsilon (s),\phi ^ \varepsilon (s)\big )\text {d}s}+\varepsilon {{\mathbb {E}}}\int _0^T{\Vert S^\varepsilon (s)\Vert ^2_ {{{\mathcal {L}}}_Q}\text {d}s}\\&\quad \le \lim \inf \limits _n{{\mathbb {E}}}\bigg \{2\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s \nabla \phi ^\varepsilon (s)),\psi (s)\big )\text {d}s}\\&\qquad +2\int _0^T{\big (\text{ div }(m\partial ^{1 -\alpha }_s\nabla \psi (s)),\phi ^\varepsilon (s)\big )\text {d}s}\bigg .\\&\qquad -2\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s\nabla \psi (s)),\psi (s) \big )\text {d}s}\\&\qquad +2\int _0^T{\big (f(s,\psi (s)),\phi ^\varepsilon (s)-\psi (s)\big )\text {d}s}\bigg .\\&\qquad \bigg .+2\int _0^T{\big (F^\varepsilon (s),\psi (s)\big )\text {d}s}\\&\qquad +\varepsilon \int _0^T{\big (2S^\varepsilon (s)-\sigma (s,\psi (s)),\sigma (s,\psi (s))\big )\text {d}s}\bigg \}.&\end{aligned}$$

and rearranging,

$$\begin{aligned}&{{\mathbb {E}}}\bigg \{2\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s\nabla \phi ^\varepsilon (s)), \phi ^\varepsilon (s)-\psi (s)\big )\text {d}s}\\&\qquad -2\int _0^T{\big (\text{ div }(m\partial ^{1-\alpha }_s \nabla \psi (s)),\phi ^\varepsilon (s)-\psi (s)\big )\text {d}s}\bigg .\\&\bigg .\qquad +2\int _0^T{\big (F^\varepsilon (s)-f(s,\psi (s)),\phi ^\varepsilon (s)-\psi (s)\big ) \text {d}s}\\&\qquad +\varepsilon \int _0^T{\Vert S^\varepsilon (s)-\sigma (s,\psi (s))\Vert ^2_{{{\mathcal {L}}}_Q} \text {d}s}\bigg \}\le 0. \end{aligned}$$

Taking \(\psi =\phi ^\varepsilon \) in the above inequality, we get \(S^\varepsilon (t)=\sigma (t,\phi ^\varepsilon (t))\). Now we get

$$\begin{aligned}&2{{\mathbb {E}}} \bigg [\int _0^T{\big ( F^\varepsilon (s)- f(s,\psi (s)),\phi ^\varepsilon (s)-\psi (s)\big )\text {d}s}\bigg ]\le 0.&\end{aligned}$$
(3.11)

Let \(\psi =\phi ^\varepsilon -\mu \tilde{\psi },\) for \(\mu >0\). From Assumption 2.1, since f is Lipschitz,

$$\begin{aligned} \big (f(s,\psi (s))- f(s,\phi ^\varepsilon (s)),\mu \tilde{\psi }(s)\big )\le \mu ^2C\Vert \tilde{\psi }(s)\Vert ^2. \end{aligned}$$

Then from (3.11),

$$\begin{aligned}&{{\mathbb {E}}} \bigg [\int _0^T{\bigg \{2\mu \langle F^\varepsilon (s)- f(s,\phi ^\varepsilon (s)),\tilde{\psi }(s)\rangle \bigg \}\text {d}s}\bigg ]\le 0.&\end{aligned}$$

Since \(\tilde{\psi }\) is arbitrary, \(F^\varepsilon (t)=f(t,\phi ^\varepsilon (t))\). Hence, from the convergence we get,

$$\begin{aligned}&\phi ^\varepsilon (T)=\phi _0+\int _0^T{\text{ div }(m\partial ^{1-\alpha }_s\nabla \phi ^\varepsilon (s))\text {d}s}\\&\qquad +\int _0^T{f(s,\phi ^\varepsilon (s))\text {d}s} +\sqrt{\varepsilon }\int _0^T{\sigma (s,\phi ^\varepsilon (s))\text {d}W(s)}. \\&{{\mathbb {E}}}\biggl (\Vert \phi ^\varepsilon \Vert _{{{\mathbb {L}}}^\infty (0,T;V)}^2\biggr )\le \varepsilon C({{\mathbb {E}}}\Vert \phi _0\Vert _V^2+\Vert f\Vert _{{{\mathbb {L}}}^\infty ([0,T],H)}). \end{aligned}$$

Hence the existence of solution is proved.

Step 3

In order to prove uniqueness, consider \(\psi ^\varepsilon \in {{\mathbb {H}}}^\alpha (0,T;V')\cap {{\mathbb {L}}}^\infty ([0,T],V)\) be another solution of (2.1). Then, \(\vartheta =\phi ^\varepsilon -\psi ^\varepsilon \) satisfies

$$\begin{aligned} \text {d}\vartheta (t)&=\text{ div }(m\partial _t^{1-\alpha }\nabla \vartheta (t))+[ f(t,\phi ^\varepsilon (t))\\&\quad -f(t,\psi ^\varepsilon (t))]\text {d}t+\sqrt{\varepsilon } [\sigma (t,\phi ^\varepsilon )-\sigma (t,\psi ^\varepsilon )\text {d}W].&\end{aligned}$$

Applying Itô’s formula, using (3.3) and Assumptions 2.1 and 2.2,

$$\begin{aligned}&\Vert \vartheta (t)\Vert ^2 +m g_{\alpha } * \Vert \nabla \vartheta (t)\Vert ^2&\\&\quad \le \Vert \vartheta (0)\Vert ^2 +m g_{\alpha } * \Vert \nabla \vartheta (0)\Vert ^2\\&\qquad +2\int _0^t{ \bigg (f(s,\phi ^\varepsilon (s))-f(s,\psi ^\varepsilon (s)),\phi ^\varepsilon (s)-\psi ^\varepsilon (s)\bigg )\text {d}s}&\\&\qquad + \varepsilon \int _0^t{\Vert \sigma (s,\phi ^\varepsilon (s))-\sigma (s,\psi ^\varepsilon (s)) \Vert _{{{\mathcal {L}}}_Q}^2\text {d}s}\\&\qquad +2\sqrt{\varepsilon }\int _0^t{\big (\vartheta (s), [\sigma (s,\phi ^\varepsilon (s))-\sigma (s,\psi ^\varepsilon (s))]\text {d}W(s)\big )}. \end{aligned}$$

Using Assumptions and Young’s inequality,

$$\begin{aligned}&\Vert \vartheta (t)\Vert ^2 +m g_{\alpha } * \Vert \nabla \vartheta (t)\Vert ^2 \\&\quad \le \Vert \vartheta (0)\Vert ^2 + m g_\alpha * \Vert (\nabla \vartheta )(0)\Vert ^2\\&\qquad +\int _0^t{\bigg [\Vert f(s,\phi ^\varepsilon (s))- f(s,\psi ^\varepsilon (s))\Vert ^2+\Vert \vartheta (s)\Vert ^2\bigg ]\text {d}s}\\&\qquad + \varepsilon K_2\int _0^t{\Vert \nabla \vartheta (s)\Vert ^2\text {d}s}\\&\qquad +2\sqrt{\varepsilon }\int _0^t{\big (\vartheta (s), \big [\sigma (s,\phi ^\varepsilon (s))-\sigma (s,\psi ^\varepsilon (s))\big ]\text {d}W(s)\big )}. \end{aligned}$$

Taking supremum over time T and then taking expectation,

$$\begin{aligned}&{{\mathbb {E}}}\sup \limits _{0\le t\le T}\bigg \{\Vert \vartheta (t)\Vert ^2 +m g_{\alpha } * \Vert \nabla \vartheta (t)\Vert ^2 \bigg \}\\&\quad \le {{\mathbb {E}}}\bigg \{\Vert \vartheta (0)\Vert ^2 + m g_\alpha * \Vert (\nabla \vartheta )(0)\Vert ^2\\&\qquad +\int _0^T{\bigg [\Vert f(s,\phi ^\varepsilon (s))- f(s,\psi ^\varepsilon (s))\Vert ^2+\Vert \vartheta (s)\Vert ^2\bigg ]\text {d}s}\bigg \}\\&\qquad + \varepsilon K_2{{\mathbb {E}}}\int _0^T{\Vert \nabla \vartheta (s)\Vert ^2\text {d}s}+2\sqrt{\varepsilon }{{\mathbb {E}}}\sup \limits _{0\le t\le T}\\&\int _0^t{\big (\vartheta (s),\big [\sigma (s,\phi ^\varepsilon (s)) -\sigma (s,\psi ^\varepsilon (s))\big ]\text {d}W(s)\big )}. \end{aligned}$$

Using Burkholder–Davis–Gundy inequality and simplifying,

$$\begin{aligned}&2\sqrt{\varepsilon }{{\mathbb {E}}}\sup \limits _{0\le t\le T}\int _0^{t}{ \!\bigg ( \big [\sigma (s,\phi ^\varepsilon (s))-\sigma (s,\psi ^\varepsilon (s))\big ]\text {d}W(s),\vartheta (s)\bigg )}\\&\quad \le 2\sqrt{\varepsilon }C_1{{\mathbb {E}}}\\&\bigg \{\sup \limits _{0\le t\le T}\Vert \vartheta (t)\Vert ^{2}\int _0^{T}{ \Vert \sigma (s,\phi ^\varepsilon (s))-\sigma (s,\psi ^\varepsilon (s))\Vert ^2_{{{\mathcal {L}}}_Q}\text {d}s}\bigg \}^{1/2}\\&\quad \le \frac{1}{2}{{\mathbb {E}}}\sup \limits _{0\le t\le T}\Vert \vartheta (t)\Vert ^{2}+\varepsilon C^2K_2{{\mathbb {E}}}\int _0^{T}{ \Vert \nabla \vartheta (s)\Vert ^2\text {d}s}.&\end{aligned}$$

Combining, we get

$$\begin{aligned}&{{\mathbb {E}}}\sup \limits _{0\le t\le T}\bigg \{\Vert \vartheta (t)\Vert ^2 +2m \Vert \nabla \vartheta (t)\Vert ^2 \bigg \}\le C(T){{\mathbb {E}}}\Vert \vartheta (0)\Vert _V^2. \end{aligned}$$

Hence \(\Vert \vartheta (t)\Vert ^2=0\) for all \(t\in [0,T]\) since \(\vartheta (0)=0\). \(\square \)

Thus, the existence and uniqueness is proved in \( {{\mathbb {H}}}^\alpha (0,T;V')\cap {{\mathbb {L}}}^\infty (0,T;V)\).

4 Conclusion

A stochastic time-fractional equation that models subdiffusion process is considered, and a study on existence and uniqueness of its solution is carried out. For this purpose, the nonlinear source term (f) and the noise coefficient (\(\sigma \)) are assumed to essentially satisfy Lipschitz continuity. A series of inequalities are used to arrive at our results. Burkholder–Davis–Gundy inequality is formally used to reduce the stochastic integral, and fractional Gronwall–Bellman-type inequality is used in the place of integer order Gronwall lemma. Having established the existence of solution, analysis of many further concepts is open. This leads to numerical computation of solution of this stochastic fractional equation. Further, one can study large and moderate deviation principles for this stochastic equation which analyses the behavior of the system for larger time.